How to Project Spherical Conics into the Plane

Abstract

We will introduce a method how to draw the orthogonal projected images of spherical conics (great circle, small circle, and conic on the sphere) in the Euclidean plane. The orthogonal projected images of spherical circles are conics in the plane. On the other hand, the orthogonal projected images of spherical conics are not conic but quartic in general. To construct these figures with basic drawing tools, stereographic projection plays an important role. The stereographic projection maps circles on the sphere to circles in the plane. Using this property, we can construct the orthogonal projected images of spherical circles in the plane. For example, the procedure to construct the orthogonal projected image of spherical small circle passing through three (orthogonal projected) points is as follows: 1.Create corresponding three stereographic projected points. 2.Draw the circle passing through these three points. 3.Take two points on the circle and create corresponding orthogonal projected points. 4.Draw the conic passing through five orthogonal projected points. As for spherical conics, the famous Pascalfs theorem (mystic hexagon) is essential. The Pascalfs theorem is also valid for spherical geometry. Applying this theorem, we can also construct the orthogonal projected images of spherical conic in the plane. The procedure to construct the orthogonal projected images of spherical conic passing through five (orthogonal projected) points is as follows:

1.Create corresponding five stereographic projected points.
2.Draw circles (stereographic projected images of great circles) passing through two of these five points.
3.Take the sixth point on the stereographic projected image of the spherical conic.
4.Construct the locus of the corresponding orthogonal projected point of the sixth point.

We will realize these constructions along with the dynamic geometry software Cabri II Plus. These constructions are very instructive to understand the importance of stereographic projection and also the great fun of conic.