Tangential angle

In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the -axis. (Note that some authors define the angle as the deviation from the direction of the curve at some fixed starting point. This is equivalent to the definition given here by the addition of a constant to the angle or by rotating the curve.)
Equations.
If a curve is given parametrically by , then the tangential angle at is defined (up to a multiple of ) by
Here, the prime symbol denotes the derivative with respect to . Thus, the tangential angle specifies the direction of the velocity vector , while the speed specifies its magnitude. The vector
is called the unit tangent vector, so an equivalent definition is that the tangential angle at is the angle such that is the unit tangent vector at .
If the curve is parametrized by arc length , so , then the definition simplifies to
In this case, the curvature is given by , where is taken to be positive if the curve bends to the left and negative if the curve bends to the right.
If the curve is given by , then we may take as the parametrization, and we may assume is between and . This produces the explicit expression
Polar tangential angle.
In polar coordinates, the polar tangential angle is defined as the angle between the tangent line to the curve at the given point and ray from the origin to the point. If denotes the polar tangential angle, then , where is as above and is, as usual, the polar angle.
If the curve is defined in polar coordinates by , then the polar tangential angle at is defined (up to a multiple of ) by
If the curve is parametrized by arc length as , , so , then the definition becomes
The logarithmic spiral can be defined a curve whose polar tangential angle is constant.