Polyconics. I. Polyellipses and optimization
A. Introduction. 1. Ellipses and hyperbolas may be defined as plane curves consisting of all points the sum, or the difference, of whose distances from two fixed points F, and Ft, the foci, is constant. This generalizes at once to a provisional definition of a polyconic with n foci Fi , ■ ■ , Fn in the plane: a plane locus consisting of all points the sum of whose signed distances to Fu ■ ■■, Fn is constant, the signatures being +1 or —1. If all n signatures are +1 the polyconic is a polyellipse or, more specifically, an n-ellipse. Thus a circle in this terminology is a 1 -ellipse and an ordinary ellipse becomes a 2-ellipse. Why bother with polyconics? There is the possible intrinsic interest, novelty, and educational value. Then, certain properties of conics themselves appear to be easier and more natural in this generalized context; for instance, the radius of curvature of a polyellipse is easier to get from first principles than is that of an ellipse. Also, the numerical and graphical treatment of polyconics introduces early, simply and naturally certain important techniques of numerical analysis, such as e.g. the variable step-length procedures. However, the most important reasons for working with polyconics appear to us to be two: a) polyconics may help to revive interest in the neglected subject of geometry, b) polyconics, and especially polyellipses, arise naturally in the treatment of an important class of optimization problems.