title: Ellipse

text: An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution.


title: Ellipse

text: Analytically, the equation of a standard ellipse centered at the origin with width formula_4 and height formula_5 is:


title: Ellipse

text: Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a cylinder is also an ellipse.


title: Ellipse

text: An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity:


title: Ellipse

text: Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the Solar System is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sunplanet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection. The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics.


title: Ellipse

text: The midpoint formula_18 of the line segment joining the foci is called the "center" of the ellipse. The line through the foci is called the "major axis", and the line perpendicular to it through the center is the "minor axis". The major axis intersects the ellipse at two "vertices" formula_19, which have distance formula_20 to the center. The distance formula_21 of the foci to the center is called the "focal distance" or linear eccentricity. The quotient formula_22 is the "eccentricity".


title: Ellipse

text: formula_25 is called the "circular directrix" (related to focus formula_26) of the ellipse. This property should not be confused with the definition of an ellipse using a directrix line below.


title: Ellipse

text: Using Dandelin spheres, one can prove that any section of a cone with a plane is an ellipse, assuming the plane does not contain the apex and has slope less than that of the lines on the cone.


title: Ellipse

text: The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the "x"-axis is the major axis, and:


title: Ellipse

text: Removing the radicals by suitable squarings and using formula_42 (see diagram) produces the standard equation of the ellipse:


title: Ellipse

text: The width and height parameters formula_45 are called the semi-major and semi-minor axes. The top and bottom points formula_46 are the "co-vertices". The distances from a point formula_40 on the ellipse to the left and right foci are formula_48 and formula_49.


title: Ellipse

text: It follows from the equation that the ellipse is "symmetric" with respect to the coordinate axes and hence with respect to the origin.


title: Ellipse

text: Throughout this article, the semi-major and semi-minor axes are denoted formula_20 and formula_51, respectively, i.e. formula_52


title: Ellipse

text: In principle, the canonical ellipse equation formula_53 may have formula_54 (and hence the ellipse would be taller than it is wide). This form can be converted to the standard form by transposing the variable names formula_55 and formula_56 and the parameter names formula_20 and formula_58


title: Ellipse

text: The length of the chord through one focus, perpendicular to the major axis, is called the "latus rectum". One half of it is the "semi-latus rectum" formula_63. A calculation shows:


title: Ellipse

text: The semi-latus rectum formula_63 is equal to the radius of curvature at the vertices (see section curvature).


title: Ellipse

text: An arbitrary line formula_66 intersects an ellipse at 0, 1, or 2 points, respectively called an "exterior line", "tangent" and "secant". Through any point of an ellipse there is a unique tangent. The tangent at a point formula_67 of the ellipse formula_68 has the coordinate equation:


title: Ellipse

text: Let formula_67 be a point on an ellipse and formula_73 be the equation of any line formula_66 containing formula_67. Inserting the line's equation into the ellipse equation and respecting formula_76 yields:


title: Ellipse

text: Using (1) one finds that formula_89 is a tangent vector at point formula_67, which proves the vector equation.


title: Ellipse

text: If formula_91 and formula_92 are two points of the ellipse such that formula_93, then the points lie on two "conjugate diameters" (see below). (If formula_62, the ellipse is a circle and "conjugate" means "orthogonal".)


title: Ellipse

text: In analytic geometry, the ellipse is defined as a quadric: the set of points formula_97 of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation


title: Ellipse

text: Then the ellipse is a non-degenerate real ellipse if and only if "C∆" < 0. If "C∆" > 0, we have an imaginary ellipse, and if "∆" = 0, we have a point ellipse.


title: Ellipse

text: The general equation's coefficients can be obtained from known semi-major axis formula_20, semi-minor axis formula_51, center coordinates formula_95, and rotation angle formula_104 (the angle from the positive horizontal axis to the ellipse's major axis) using the formulae:


title: Ellipse

text: These expressions can be derived from the canonical equation formula_68 by an affine transformation of the coordinates formula_40:


title: Ellipse

text: Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations:


title: Ellipse

text: The parameter "t" (called the "eccentric anomaly" in astronomy) is not the angle of formula_112 with the "x"-axis, but has a geometric meaning due to Philippe de La Hire (see "Drawing ellipses" below).


title: Ellipse

text: For formula_118 this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing formula_119 The left vertex is the limit formula_120


title: Ellipse

text: Alternately, if the parameter formula_121 is considered to be a point on the real projective line formula_122, then the corresponding rational parametrization is


title: Ellipse

text: Rational representations of conic sections are commonly used in computer-aided design (see Bezier curve).


title: Ellipse

text: A parametric representation, which uses the slope formula_125 of the tangent at a point of the ellipse


title: Ellipse

text: Here formula_125 is the slope of the tangent at the corresponding ellipse point, formula_133 is the upper and formula_134 the lower half of the ellipse. The verticesformula_135, having vertical tangents, are not covered by the representation.


title: Ellipse

text: The equation of the tangent at point formula_136 has the form formula_137. The still unknown formula_138 can be determined by inserting the coordinates of the corresponding ellipse point formula_136:


title: Ellipse

text: This description of the tangents of an ellipse is an essential tool for the determination of the orthoptic of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae.


title: Ellipse

text: An affine transformation of the Euclidean plane has the form formula_142, where formula_143 is a regular matrix (with non-zero determinant) and formula_144 is an arbitrary vector. If formula_145 are the column vectors of the matrix formula_143, the unit circle formula_147, formula_148, is mapped onto the ellipse:


title: Ellipse

text: Here formula_144 is the center and formula_151 are the directions of two conjugate diameters, in general not perpendicular.


title: Ellipse

text: (If formula_155, then formula_156.) This is derived as follows. The tangent vector at point formula_157 is:


title: Ellipse

text: Solving the parametric representation for formula_170 by Cramer's rule and using formula_171, one obtains the implicit representation


title: Ellipse

text: For formula_178 one obtains a parametric representation of the standard ellipse rotated by angle formula_104:


title: Ellipse

text: The definition of an ellipse in this section gives a parametric representation of an arbitrary ellipse, even in space, if one allows formula_182 to be vectors in space.


title: Ellipse

text: In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate formula_104 measured from the major axis, the ellipse's equation is


title: Ellipse

text: If instead we use polar coordinates with the origin at one focus, with the angular coordinate formula_186 still measured from the major axis, the ellipse's equation is


title: Ellipse

text: where the sign in the denominator is negative if the reference direction formula_186 points towards the center (as illustrated on the right), and positive if that direction points away from the center.


title: Ellipse

text: In the slightly more general case of an ellipse with one focus at the origin and the other focus at angular coordinate formula_189, the polar form is


title: Ellipse

text: The angle formula_104 in these formulas is called the true anomaly of the point. The numerator of these formulas is the semi-latus rectum formula_192.


title: Ellipse

text: Each of the two lines parallel to the minor axis, and at a distance of formula_193 from it, is called a "directrix" of the ellipse (see diagram).


title: Ellipse

text: The proof for the pair formula_196 follows from the fact that formula_197 and formula_198 satisfy the equation


title: Ellipse

text: The converse is also true and can be used to define an ellipse (in a manner similar to the definition of a parabola):


title: Ellipse

text: The extension to formula_2, which is the eccentricity of a circle, is not allowed in this context in the Euclidean plane. However, one may consider the directrix of a circle to be the line at infinity in the projective plane.


title: Ellipse

text: The directrix formula_201 has equation formula_213. With formula_214, the relation formula_215 produces the equations


title: Ellipse

text: This is the equation of an "ellipse" (formula_220), or a "parabola" (formula_3), or a "hyperbola" (formula_209). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).


title: Ellipse

text: If formula_220, introduce new parameters formula_224 so that formula_225, and then the equation above becomes


title: Ellipse

text: Because of formula_229 point formula_230 of directrix formula_231 (see diagram) and focus formula_30 are inverse with respect to the circle inversion at circle formula_233 (in diagram green). Hence formula_230 can be constructed as shown in the diagram. Directrix formula_231 is the perpendicular to the main axis at point formula_230.


title: Ellipse

text: Because the tangent is perpendicular to the normal, the statement is true for the tangent and the supplementary angle of the angle between the lines to the foci (see diagram), too.


title: Ellipse

text: Let formula_243 be the point on the line formula_244 with the distance formula_4 to the focus formula_26, formula_20 is the semi-major axis of the ellipse. Let line formula_248 be the bisector of the supplementary angle to the angle between the lines formula_242. In order to prove that formula_248 is the tangent line at point formula_14, one checks that any point formula_252 on line formula_248 which is different from formula_14 cannot be on the ellipse. Hence formula_248 has only point formula_14 in common with the ellipse and is, therefore, the tangent at point formula_14.


title: Ellipse

text: From the diagram and the triangle inequality one recognizes that formula_258 holds, which means: formula_259 . The equality formula_260 is true from the Angle bisector theorem because formula_261 and formula_262 . But if formula_252 is a point of the ellipse, the sum should be formula_4.


title: Ellipse

text: The rays from one focus are reflected by the ellipse to the second focus. This property has optical and acoustic applications similar to the reflective property of a parabola (see whispering gallery).


title: Ellipse

text: An affine transformation preserves parallelism and midpoints of line segments, so this property is true for any ellipse. (Note that the parallel chords and the diameter are no longer orthogonal.)


title: Ellipse

text: Two diameters formula_265 of an ellipse are "conjugate" if the midpoints of chords parallel to formula_266 lie on formula_267


title: Ellipse

text: any pair of points formula_273 belong to a diameter, and the pair formula_274 belong to its conjugate diameter.


title: Ellipse

text: For the common parametric representation formula_275 of the ellipse with equation formula_276 one gets: The points


title: Ellipse

text: The two points formula_295 are on conjugate diameters (see previous section). From trigonometric formulae one obtains formula_296 and


title: Ellipse

text: and from the diagram it can be seen that the area of the parallelogram is 8 times that of formula_300. Hence


title: Ellipse

text: For the ellipse formula_276 the intersection points of "orthogonal" tangents lie on the circle formula_303.


title: Ellipse

text: This circle is called "orthoptic" or director circle of the ellipse (not to be confused with the circular directrix defined above).


title: Ellipse

text: Ellipses appear in descriptive geometry as images (parallel or central projection) of circles. There exist various tools to draw an ellipse. Computers provide the fastest and most accurate method for drawing an ellipse. However, technical tools ("ellipsographs") to draw an ellipse without a computer exist. The principle of ellipsographs were known to Greek mathematicians such as Archimedes and Proklos.


title: Ellipse

text: If there is no ellipsograph available, one can draw an ellipse using an approximation by the four osculating circles at the vertices.


title: Ellipse

text: For any method described below, knowledge of the axes and the semi-axes is necessary (or equivalently: the foci and the semi-major axis).


title: Ellipse

text: If this presumption is not fulfilled one has to know at least two conjugate diameters. With help of Rytz's construction the axes and semi-axes can be retrieved.


title: Ellipse

text: The following construction of single points of an ellipse is due to de La Hire. It is based on the standard parametric representation formula_304 of an ellipse:


title: Ellipse

text: The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two drawing pins, a length of string, and a pencil. In this method, pins are pushed into the paper at two points, which become the ellipse's foci. A string is tied at each end to the two pins; its length after tying is formula_4. The tip of the pencil then traces an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, gardeners use this procedure to outline an elliptical flower bed—thus it is called the "gardener's ellipse".


title: Ellipse

text: The two following methods rely on the parametric representation (see section "parametric representation", above):


title: Ellipse

text: This representation can be modeled technically by two simple methods. In both cases center, the axes and semi axes formula_312 have to be known.


title: Ellipse

text: The point, where the semi axes meet is marked by formula_14. If the strip slides with both ends on the axes of the desired ellipse, then point formula_14 traces the ellipse. For the proof one shows that point formula_14 has the parametric representation formula_304, where parameter formula_318 is the angle of the slope of the paper strip.


title: Ellipse

text: A technical realization of the motion of the paper strip can be achieved by a Tusi couple (see animation). The device is able to draw any ellipse with a "fixed" sum formula_313, which is the radius of the large circle. This restriction may be a disadvantage in real life. More flexible is the second paper strip method.


title: Ellipse

text: A variation of the paper strip method 1 uses the observation that the midpoint formula_320 of the paper strip is moving on the circle with center formula_321 (of the ellipse) and radius formula_322. Hence, the paperstrip can be cut at point formula_320 into halves, connected again by a joint at formula_320 and the sliding end formula_325 fixed at the center formula_321 (see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged. This variation requires only one sliding shoe.


title: Ellipse

text: One marks the point, which divides the strip into two substrips of length formula_51 and formula_329. The strip is positioned onto the axes as described in the diagram. Then the free end of the strip traces an ellipse, while the strip is moved. For the proof, one recognizes that the tracing point can be described parametrically by formula_304, where parameter formula_318 is the angle of slope of the paper strip.


title: Ellipse

text: Similar to the variation of the paper strip method 1 a "variation of the paper strip method 2" can be established (see diagram) by cutting the part between the axes into halves.


title: Ellipse

text: The diagram shows an easy way to find the centers of curvature formula_336 at vertex formula_337 and co-vertex formula_338, respectively:


title: Ellipse

text: The following method to construct single points of an ellipse relies on the Steiner generation of a conic section:


title: Ellipse

text: For the generation of points of the ellipse formula_68 one uses the pencils at the vertices formula_332. Let formula_352 be an upper co-vertex of the ellipse and formula_353.


title: Ellipse

text: formula_14 is the center of the rectangle formula_355. The side formula_356 of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal formula_357 as direction onto the line segment formula_358 and assign the division as shown in the diagram. The parallel projection together with the reverse of the orientation is part of the projective mapping between the pencils at formula_337 and formula_360 needed. The intersection points of any two related lines formula_361 and formula_362 are points of the uniquely defined ellipse. With help of the points formula_363 the points of the second quarter of the ellipse can be determined. Analogously one obtains the points of the lower half of the ellipse.


title: Ellipse

text: Steiner generation can also be defined for hyperbolas and parabolas. It is sometimes called a "parallelogram method" because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.


title: Ellipse

text: The ellipse is a special case of the hypotrochoid when formula_364, as shown in the adjacent image. The special case of a moving circle with radius formula_365 inside a circle with radius formula_364 is called a Tusi couple.


title: Ellipse

text: A circle with equation formula_367 is uniquely determined by three points formula_368 not on a line. A simple way to determine the parameters formula_369 uses the "inscribed angle theorem" for circles:


title: Ellipse

text: Usually one measures inscribed angles by a degree or radian "θ," but here the following measurement is more convenient:


title: Ellipse

text: At first the measure is available only for chords not parallel to the y-axis, but the final formula works for any chord.


title: Ellipse

text: Using vectors, dot products and determinants this formula can be arranged more clearly, letting formula_384:


title: Ellipse

text: This section, we consider the family of ellipses defined by equations formula_389 with a "fixed" eccentricity formula_1. It is convenient to use the parameter:


title: Ellipse

text: where "q" is fixed and formula_393 vary over the real numbers. (Such ellipses have their axes parallel to the coordinate axes: if formula_394, the major axis is parallel to the "x"-axis; if formula_395, it is parallel to the "y"-axis.)


title: Ellipse

text: For this family of ellipses, one introduces the following q-analog angle measure, which is "not" a function of the usual angle measure "θ":


title: Ellipse

text: At first the measure is available only for chords which are not parallel to the y-axis. But the final formula works for any chord. The proof follows from a straightforward calculation. For the direction of proof given that the points are on an ellipse, one can assume that the center of the ellipse is the origin.


title: Ellipse

text: Any ellipse can be described in a suitable coordinate system by an equation formula_68. The equation of the tangent at a point formula_413 of the ellipse is formula_414 If one allows point formula_413 to be an arbitrary point different from the origin, then


title: Ellipse

text: Such a relation between points and lines generated by a conic is called "pole-polar relation" or "polarity". The pole is the point; the polar the line.


title: Ellipse

text: except for the section on the area enclosed by a tilted ellipse, where the generalized form of Eq.() will be given.


title: Ellipse

text: where formula_20 and formula_51 are the lengths of the semi-major and semi-minor axes, respectively. The area formula formula_435 is intuitive: start with a circle of radius formula_51 (so its area is formula_437) and stretch it by a factor formula_438 to make an ellipse. This scales the area by the same factor: formula_439 However, using the same approach for the circumference would be fallacious – compare the integrals formula_440 and formula_441. It is also easy to rigorously prove the area formula using integration as follows. Equation () can be rewritten as formula_442 For formula_443 this curve is the top half of the ellipse. So twice the integral of formula_444 over the interval formula_445 will be the area of the ellipse:


title: Ellipse

text: The area can also be expressed in terms of eccentricity and the length of the semi-major axis as formula_452 (obtained by solving for flattening, then computing the semi-minor axis).


title: Ellipse

text: So far we have dealt with "erect" ellipses, whose major and minor axes are parallel to the formula_55 and formula_454 axes. However, some applications require "tilted" ellipses. In charged-particle beam optics, for instance, the enclosed area of an erect or tilted ellipse is an important property of the beam, its "emittance". In this case a simple formula still applies, namely


title: Ellipse

text: where formula_455, formula_456 are intercepts and formula_457, formula_458 are maximum values. It follows directly from Apollonios's theorem.


title: Ellipse

text: where again formula_20 is the length of the semi-major axis, formula_462 is the eccentricity, and the function formula_463 is the complete elliptic integral of the second kind,


title: Ellipse

text: The circumference of the ellipse may be evaluated in terms of formula_465 using Gauss's arithmetic-geometric mean; this is a quadratically converging iterative method (see here for details).


title: Ellipse

text: where formula_467 is the double factorial (extended to negative odd integers by the recurrence relation formula_468, for formula_469). This series converges, but by expanding in terms of formula_470 James Ivory and Bessel derived an expression that converges much more rapidly:


title: Ellipse

text: Srinivasa Ramanujan gave two close approximations for the circumference in §16 of "Modular Equations and Approximations to formula_347"; they are


title: Ellipse

text: where formula_475 takes on the same meaning as above. The errors in these approximations, which were obtained empirically, are of order formula_476 and formula_477 respectively.


title: Ellipse

text: More generally, the arc length of a portion of the circumference, as a function of the angle subtended (or -coordinates of any two points on the upper half of the ellipse), is given by an incomplete elliptic integral. The upper half of an ellipse is parameterized by


title: Ellipse

text: Some lower and upper bounds on the circumference of the canonical ellipse formula_486 with formula_487 are


title: Ellipse

text: Here the upper bound formula_489 is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound formula_490 is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and the minor axes.


title: Ellipse

text: If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves of that disturbance, after reflecting off the walls, converge simultaneously to a single point: the "second focus". This is a consequence of the total travel length being the same along any wall-bouncing path between the two foci.


title: Ellipse

text: Similarly, if a light source is placed at one focus of an elliptic mirror, all light rays on the plane of the ellipse are reflected to the second focus. Since no other smooth curve has such a property, it can be used as an alternative definition of an ellipse. (In the special case of a circle with a source at its center all light would be reflected back to the center.) If the ellipse is rotated along its major axis to produce an ellipsoidal mirror (specifically, a prolate spheroid), this property holds for all rays out of the source. Alternatively, a cylindrical mirror with elliptical cross-section can be used to focus light from a linear fluorescent lamp along a line of the paper; such mirrors are used in some document scanners.


title: Ellipse

text: Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear a person standing at the other focus remarkably well. The effect is even more evident under a vaulted roof shaped as a section of a prolate spheroid. Such a room is called a "whisper chamber". The same effect can be demonstrated with two reflectors shaped like the end caps of such a spheroid, placed facing each other at the proper distance. Examples are the National Statuary Hall at the United States Capitol (where John Quincy Adams is said to have used this property for eavesdropping on political matters); the Mormon Tabernacle at Temple Square in Salt Lake City, Utah; at an exhibit on sound at the Museum of Science and Industry in Chicago; in front of the University of Illinois at Urbana–Champaign Foellinger Auditorium; and also at a side chamber of the Palace of Charles V, in the Alhambra.


title: Ellipse

text: In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun [approximately] at one focus, in his first law of planetary motion. Later, Isaac Newton explained this as a corollary of his law of universal gravitation.


title: Ellipse

text: More generally, in the gravitational two-body problem, if the two bodies are bound to each other (that is, the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. The orbit of either body in the reference frame of the other is also an ellipse, with the other body at the same focus.


title: Ellipse

text: Keplerian elliptical orbits are the result of any radially directed attraction force whose strength is inversely proportional to the square of the distance. Thus, in principle, the motion of two oppositely charged particles in empty space would also be an ellipse. (However, this conclusion ignores losses due to electromagnetic radiation and quantum effects, which become significant when the particles are moving at high speed.)


title: Ellipse

text: Also, in terms of formula_500 and formula_501, the semi-major axis formula_20 is their arithmetic mean, the semi-minor axis formula_51 is their geometric mean, and the semi-latus rectum formula_63 is their harmonic mean. In other words,


title: Ellipse

text: The general solution for a harmonic oscillator in two or more dimensions is also an ellipse. Such is the case, for instance, of a long pendulum that is free to move in two dimensions; of a mass attached to a fixed point by a perfectly elastic spring; or of any object that moves under influence of an attractive force that is directly proportional to its distance from a fixed attractor. Unlike Keplerian orbits, however, these "harmonic orbits" have the center of attraction at the geometric center of the ellipse, and have fairly simple equations of motion.


title: Ellipse

text: In electronics, the relative phase of two sinusoidal signals can be compared by feeding them to the vertical and horizontal inputs of an oscilloscope. If the Lissajous figure display is an ellipse, rather than a straight line, the two signals are out of phase.


title: Ellipse

text: Two non-circular gears with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a link chain or timing belt, or in the case of a bicycle the main chainring may be elliptical, or an ovoid similar to an ellipse in form. Such elliptical gears may be used in mechanical equipment to produce variable angular speed or torque from a constant rotation of the driving axle, or in the case of a bicycle to allow a varying crank rotation speed with inversely varying mechanical advantage.


title: Ellipse

text: An example gear application would be a device that winds thread onto a conical bobbin on a spinning machine. The bobbin would need to wind faster when the thread is near the apex than when it is near the base.


title: Ellipse

text: In statistics, a bivariate random vector formula_509 is jointly elliptically distributed if its iso-density contours—loci of equal values of the density function—are ellipses. The concept extends to an arbitrary number of elements of the random vector, in which case in general the iso-density contours are ellipsoids. A special case is the multivariate normal distribution. The elliptical distributions are important in finance because if rates of return on assets are jointly elliptically distributed then all portfolios can be characterized completely by their mean and variance—that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return.


title: Ellipse

text: Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the MacIntosh QuickDraw API, and Direct2D on Windows. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. M. L. V. Pitteway extended Bresenham's algorithm for lines to conics in 1967. Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.


title: Ellipse

text: In 1970 Danny Cohen presented at the "Computer Graphics 1970" conference in England a linear algorithm for drawing ellipses and circles. In 1971, L. B. Smith published similar algorithms for all conic sections and proved them to have good properties. These algorithms need only a few multiplications and additions to calculate each vector.


title: Ellipse

text: It is beneficial to use a parametric formulation in computer graphics because the density of points is greatest where there is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent "jaggedness" of the approximation.


title: Ellipse

text: Composite Bézier curves may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as an affine transformation of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituent Bézier curves behave appropriately under such transformations.


title: Ellipse

text: It is sometimes useful to find the minimum bounding ellipse on a set of points. The ellipsoid method is quite useful for solving this problem.


title: Cassian Andor

text: Cassian Andor is a fictional character in the "Star Wars" franchise, primarily portrayed by actor Diego Luna. Introduced in the feature film "Rogue One" (2016) as a supporting character, he is the protagonist of the ongoing prequel television series "Andor" (2022–present).


title: Cassian Andor

text: He begins his life as an orphan on the abandoned mining planet Kenari. After being adopted and taken from his home by scavengers, he grows up on the industrial planet Ferrix. As a young man, he smuggles stolen Imperial ship components before joining the Rebel Alliance. During the events of "Rogue One", he is a pilot, intelligence officer for the Rebel Alliance, and leader of Rogue One, a rebel unit that steals the plans to the Death Star, a weapon powerful enough to destroy a planet.


title: Cassian Andor

text: The character has received a mixed response from critics, with praise for Luna's performance but some criticism for his writing. As the first Mexican lead actor and one of the first Latino leads in a "Star Wars" film, his introduction in "Rogue One" was considered to be a milestone in the history of Latino representation on film.


title: Cassian Andor

text: Though not initially named Cassian Andor, a "Cassian Andor-type character" appeared in the original treatment of "Rogue One" written by John Knoll, chief creative officer at Industrial Light & Magic and in the first draft of the script written by Gary Whitta. The character was created as a member of Rogue One, then commanded by a Rebel Alliance sergeant version of Jyn Erso. The original intention was to kill all members of the Rogue One team, including Cassian. However, fearing that Disney would not allow the ending, Knoll and Whitta wrote that Cassian escapes Scarif with the Death Star plans alongside Jyn Erso and, though their ship is destroyed by Darth Vader after they transfer the plans to Princess Leia, they narrowly survive in an escape pod. Whitta stated that having to "jump through so many hoops" to ensure survival indicated that Cassian and Jyn should die on Scarif. Producer Kathleen Kennedy and Disney ultimately approved the ending in which the entire crew dies. The main purpose of this action was believed to be for the characters to be ultimately replaced after the film by the crew of the original trilogy.


title: Cassian Andor

text: In May 2015, it was announced that Diego Luna had been cast in a lead role in "Rogue One". Director Gareth Edwards wanted Cassian to be warm and likeable rather than the typical stoic, brooding action hero, leading to Edwards's decision to cast Luna early in the casting process. Edwards was impressed by Luna's background in small, character driven productions, as he wanted the film to tell a more personal story than previous "Star Wars" instalments. Luna felt that his casting, and the casting of other non-white men in the film's leading roles, reflected a "modern approach" and a world in which "racial and cultural diversity is in fact making us richer and more interesting."


title: Cassian Andor

text: Luna spoke in the role with his native Mexican accent, an exception for the film franchise, which has mostly featured United States, Canadian and British accents. Luna stated that his accent was not an issue with the film's producers, who were happy with it. Luna also dubbed Andor's dialogue for the Spanish-language version of the film; the dub was recorded in Mexico City months before the film's theatrical release.


title: Cassian Andor

text: While filming "Rogue One", Luna developed a personal backstory for Cassian to help him get into character. In an interview with "Entertainment Weekly", he stated that he incorporated Mexican cultural influences into the character's backstory. He imagined the character to come from a "marginalized" background as a way to explain his unique accent in the Star Wars universe. He stated this backstory resonated with "Andor" showrunner Tony Gilroy, who also wanted to explain the character's distinctive accent.


title: Cassian Andor

text: In November 2018, it was reported that Luna would reprise his role in the Disney+ series "Andor", which was released in 2022. A young version of Cassian is portrayed by Antonio Viña in the show's first season. Luna once again dubbed the character for the Spanish-language version of "Andor", which also features co-star Adria Arjona voicing her own character.


title: Cassian Andor

text: For "Andor", Luna and the writers of the series both sought to expand on Cassian's backstory and explore how he became the rebel leader portrayed in "Rogue One". At the beginning of the series, Cassian is more childish and shortsighted than the character in "Rogue One". At the beginning of his story, Gilroy imagined Cassian as being motivated by anger rather than ideology. According to Gilroy, the first season of "Andor" takes a "completely disillusioned and completely self-interested" character and follows his development into someone fully dedicated to the Rebellion's cause. Luna said that the series was ultimately about change and transformation.


title: Cassian Andor

text: In "Andor", Cassian is established to be a refugee who was separated from his culture and community on the indigenous planet of Kenari by the Empire's oppressive regime. Unlike most humans in the "Star Wars" universe, the inhabitants of Kenari do not speak "Galactic Basic", the common language of the Galaxy, explaining Cassian's accent. The series retcons Cassian's prior backstory as established in "", which claimed that he was born on Fest. In the series, he has falsified documents which claim Fest as his birthplace.


title: Cassian Andor

text: Luna and Gilroy saw this backstory as a way to establish Cassian's alienation and resentment for the Empire. Gilroy stated that the loss of his home explained Cassian's coldblooded commitment to the Rebellion, saying he wanted to "allow [Cassian] to do some very unpleasant things along the way and yet you still not reject him." Cassian's unjust incarceration and the execution of his adopted father contributes to his hatred of the Empire, and he dedicates his life to resisting the Empire after being imprisoned on Narkina-5. By the end of the first season, Cassian has gradually transformed into a committed rebel.


title: Cassian Andor

text: "Rogue One: A Star Wars Story" was released in December 2016, featuring Luna in the role of Cassian Andor, a man who had been fighting in the Rebellion since he was six years old. In the film, Cassian learns about the defection of Imperial pilot Bodhi Rook and the Death Star superweapon from Rebel informant Tivik, whom he kills to avoid his capture since he is unable to escape due to an injured arm. Cassian is later secretly ordered to find and assassinate scientist Galen Erso, who was captured by Director Orson Krennic and forced to help build the Death Star.


title: Cassian Andor

text: Andor and his co-pilot, reprogrammed Imperial security droid K-2SO, lead a mission to spring Galen's daughter Jyn Erso from Imperial custody so that they can make contact with Saw Gerrera. Then Andor, K-2S0, and Jyn travel to Jedha to visit Saw and obtain a message from Galen which was carried by Rook. Later, the Rebels arrive on Eadu after learning that Galen is there; Cassian hesitantly chooses not to kill him. He ultimately leads a team with Jyn to steal the plans for the Death Star on Scarif, where he infiltrates its Imperial data center with Jyn and K-2SO. After K-2SO sacrifices himself, Jyn and Cassian obtain the schematics, but they are ambushed by Krennic, who shoots Cassian. However, Andor recovers and shoots and wounds Krennic before Jyn successfully transmits the plans to the Rebel forces in orbit. The Death Star then fires upon the planet, killing Cassian and Jyn as they embrace.


title: Cassian Andor

text: Luna reprises his role as Andor in "Andor", a live-action "spy thriller" series for Disney+, which is set five years before "Rogue One", and which premiered September 21, 2022.


title: Cassian Andor

text: In the first season of "Andor", it is revealed that Cassian was an orphan named Kassa from Kenari, a former mining planet abandoned by the Empire whose indigenous inhabitants lead a tribal lifestyle. He was adopted by scavengers Clem and Maarva and raised on Ferrix, a planet whose primary activity is heavy industry. At the age of 13, Cassian witnessed Clem being killed by Imperials. In retaliation, he attacked a group of stormtroopers and was sent to an Imperial youth center for three years.


title: Cassian Andor

text: Cassian later began stealing ship components and other valuable technology from the Empire with his former lover Bix Caleen. These activities bring him to the attention of the rebel leader Luthen Rael. Cassian, who is wanted for the murder of two corporate security guards, agrees to join Luthen's operation. He is sent to the planet of Aldhani to help a group of rebels steal the Imperial payroll stored there. After the heist, Cassian took his payment and returned to Ferrix for Maarva and their droid B2EMO, but Maarva refuses to leave the planet where she grew up. She reveals to Cassian that she is a rebel, and that reports of the attack on Aldhani inspired her to join the cause. Cassian reluctantly leaves Maarva behind, and goes to the tropical planet Niamos.


title: Cassian Andor

text: Unbeknownst to him, Luthen has decided to kill Cassian to keep his own involvement with the rebellion secret, and has sent two of Cassian's teammates from Aldhani to kill him. ISB supervisor Dedra Meero is also searching for him as she believes that he is connected to a rebel leader codenamed Axis, whom she is unaware is Luthen. Both the Empire and the Rebellion plant spies on Ferrix in an attempt to trap Cassian.


title: Cassian Andor

text: While on Niamos, Cassian is harassed by a sandtrooper who falsely accuses him of vandalism. He is arrested and sentenced to six years on the prison planet Narkina-5. Conditions on Narkina-5 are brutal; prisoners work 12 hour shifts building components for an unknown machine, and electric shocks delivered through the metal floor are used to keep them in line. After it is revealed that none of the prisoners will be released, and that the prison electrocuted one hundred men, Cassian puts his escape plan into motion. He and fellow prisoner Kino Loy lead a mass escape attempt that frees all five thousand prisoners.


title: Cassian Andor

text: Upon returning to Niamos, he attempts to call Ferrix and learns that Maarva died while he was in prison. Luthen travels to Ferrix once he hears of this, as he believes that Cassian will return there. Despite the danger involved in returning, Cassian decides to visit Ferrix for Maarva's funeral. He learns that Bix has been tortured and imprisoned in the makeshift Imperial headquarters on Ferrix, and rescues her while the people of Ferrix stage a local rebellion. After ensuring that Bix, B2EMO and his friend Brasso can safely escape Ferrix, he goes to Luthen's ship and offers himself up. Rather than kill him, Luthen agrees to take him back into the Rebellion.


title: Cassian Andor

text: Cassian appears in a number of related works, some of which differ from the film and television canon. He appears in the by Alexander Freed. In the novel, he is established to have come from a family of Separatists who resisted the Galactic Republic during the Clone Wars. In the "Ultimate Star Wars New Edition", he is described as having been "battle-hardened" by his Separatist action during the Clone Wars, before being recruited into the Rebellion by General Davits Draven. "Star Wars: Rogue One: The Ultimate Visual Guide" describes his involvement with insurrectionary cells and anarchist movements in his youth. He also appears in "Star Wars: Secrets of the Empire", a virtual reality experience produced by ILMxLAB and The VOID for the Disneyland Resort and Walt Disney World; Luna reprises his role as Cassian to provide mission briefings and deliver orders throughout the experience.


title: Cassian Andor

text: The character's appearances in "Rogue One" and "Andor" have received mixed reviews from critics, who were divided on the quality of his narrative. Luna's performance in "Rogue One" garnered a mixed reception, while his performance in "Andor" was well received.


title: Cassian Andor

text: Justin Chang of the "Los Angeles Times" praised Luna's "flinty charisma" in "Rogue One", and David Ehrlich of "IndieWire" wrote that the character was "a bit lost between archetypes, but he’s charming enough." Ann Hornaday of "The Washington Post" wrote that "the fey, soft-spoken Luna is particularly ill-suited to play a rakish man of adventure". Praising Felicity Jones in the role of Jyn Erso, Chris Nashawaty wrote in "Entertainment Weekly", "I wish Luna had a little more personality, a little more Han Solo swagger, to match her."


title: Cassian Andor

text: Many critics and authors commented on the character's ruthless and cynical personality, especially in comparison to other "Star Wars" protagonists. Cassian's murder of an informant at the beginning of "Rogue One" is often considered to be a defining moment for the character. Eric Goldman of "IGN" wrote that "Luna projects innate charisma mixed with ongoing unease, as we see [Cassian] participate in actions he's not always proud of in service of the greater good."


title: Cassian Andor

text: Luna, reflecting on the events of "Rogue One", has stated that Cassian's willingness to kill is a symptom of living under oppression.


title: Cassian Andor

text: Terrance MacMullan in "Rebellions are Built on Hope", describes Cassian as a "complex" but ultimately good character, whose faith in his comrades is a core part of the film's philosophy. To MacMullan, Cassian's murder of the informant distinguishes "Rogue One"'s more nuanced philosophy from the clear good against evil conflicts of the Original Trilogy. Macmullan commented that:Having to witness the daring Cassian Andor murder a fellow Rebel in cold blood nearly inverts the famous line from the crawl of "Revenge of The Sith": while the battles of the Clone Wars taught us that "there are heroes on both sides," "Rogue One" makes us wonder if "there are villains on both sides."


title: Cassian Andor

text: The character's portrayal in "Andor" received mixed reviews from critics, with some praising his character development over the first season, while others considered his narrative to be a weak point of the otherwise well received show. Sarah Odman, in a review for "The Hollywood Reporter", praised Cassian's transformation "from a disenfranchised criminal to a man who looks more like the ruthless Rebel leader they remember."


title: Cassian Andor

text: Roxana Hadadi of "Vulture" and Joshua Rivera of "Polygon" both praised the character's expanded history and Luna's performance. Shirley Li, writing for "The Atlantic", commented that the series "depicts how an ordinary, disillusioned character can undergo a political awakening in just a few short years."


title: Cassian Andor

text: Critic Alan Sepinwall of "Rolling Stone" described this arc as a "hero's journey," while opining that it was the least compelling part of "Andor"'s first season. He wrote that the character is "never dramatized in a particularly interesting way beyond [Luna's] performance." Mike Hale, writing for "The New York Times", made similar criticisms of the character, who he described as a thinly written "Humphrey Bogart-style cynical romantic." Ryan Britt, in a review for "Den of Geek", wrote that the character had unclear motivations and a lack of agency in the first season that made its finale unsatisfying.


title: Cassian Andor

text: Some critics have considered the story arc in which Cassian is imprisoned on Narkina-5 to be a commentary on exploitation in the prison-industrial complex. Tracy Brown and Jamil Smith of the "Los Angeles Times" felt that the arc was a timely exploration of mass incarceration.


title: Cassian Andor

text: Cassian is the first Star Wars character portrayed by a Mexican actor, and the one of first Latino lead characters in a "Star Wars" film. His appearances in "Rogue One" and "Andor" are considered milestones in the history of racial diversity and Latino representation in the Star Wars franchise. Yara Simón, in an article for "Remezcla", described Cassian as "the Latino hero we deserve, and the Latino actor who has had the most prominent role in any of the eight "Star Wars" films released in the last 39 years."


title: Cassian Andor

text: After the release of the theatrical trailer for "Rogue One", the film was praised for not masking Luna's Mexican accent. Caroline Framske of "Vox" wrote that "getting to see a "Star Wars" hero save the day with a distinctive Mexican accent is a huge deal — especially for audiences who rarely see themselves reflected onscreen in a role that isn’t a tiny part or, worse, the butt of some joke."


title: Cassian Andor

text: Writing for "Latino Rebels", Cristina Escobar wrote that "the question of whether Cassian is Latino is a thorny one." She observed that geopolitical categories like Latinx do not exist in the science fiction universe of "Star Wars", and wrote that essentializing the Latino experience "reduce[d] [Latin people] to a set of traits we don’t all share."


title: Cassian Andor

text: In "Why We Love Star Wars: The Great Moments That Built a Galaxy Far, Far Away", Ken Napzok described Cassian's characterization in Freed's "Rogue One" novelization as part of the more nuanced and realistic portrayal of the Rebellion in later "Star Wars" instalments. In a review of the novel for "New York Daily News", Sean Keane wrote that "Fans will be happy to see that [Cassian] carries the weight of his questionable actions throughout the story."


title: Metaphor

text: A metaphor is a figure of speech that, for rhetorical effect, directly refers to one thing by mentioning another. It may provide (or obscure) clarity or identify hidden similarities between two different ideas. Metaphors are often compared with other types of figurative language, such as antithesis, hyperbole, metonymy, and simile. One of the most commonly cited examples of a metaphor in English literature comes from the "All the world's a stage" monologue from "As You Like It":


title: Metaphor

text: This quotation expresses a metaphor because the world is not literally a stage, and most humans are not literally actors and actresses playing roles. By asserting that the world is a stage, Shakespeare uses points of comparison between the world and a stage to convey an understanding about the mechanics of the world and the behavior of the people within it.


title: Metaphor

text: In the ancient Hebrew psalms (around 1000 B.C.), one finds already vivid and poetic examples of metaphor such as, "The Lord is my rock, my fortress and my deliverer; my God is my rock, in whom I take refuge, my shield and the horn of my salvation, my stronghold” and “The Lord is my shepherd, I shall not want”. Some recent linguistic theories view all language in essence as metaphorical.


title: Metaphor

text: The word metaphor itself is a metaphor, coming from a Greek term meaning "transference (of ownership)". The user of a metaphor alters the reference of the word, "carrying" it from one semantic "realm" to another. The new meaning of the word might be derived from an analogy between the two semantic realms, but also from other reasons such as the distortion of the semantic realm - for example in sarcasm.


title: Metaphor

text: The English word "metaphor" derives from the 16th-century Old French word "métaphore", which comes from the Latin "metaphora", "carrying over", and in turn from the Greek μεταφορά ("metaphorá"), "transference (of ownership)", from μεταφέρω ("metapherō"), "to carry over", "to transfer" and that from μετά ("meta"), "behind", "along with", "across" + φέρω ("pherō"), "to bear", "to carry".


title: Metaphor

text: "The Philosophy of Rhetoric" (1937) by rhetorician I. A. Richards describes a metaphor as having two parts: the tenor and the vehicle. The tenor is the subject to which attributes are ascribed. The vehicle is the object whose attributes are borrowed. In the previous example, "the world" is compared to a stage, describing it with the attributes of "the stage"; "the world" is the tenor, and "a stage" is the vehicle; "men and women" is the secondary tenor, and "players" is the secondary vehicle.


title: Metaphor

text: Other writers employ the general terms 'ground' and 'figure' to denote the tenor and the vehicle. Cognitive linguistics uses the terms 'target' and 'source', respectively.


title: Metaphor

text: Psychologist Julian Jaynes coined the terms 'metaphrand' and 'metaphier', plus two new concepts, 'paraphrand' and 'paraphier'.


title: Metaphor

text: 'Metaphrand' is equivalent to the metaphor-theory terms 'tenor', 'target', and 'ground'. 'Metaphier' is equivalent to the metaphor-theory terms 'vehicle', 'figure', and 'source'. In a simple metaphor, an obvious attribute of the metaphier exactly characterizes the metaphrand (e.g. the ship plowed the seas). With an inexact metaphor, however, a metaphier might have associated attributes or nuances – its paraphiers – that enrich the metaphor because they "project back" to the metaphrand, potentially creating new ideas – the paraphrands – associated thereafter with the metaphrand or even leading to a new metaphor. For example, in the metaphor "Pat is a tornado", the metaphrand is "Pat", the metaphier is "tornado". As metaphier, "tornado" carries paraphiers such as power, storm and wind, counterclockwise motion, and danger, threat, destruction, etc. The metaphoric meaning of "tornado" is inexact: one might understand that 'Pat is powerfully destructive' through the paraphrand of physical and emotional destruction; another person might understand the metaphor as 'Pat can spin out of control'. In the latter case, the paraphier of 'spinning motion' has become the paraphrand 'psychological spin', suggesting an entirely new metaphor for emotional unpredictability, a possibly apt description for a human being hardly applicable to a tornado.


title: Metaphor

text: Based on his analysis, Jaynes claims that metaphors not only enhance description, but "increase enormously our powers of perception...and our understanding of [the world], and literally create new objects".


title: Metaphor

text: Metaphors are most frequently compared with similes. It is said, for instance, that a metaphor is 'a condensed analogy' or 'analogical fusion' or that they 'operate in a similar fashion' or are 'based on the same mental process' or yet that 'the basic processes of analogy are at work in metaphor'. It is also pointed out that 'a border between metaphor and analogy is fuzzy' and 'the difference between them might be described (metaphorically) as the distance between things being compared'. A metaphor asserts the objects in the comparison are identical on the point of comparison, while a simile merely asserts a similarity through use of words such as "like" or "as". For this reason a common-type metaphor is generally considered more forceful than a simile.


title: Metaphor

text: Metaphor is distinct from metonymy, both constituting two fundamental modes of thought. Metaphor works by bringing together concepts from different conceptual domains, whereas metonymy uses one element from a given domain to refer to another closely related element. A metaphor creates new links between otherwise distinct conceptual domains, whereas a metonymy relies on pre-existent links within them.


title: Metaphor

text: For example, in the phrase "lands belonging to the crown", the word "crown" is a metonymy because some monarchs do indeed wear a crown, physically. In other words, there is a pre-existent link between "crown" and "monarchy". On the other hand, when Ghil'ad Zuckermann argues that the Israeli language is a "phoenicuckoo cross with some magpie characteristics", he is using a metaphor. There is no physical link between a language and a bird. The reason the metaphors "phoenix" and "cuckoo" are used is that on the one hand hybridic "Israeli" is based on Hebrew, which, like a phoenix, rises from the ashes; and on the other hand, hybridic "Israeli" is based on Yiddish, which like a cuckoo, lays its egg in the nest of another bird, tricking it to believe that it is its own egg. Furthermore, the metaphor "magpie" is employed because, according to Zuckermann, hybridic "Israeli" displays the characteristics of a magpie, "stealing" from languages such as Arabic and English.


title: Metaphor

text: A dead metaphor is a metaphor in which the sense of a transferred image has become absent. The phrases "to grasp a concept" and "to gather what you've understood" use physical action as a metaphor for understanding. The audience does not need to visualize the action; dead metaphors normally go unnoticed. Some distinguish between a dead metaphor and a cliché. Others use "dead metaphor" to denote both.


title: Metaphor

text: A mixed metaphor is a metaphor that leaps from one identification to a second inconsistent with the first, e.g.:


title: Metaphor

text: An extended metaphor, or conceit, sets up a principal subject with several subsidiary subjects or comparisons. In the above quote from "As You Like It", the world is first described as a stage and then the subsidiary subjects men and women are further described in the same context.


title: Metaphor

text: An implicit metaphor has no specified tenor, although the vehicle is present. M. H. Abrams offers the following as an example of an implicit metaphor: "That reed was too frail to survive the storm of its sorrows". The reed is the vehicle for the implicit tenor, someone's death, and the "storm" is the vehicle for the person's "sorrows".


title: Metaphor

text: Metaphor can serve as a device for persuading an audience of the user's argument or thesis, the so-called rhetorical metaphor.


title: Metaphor

text: Aristotle writes in his work the "Rhetoric" that metaphors make learning pleasant: "To learn easily is naturally pleasant to all people, and words signify something, so whatever words create knowledge in us are the pleasantest." When discussing Aristotle's "Rhetoric", Jan Garret stated "metaphor most brings about learning; for when [Homer] calls old age "stubble", he creates understanding and knowledge through the genus, since both old age and stubble are [species of the genus of] things that have lost their bloom." Metaphors, according to Aristotle, have "qualities of the exotic and the fascinating; but at the same time we recognize that strangers do not have the same rights as our fellow citizens".


title: Metaphor

text: Educational psychologist Andrew Ortony gives more explicit detail: "Metaphors are necessary as a communicative device because they allow the transfer of coherent chunks of characteristics -- perceptual, cognitive, emotional and experiential -- from a vehicle which is known to a topic which is less so. In so doing they circumvent the problem of specifying one by one each of the often unnameable and innumerable characteristics; they avoid discretizing the perceived continuity of experience and are thus closer to experience and consequently more vivid and memorable."


title: Metaphor

text: As a characteristic of speech and writing, metaphors can serve the poetic imagination. This allows Sylvia Plath, in her poem "Cut", to compare the blood issuing from her cut thumb to the running of a million soldiers, "redcoats, every one"; and enabling Robert Frost, in "The Road Not Taken", to compare a life to a journey.


title: Metaphor

text: Sonja K. Foss characterizes metaphors as "nonliteral comparisons in which a word or phrase from one domain of experience is applied to another domain".


title: Metaphor

text: She argues that since reality is mediated by the language we use to describe it, the metaphors we use shape the world and our interactions to it.


title: Metaphor

text: Some theorists have suggested that metaphors are not merely stylistic, but that they are cognitively important as well. In "Metaphors We Live By", George Lakoff and Mark Johnson argue that metaphors are pervasive in everyday life, not just in language, but also in thought and action. A common definition of metaphor can be described as a comparison that shows how two things that are not alike in most ways are similar in another important way. They explain how a metaphor is simply understanding and experiencing one kind of thing in terms of another, called a "conduit metaphor". A speaker can put ideas or objects into containers, and then send them along a conduit to a listener who removes the object from the container to make meaning of it. Thus, communication is something that ideas go into, and the container is separate from the ideas themselves. Lakoff and Johnson give several examples of daily metaphors in use, including "argument is war" and "time is money". Metaphors are widely used in context to describe personal meaning. The authors suggest that communication can be viewed as a machine: "Communication is not what one does with the machine, but is the machine itself."


title: Metaphor

text: Experimental evidence shows that "priming" people with material from one area will influence how they perform tasks and interpret language in a metaphorically related area.


title: Metaphor

text: Cognitive linguists emphasize that metaphors serve to facilitate the understanding of one conceptual domain—typically an abstraction such as "life", "theories" or "ideas"—through expressions that relate to another, more familiar conceptual domain—typically more concrete, such as "journey", "buildings" or "food". For example: we "devour" a book of "raw" facts, try to "digest" them, "stew" over them, let them "simmer on the back-burner", "regurgitate" them in discussions, and "cook" up explanations, hoping they do not seem "half-baked".


title: Metaphor

text: Lakoff and Johnson greatly contributed to establishing the importance of conceptual metaphor as a framework for thinking in language, leading scholars to investigate the original ways in which writers used novel metaphors and question the fundamental frameworks of thinking in conceptual metaphors.


title: Metaphor

text: From a sociological, cultural, or philosophical perspective, one asks to what extent ideologies maintain and impose conceptual patterns of thought by introducing, supporting, and adapting fundamental patterns of thinking metaphorically. To what extent does the ideology fashion and refashion the idea of the nation as a container with borders? How are enemies and outsiders represented? As diseases? As attackers? How are the metaphoric paths of fate, destiny, history, and progress represented? As the opening of an eternal monumental moment (German fascism)? Or as the path to communism (in Russian or Czech for example)?


title: Metaphor

text: Some cognitive scholars have attempted to take on board the idea that different languages have evolved radically different concepts and conceptual metaphors, while others hold to the Sapir-Whorf hypothesis. German philologist Wilhelm von Humboldt contributed significantly to this debate on the relationship between culture, language, and linguistic communities. Humboldt remains, however, relatively unknown in English-speaking nations. Andrew Goatly, in "Washing the Brain", takes on board the dual problem of conceptual metaphor as a framework implicit in the language as a system and the way individuals and ideologies negotiate conceptual metaphors. Neural biological research suggests some metaphors are innate, as demonstrated by reduced metaphorical understanding in psychopathy.


title: Metaphor

text: James W. Underhill, in "Creating Worldviews: Ideology, Metaphor & Language" (Edinburgh UP), considers the way individual speech adopts and reinforces certain metaphoric paradigms. This involves a critique of both communist and fascist discourse. Underhill's studies are situated in Czech and German, which allows him to demonstrate the ways individuals are thinking both within and resisting the modes by which ideologies seek to appropriate key concepts such as "the people", "the state", "history", and "struggle".


title: Metaphor

text: Though metaphors can be considered to be "in" language, Underhill's chapter on French, English and ethnolinguistics demonstrates that we cannot conceive of language or languages in anything other than metaphoric terms.


title: Metaphor

text: Metaphors can map experience between two nonlinguistic realms. Musicologist Leonard B. Meyer demonstrated how purely rhythmic and harmonic events can express human emotions. It is an open question whether synesthesia experiences are a sensory version of metaphor, the "source" domain being the presented stimulus, such as a musical tone, and the target domain, being the experience in another modality, such as color.


title: Metaphor

text: Art theorist Robert Vischer argued that when we look at a painting, we "feel ourselves into it" by imagining our body in the posture of a nonhuman or inanimate object in the painting. For example, the painting "The Lonely Tree" by Caspar David Friedrich shows a tree with contorted, barren limbs. Looking at the painting, we imagine our limbs in a similarly contorted and barren shape, evoking a feeling of strain and distress. Nonlinguistic metaphors may be the foundation of our experience of visual and musical art, as well as dance and other art forms.


title: Metaphor

text: In historical onomasiology or in historical linguistics, a metaphor is defined as a semantic change based on a similarity in form or function between the original concept and the target concept named by a word.


title: Metaphor

text: For example, mouse: "small, gray rodent with a long tail" → "small, gray computer device with a long cord".


title: Metaphor

text: Aristotle's discusses the creation of metaphors at the end of his "Poetics": "But the greatest thing by far is to be a master of metaphor. It is the one thing that cannot be learnt from others; and it is also a sign of genius, since a good metaphor implies an intuitive perception of the similarity in dissimilars."


title: Metaphor

text: Baroque literary theorist Emanuele Tesauro defines the metaphor "the most witty and acute, the most strange and marvelous, the most pleasant and useful, the most eloquent and fecund part of the human intellect". There is, he suggests, something divine in metaphor: the world itself is God's poem and metaphor is not just a literary or rhetorical figure but an analytic tool that can penetrate the mysteries of God and His creation.


title: Metaphor

text: Friedrich Nietzsche makes metaphor the conceptual center of his early theory of society in "On Truth and Lies in the Non-Moral Sense". Some sociologists have found his essay useful for thinking about metaphors used in society and for reflecting on their own use of metaphor. Sociologists of religion note the importance of metaphor in religious worldviews, and that it is impossible to think sociologically about religion without metaphor.


title: Nigel Mansell

text: Nigel Ernest James Mansell, (; born 8 August 1953) is a British retired racing driver who won both the Formula One World Championship (1992) and the CART Indy Car World Series (1993). Mansell was the reigning F1 champion when he moved over to CART, becoming the first person to win the CART title in his debut season, and making him the only person to hold both the World Drivers' Championship and the American open-wheel National Championship simultaneously.


title: Nigel Mansell

text: His career in Formula One spanned 15 seasons, with his final two full seasons of top-level racing being spent in the CART series. Mansell is the second most successful British Formula One driver of all time in terms of race wins with 31 victories, (behind Lewis Hamilton with wins), and is eighth overall on the Formula One race winners list, behind Hamilton, Michael Schumacher, Sebastian Vettel, Alain Prost, Ayrton Senna, Max Verstappen and Fernando Alonso. He held the record for the most poles set in a single season, which was broken in by Sebastian Vettel. He was rated in the top 10 Formula One drivers of all time by longtime Formula One commentator Murray Walker. In 2008, ESPN.com ranked him 24th on their "Top 25 Drivers of All Time" list. He was also ranked No. 9 of the 50 greatest F1 drivers of all time by the Times Online on a list that also included such drivers as Prost, Senna, Jackie Stewart and Jim Clark.


title: Nigel Mansell

text: Mansell raced in the Grand Prix Masters series in 2005, and won the championship title. He later signed a one-off race deal for the Scuderia Ecosse GT race team to drive their number 63 Ferrari F430 GT2 car at Silverstone on 6 May 2007. He has since competed in additional sports car races with his sons Leo and Greg, including the 2010 24 Hours of Le Mans, and was inducted to the International Motorsports Hall of Fame in 2005.


title: Nigel Mansell

text: Nigel Ernest James Mansell was born on 8 August 1953 in Upton-upon-Severn, Worcestershire, where his parents, Eric and Joyce Mansell, ran a tea shop. He grew up in Hall Green, Birmingham.


title: Nigel Mansell

text: Mansell had a fairly slow start to his racing career, using his own money to help work his way up the ranks. After considerable success in kart racing, he moved to the Formula Ford series to the disapproval of his father. In 1976, Mansell won six of the nine races he took part in, including his debut event at Mallory Park. He entered 42 races the following year and won 33 to become the 1977 British Formula Ford champion, despite suffering a broken neck in a qualifying session at Brands Hatch. Doctors told him he had been perilously close to quadriplegia, that he would be confined for six months and would never drive again. Mansell discharged himself from the hospital and returned to racing. Three weeks before the accident he had resigned from his job as an aerospace engineer, having previously sold most of his personal belongings to finance his foray into Formula Ford. Later that year he was given the chance to race a Lola T570 Formula 3 car at Silverstone. He finished fourth and decided that he was ready to move into the higher formula.


title: Nigel Mansell

text: Mansell raced in Formula Three in 1978–1980. Mansell's first season in Formula Three started with a pole position and a second-place finish. However, the car was not competitive, as a commercial deal with Unipart required his team to use Triumph Dolomite engines that were vastly inferior to the Toyota engines used by the leading teams. After three seventh-place finishes and a fourth in his last race, he parted from the team. The next season saw him drive for David Price Racing. Following a first win in the series at Silverstone in March, he went on to finish eighth in the championship. His racing was consistent, but a collision with Andrea de Cesaris resulted in a huge cartwheeling crash which he was lucky to survive. Again he was hospitalised, this time with broken vertebrae.


title: Nigel Mansell

text: His driving was noticed by Colin Chapman, owner of Lotus, and shortly after his accident, hiding the extent of his injury with painkillers, Mansell performed well enough during a tryout at the Paul Ricard circuit with Lotus, where he was pitted against a number of other drivers to determine who was going to take the second seat for the 1980 season alongside Mario Andretti, as Argentine Carlos Reutemann was leaving to go to Williams. Driving a 79, the seat eventually went to Italian driver Elio de Angelis, but Mansell was selected to become a test driver for the Norfolk-based Formula One team.


title: Nigel Mansell

text: Mansell's skill as a test driver, including setting the fastest lap around Silverstone in a Lotus car at the time, impressed Chapman enough to give him a trio of starts in F1 in , driving a development version of the Lotus 81 used by the team, the Lotus 81B. In his Formula One debut at the 1980 Austrian Grand Prix, a fuel leak in the cockpit that developed shortly before the start of the race left him with painful first and second degree burns on his buttocks. An engine failure forced him to retire from that race and his second, however an accident at his third event at Imola meant he failed to qualify. Team leader Mario Andretti wrote his car off in a start-line accident during the Canadian Grand Prix at Montreal, so Mansell had to give up his car for Andretti to compete in for his home race, the final race of the season at Watkins Glen in the United States. Andretti announced he was leaving to move to Alfa Romeo at the end of the season leaving Lotus with a vacant race seat.


title: Nigel Mansell

text: During the season, Mansell planned to race in the 24 Hours of Le Mans sportscar event in order to earn extra money. At the time Mansell was paid £50,000 a year and was offered £10,000 to take part in Le Mans. Chapman believed that by entering the Le Mans race, Mansell was exposing himself to unnecessary risk and paid him £10,000 not to take part in the race. Chapman extended Mansell's contract to the end of the season in a deal that made him a millionaire.


title: Nigel Mansell

text: As a result of the gestures, such as described above, Mansell became very close to Chapman, who made him equal number one in the team with de Angelis, and was devastated by Chapman's sudden death in 1982. In his autobiography Mansell stated that when Chapman died, "The bottom dropped out of my world. Part of me died with him. I had lost a member of my family." Following Chapman's death, relationships at Lotus became strained, as replacement team principal Peter Warr did not have a high regard for him as a driver or person. Warr in his book titled "Team Lotus – My View From The Pit Wall" stated about the 1982 season:


title: Nigel Mansell

text: "After carefully analysing the season just finished, it was completely clear who was the number one. It was Elio. He was faster, he had out-qualified Nigel ten times to three. Elio had seven points scoring finishes to Nigel's two and in addition to Elio's win in Austria, had a tally of more than three times the number of points gained by Nigel. What is more, the margin by which Elio eclipsed his team-mate in qualifying overall was a huge 4.5 per cent. And all this in the year when, as near as can be reasonably achieved, the two drivers were given equal equipment and treatment."


title: Nigel Mansell

text: De Angelis was then promoted back to outright number one for the 1983 season. This was demonstrated by the fact that he had exclusive use of the quick but unreliable Renault turbo-charged 93T for the whole season, and Mansell did not get to drive a turbocharged car until the ninth round, the British Grand Prix at Silverstone, a race where he climbed from 16th to second and eventually finished fourth in a brand-new, hastily designed 94T. Due to their tumultuous relationship and a lack of decent results, Warr was not keen on honouring the last year of the contract that Mansell had signed with Chapman. However, with encouragement from Lotus's sponsors, John Player Special (who allegedly preferred a British driver), and with the only other remaining top British driver (Derek Warwick, after John Watson's retirement) already confirmed to be joining the factory Renault team, it was announced Mansell would be staying with the team.


title: Nigel Mansell

text: In 1984, Mansell finished in the championship top 10 for the first time, and took his first career pole position but still finished behind teammate de Angelis, who finished third, in the championship. At the 1984 Monaco Grand Prix Mansell surprised many by overtaking Alain Prost in a wet race for the lead, but soon after retired from the race after losing control on the slippery painted lines on the road surface on the run up the hill on lap 15. Late in the season, Lotus announced the recruitment of Ayrton Senna for the following year, leaving Mansell with no race seat at Lotus. After receiving offers from Arrows and Williams, and first turning down Williams's offer, it was announced before the Dutch Grand Prix that he would indeed be joining Williams.


title: Nigel Mansell

text: Mansell was remembered by many that year when he collapsed while pushing his car to the finish line after the transmission failed on the last lap of the 1984 Dallas Grand Prix. The race was one of the hottest on record, and after two hours of driving in 104 °F (about 40 °C) conditions Mansell fainted while pushing his car over the line to salvage a sixth-place finish (and thus one championship point) in a race of which he had led half, having started from pole.


title: Nigel Mansell

text: In his autobiography, Mansell claimed that his final race with the Lotus team—the 1984 Portuguese Grand Prix—was heavily compromised by Warr's unwillingness to give Mansell the brake pads he desired for the race. With 18 laps of the race remaining, and with Mansell in second position, the brakes on his car failed. On Mansell's departure, Warr was quoted as saying "He'll never win a Grand Prix as long as I have a hole in my arse".


title: Nigel Mansell

text: In Frank Williams hired Mansell to drive alongside Keke Rosberg as part of the Williams team, Mansell later saying "We have the greatest respect for each other." Mansell was given the "Red 5" number on his car, which he carried on subsequent Williams and Newman/Haas cars and which was brought to the public's attention mainly through commentator Murray Walker for the BBC.


title: Nigel Mansell

text: Rosberg, the World Champion who was heading into his fourth season with the team, was initially against Williams signing Mansell based on the clash the pair had at Dallas the previous year (Rosberg won that race and in an interview while on the podium publicly berated Mansell's blocking tactics while leading early in the race, which earned Rosberg a round of boos from the crowd who had appreciated Mansell's courage in trying to push his Lotus to the finish in the extreme heat). Other factors were what Rosberg later said in a 1986 interview was second-hand information about Mansell which ultimately proved to be false. The drivers found they got along well and from early in the pre-season formed a good working and personal relationship.


title: Nigel Mansell

text: 1985 initially appeared to provide more of the same for Mansell, although he was closer to the pace than before, especially as the Honda engines became more competitive by mid-season.


title: Nigel Mansell

text: During practice for the 1985 French Grand Prix, Mansell unwillingly broke the record for the highest speed crash in Formula One history. At the end of the Paul Ricard Circuit's 1.8 km long Mistral Straight he went off at the fast Courbe de Signes at over in his Williams FW10. Mansell suffered a concussion, which kept him out of the race. Teammate Rosberg claimed the pole for the race and finished second behind the Brabham-BMW of Nelson Piquet.


title: Nigel Mansell

text: Mansell achieved second place at the Belgian Grand Prix at Spa-Francorchamps, and followed this with his first victory in 72 starts at the European Grand Prix at Brands Hatch in England. He achieved a second straight victory at the South African Grand Prix in Kyalami. These triumphs helped turn Mansell into a Formula One star.


title: Nigel Mansell

text: Going into , the Williams-Honda team had a car, the FW11 which was capable of winning regularly, and Mansell had a new confidence that led to establishing himself as a potential World Championship contender. He also had a new teammate in twice World Champion Nelson Piquet who had joined Williams looking to be a regular winner and contender again after the Brabham-BMWs had become increasingly unreliable and uncompetitive. The Brazilian publicly described Mansell as "an uneducated blockhead" and had also criticised Mansell's wife Roseanne's looks, later retracting these statements following threats of legal action.


title: Nigel Mansell

text: Mansell won five Grand Prix in 1986 and also played a part in one of the closest finishes in Formula One history, finishing second to Ayrton Senna in the Spanish Grand Prix at Jerez by a mere 0.014 seconds (Mansell later jokingly said they should give himself and Senna 7½ points each). The 1986 season was led mostly by Mansell in championship points, and it went down to the wire in Adelaide, Australia for the Australian Grand Prix with Prost, Piquet and Mansell all still in contention for the title. The equation was simple, Prost and Piquet needed to win and have Mansell finish no higher than fourth. After aiming for a third-place finish which would guarantee him the title, Mansell would narrowly miss out on winning it after his left-rear tyre exploded in spectacular fashion on the main straight with only 19 laps of the race to go. In a 2012 interview for Sky Sports Legends of F1 Mansell revealed that, had he hit the wall rather than wrestling the car safely to a halt in the run-off area at the end of the straight, the stewards would most likely have red flagged the race. As the race was over two thirds distance, he would have kept his position and won his first F1 world title. Instead Mansell ended the season as runner-up to Alain Prost. His efforts in 1986 led to his being voted the BBC Sports Personality of the Year.


title: Nigel Mansell

text: 1986 proved to be a tough year for both Mansell and the Williams team, off the track at least. After a pre-season test session at the Paul Ricard Circuit in the south of France, team owner Frank Williams was involved in a horrific road accident which left him a tetraplegic. Williams would not return to the scene until making a surprise appearance at the British Grand Prix at Brands Hatch where Mansell and Piquet finished 1–2. Williams's absence from the day-to-day running of the team actually created tension between the team and engine supplier Honda. The Japanese giant regarded dual World Champion Nelson Piquet as the team's number one driver (they were reportedly paying the bulk of Piquet's multimillion-dollar retainer) and were reportedly unhappy that the team's co-owner and Technical Director Patrick Head did not rein in Mansell during races and allowed him to take both points and wins from Piquet. As it was, while Williams dominated the Constructors' Championship, the two drivers took enough points from each other to allow McLaren's Alain Prost to sneak through and win the Drivers' Championship.


title: Nigel Mansell

text: Before the season had started Honda had reportedly attempted to persuade Frank Williams to replace Mansell with their own test driver Satoru Nakajima. Williams, who was always more interested in the Constructors' title than the Drivers' as it showed that his team was the best, refused to do this, rightly believing that having two proven F1 winners in Piquet and Mansell would better-serve the team than Nakajima would as an F1 rookie.


title: Nigel Mansell

text: Six more wins followed in , including an emotional and hugely popular victory at Silverstone for the British Grand Prix in which he came back from 28 seconds behind in 30 laps to beat teammate Piquet, with his car running out of fuel on the slowing down lap. A serious qualifying accident at Suzuka in Japan for the penultimate race of the season severely injured Mansell's back (a spinal concussion). Trying to beat Piquet's lap time, Mansell made a mistake and hit the guardrail. As a result of Mansell's absence from the remaining two races, Piquet became champion for the third time. Piquet called his win over Mansell "a win of intelligence over stupidity". The Brazilian also added that he won because he was more consistent than his teammate, racking up points and podiums where Mansell often ran into trouble. Piquet's was a percentage driving policy which worked well in the ultra-competitive Williams-Honda, whereas Mansell was a hard charger who many felt often pushed his luck too far.


title: Nigel Mansell

text: In , for the first time in his career, Mansell was a team's first driver, having won more races in the previous two seasons than any other driver. However, Williams lost the turbo power of Honda to McLaren, and had to settle with a naturally aspirated Judd V8 engine in its first season in F1. A dismal season followed, which saw Mansell's Williams team experiment with a terribly unreliable (but extremely innovative) active suspension system (the system had worked well when introduced by the team in 1987 where it could draw on approximately 5% of the reported produced by the Honda turbo, but struggled with the Judd V8). Mansell would complete only two of the 14 races in which he appeared in 1988, both being podium finishes. One of these was a second place at the British Grand Prix at Silverstone where, overnight, the team had stopped using its active suspension (after months of Patrick Head telling Mansell and teammate Riccardo Patrese that it would take many months of work to do so), and reverted to a passive suspension set-up.


title: Nigel Mansell

text: Mansell contracted chickenpox in the summer of 1988 and after a competitive (but ill-advised) drive in the very hot conditions of the 1988 Hungarian Grand Prix the illness became worse, forcing him to miss the next two Grands Prix in Belgium where he was replaced by Martin Brundle, and Italy where he was replaced by Frenchman Jean-Louis Schlesser. By missing the Italian Grand Prix at Monza, Mansell missed the traditional welcome by the Tifosi for a newly signed Ferrari driver after he had announced he would be leaving Williams to join the Maranello-based team for .


title: Nigel Mansell

text: In preparation for the season, Mansell became the last Ferrari driver to be personally selected by Enzo Ferrari before his death in August 1988, an honour Mansell described as "one of the greatest in my entire career". Enzo Ferrari presented a 1989 Ferrari F40 as a gift to Mansell. In Italy he became known as "il leone" ("the lion") by the "tifosi" because of his fearless driving style. The season was one of change in the sport, with the banning of turbo engines by the FIA and the introduction of the electronically controlled semi-automatic transmission by Ferrari.


title: Nigel Mansell

text: A tough followed with Ferrari, in which his car suffered more reliability problems, forcing him to retire from seven races. He was paired with Alain Prost, (who was also the reigning World Champion), and who took over as the team's lead driver. Mansell recalls one incident where at the 1990 British Grand Prix, the car he drove did not handle the same as in the previous race where he had taken pole position. On confronting the mechanics, it transpired that Prost saw Mansell as having a superior car and as a result, they were swapped without telling Mansell. After retiring from the race, he announced he was retiring from the sport altogether at the end of the season. This, combined with the fact that Frenchman Prost was not only a triple World Champion and the winner of more Grands Prix than anyone in history, but also spoke fluent Italian, whereas Mansell's Italian was only conversational at best, gave Prost greater influence within the Maranello-based team. According to Prost, Mansell only attended two or three mechanical briefings throughout the season, preferring playing golf. One notable highlight of the season was a daring pass on Gerhard Berger around the daunting high speed Peraltada corner that was later renamed in his honour. Approaching the corner for the penultimate time Mansell was bobbing from side to side in Berger's mirrors. Heading into one of the quickest corners on the calendar at the time, where the Ferraris had registered forces of 4.7g during practice, Mansell launched to the outside of Berger and flashed past to take second place.


title: Nigel Mansell

text: Mansell's retirement plans were halted when Frank Williams stepped in. Mansell's return to Williams was not straightforward. He would agree to return only if a list of demands were met, including undisputed number one status over Riccardo Patrese (who had remained with the team through 1989 and 1990), guarantees of support in a wide variety of areas with each guarantee in writing, and assurances from suppliers such as Renault and Elf that they would do everything necessary to help him win. Frank Williams said the demands were 'impossible'; Mansell concluded that if that were the case he would be happy to retire. Three weeks later the impossible had happened and Mansell was a Williams driver. Williams signed Mansell on 1 October 1990 after Mansell was assured the contract stated that he would be the focus of the team, having experienced being the 'Number Two' driver at Ferrari. Mansell would be paid £4.6 million a season, a deal which made him the highest paid British sportsman at the time.


title: Nigel Mansell

text: His second stint with Williams was even better than the first. Back in the familiar 'Red 5', he won five races in , including the Spanish Grand Prix. In this race he went wheel-to-wheel with Ayrton Senna, with only centimetres to spare, at over on the main straight. Quite a different spectacle was offered following Mansell's victory in the British Grand Prix at Silverstone. Senna's car had stopped on the final lap, but, rather than leave his rival stranded out on the circuit (the two had come to blows in the pits following their first lap tangle during the 1987 Belgian Grand Prix and were hardly close friends), Mansell pulled over on his victory lap and allowed Senna to ride on the Williams sidepod back to the pits.


title: Nigel Mansell

text: The Williams team's decision to develop their new semi-automatic gearbox by racing with it at the start of the season was at the cost of points in the opening rounds of the championship. Senna was on 40 points with four straight wins to open the season by the time Mansell gained his first finish with a second in Monaco. Mansell then had the next race in Canada practically won when his Williams FW14 stopped half a lap from the finish with what was reported to be transmission failure, though it was claimed by designer Adrian Newey that Mansell had let his engine revs drop too low while he was waving to the crowd in celebration and stalled his engine. This handed Nelson Piquet his 23rd and last F1 race win. Despite a good mid-season, which included a hat-trick of victories, Senna's consistency and Mansell's retirements at key races meant that he finished second in the Championship for the third time in his career, this time behind Senna.


title: Nigel Mansell

text: Mansell started the season with five straight victories (a record not equalled until Michael Schumacher in ). At the sixth round of the season in Monaco, he took pole and dominated much of the race. However, with seven laps remaining, Mansell suffered a loose wheel nut and was forced into the pits, emerging behind Ayrton Senna's McLaren-Honda. Mansell, on fresh tyres, set a lap record almost two seconds quicker than Senna's and closed from 5.2 to 1.9 seconds in only two laps. The pair duelled around Monaco for the final four laps but Mansell could find no way past, finishing just 0.2 seconds behind the Brazilian. Mansell became the most successful British driver of all time when he won the British Grand Prix at Silverstone, as he surpassed Jackie Stewart's record of 27 wins with his 28th. Mansell was finally crowned Formula One World Champion at the age of 39 early in the season at the Hungarian Grand Prix, the 11th round of that season, where his second-place finish clinched the Drivers' Championship, securing the title in the fewest Grands Prix since the 16-race season format started. Mansell also set the then-record for the most wins in one season (9); both records stood until broken by Schumacher in 2002. He managed 14 pole positions that year, a record only broken by Sebastian Vettel in 2011 in the Brazilian Grand Prix on 26 November. He also held the record for the most races before becoming World Champion with 180 races; this record was broken by Nico Rosberg in 2016 with 206 races.


title: Nigel Mansell

text: Other Formula One records set in 1992 that he still holds are the highest percentage of pole positions in a season (88%), most wins from pole position in a season (nine) and most runner-up championship finishes before becoming World Champion (three). Mansell also holds the record for obtaining pole position and scoring the fastest lap and subsequently retiring from the race (1987 German Grand Prix, 1990 British Grand Prix, 1992 Japanese Grand Prix, and 1992 Italian Grand Prix). He is the driver having the most wins (31) without ever winning Monaco.


title: Nigel Mansell

text: He won the BBC Sports Personality of the Year award again in 1992, one of only four people to have won the award twice.


title: Nigel Mansell

text: During this season, Mansell gained a reputation for a psychological competitiveness and mind games. After the announcement by the FIA that the organization would be weighing drivers, Mansell, known for ignoring diet, starved and dehydrated himself the day prior to weigh-ins. This measure led to Mansell weighing less than his teammate Riccardo Patrese by half a kilogram.


title: Nigel Mansell

text: Despite being world champion, Mansell had a public disagreement with Williams. In his autobiography Mansell writes that this was because of a deal made at the previous Hungarian Grand Prix, which Williams reneged on, and the prospect of his former teammate Alain Prost, who had sat out the 1992 season, joining the Renault-powered team.


title: Nigel Mansell

text: Despite Mansell's early and subsequent success during 1992, his contract was due to expire at the end of the season.


title: Nigel Mansell

text: According to Patrick Head (in an interview many years later), Mansell pushed for a contract extension to be agreed early in the season. Despite this being unusual for the team, Mansell's perseverance purportedly paid off, and met with both Frank Williams and Patrick Head at the Williams Motorhome in Jerez, where they discussed and subsequently shook hands on a deal for a further two seasons (1993 and 1994). They followed this up with a written contract sent to Nigel's home address in the Isle of Man for final review and signature. According to Head, Mansell procrastinated on returning the signed agreement however, whilst winning back-to-back races over the upcoming months, finally resulting in Mansell asking for more money, which infuriated and frustrated the Williams leadership team.


title: Nigel Mansell

text: According to Mansell however, Williams had initially neglected to tell him that Prost had signed for 1993 at only the second race of the 1992 season in Mexico, a position that Mansell felt would be similar to their days together at Ferrari in 1990. To boot, Ayrton Senna had expressed a strong desire to drive for Williams and even offered to drive for them at no salary (only to later be rebuffed as Prost, whose rivalry with the Brazilian was more intense as the one he had with Mansell, had a clause written into his contract which enabled him to block Senna's effort). Williams decided that there was little sense in paying the high fees Mansell went on to demand, and told him that he needed to sign on the original deal or Senna was ready to instead. When Mansell discovered this to be untrue and a ruse, he decided to move on and called a press conference to announce his retirement. An 11th hour offer was made to him at the Italian Grand Prix, but by then the damage was done - Mansell retired from F1.


title: Nigel Mansell

text: Mansell then signed with Newman/Haas Racing to pair with Mario Andretti in the CART series, replacing Mario's son Michael who moved to F1 and McLaren. At the season opener at Surfers Paradise, Australia, he became the first "rookie" to take pole position and win his first race. A few weeks later however, he suffered a substantial crash at the Phoenix International Raceway, severely injuring his back. At the 1993 Indianapolis 500, Mansell would lead the race only to finish third after losing the lead to Emerson Fittipaldi and Arie Luyendyk after a poor restart. On his 40th birthday, however, Mansell would avenge his loss at Indianapolis to score a 200-mile race victory at New Hampshire International Speedway, perhaps his most exciting CART victory. He would go on to score five wins for the 1993 CART season, which, with more high-placed finishes, was good enough to earn him the championship. This enabled Mansell to become the only driver in history to hold both the Formula One and CART championships at the same time; when he won the 1993 CART championship he was still the reigning F1 world champion, the 1993 F1 championship not yet having been decided.


title: Nigel Mansell

text: Following this successful season in CART, Mansell received several awards, including a Gold Medal from the Royal Automobile Club and the 1994 ESPY Award for Best Driver.


title: Nigel Mansell

text: His Newman/Haas car was much less reliable the following year, 1994, and results suffered. It was during this season that Mansell "wore out his welcome" in the United States with glimpses of rude behaviour, particularly after he was knocked out of the Indianapolis 500. After the crash, he stormed out of the track hospital, and refused medical care. When reporter Dr. Jerry Punch asked Mansell if he had spoken with Dennis Vitolo, the driver who had crashed into him, Mansell replied, "You speak to him" and shoved the camera away. Subsequently, Mansell was due to sign autographs at a K-mart store (the primary sponsor of his car), but because of a lack of demand the event was cancelled. Mansell was also the catalyst for the breakdown in the relationship between himself and Mario Andretti. Mario has since remarked "I guess if Ronnie Peterson was the best team-mate I ever had, Nigel Mansell was the worst" and "I had a lot of respect for him as a driver, but not as a man".