How to Project Spherical Conics into the Plane
Yoichi Maeda maeda@keyaki.cc.utokai.ac.jp
Mathematics/Sciencs Tokai University Japan
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 Pascalfs theorem (mystic hexagon) is essential.
The Pascalfs 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.
