A professor insists it's possible to map circles on the surface of a sphere to circles in the plane so that certain criteria are satisfied. I'm not so sure.
1) Start with a unit sphere centered at the origin. 2) Consider a set of right circular cones with the apex at the origin and with various apex angles. 3) The intersections of the cones with the unit sphere are circles some of which intersect. 4) Cones can be characterized by a unit vector representing an axis corresponding to the center of the circle of intersection, and an apex angle corresponding to the radius of that circle.
Is it possible to map the circles on the sphere to the plane so that they satisfy the following criteria: A) Circles are mapped from the surface of the sphere to CIRCLES on the plane. Prof insists on his circles. B) Circles which intersect on the sphere intersect in the plane. C) Whatever 2D coordinates are used in the plane, they or some function of them has some bearing on their position and size on the sphere. It isn't necessary for the X coordinate to increase with longitude or latitude, it could vary with some function of the apex angle. Size parameters can be swapped out with position parameters if need be. D) Any mapping procedure from the sphere to the plane needs to be uniform, if X is the cosine of the azimuthal angle and Y the sine of the axial angle for one cone, that's the rule for all the others. No variable rescaling or such.
Current understanding and things I've tried:
Without loss of generality we can focus on those cones with a positive Z component. The axis of one of these cones is (ax,ay,az). Since it's a unit vector, the sum of their squares is 1. We only care about positive values of az. az is therefor determined once you know ax and ay, so we have 2 degrees of freedom for the axis. Cones can also be differentiated by their apex angles corresponding to the radius of the intersections. This is a third degree of freedom.
Each cone can then be specified by (ax,ay, alpha) with ax and ay running from -1 to 1 and alpha running from 0 to pi/2.
Attempts: 1) Suppress the Z component, you get coordinates for the plane. Satisfies all criteria but the preservation of shape.
2) Some conformal mappings. Distorted figures as might be expected.
Guesses: 1)The curvature of the sphere implies necessary distortions when mapped to the plane, so no such mapping is possible. I'm not sure how to prove this.
2) A circle in the plane has 3 degrees of freedom, x-coordinate of center, y-coordinate of center, and radius. As stated before, the cones have 3 degrees of freedom as well. Does a mapping with the center coordinates and radius being multi-variable functions of the cone parameters exist that satisfies the above mentioned rules exist?
This automatically maps circles to circles and intersections to intersections. How well do position parameters of the circle correspond to position parameters of the cones? Are there problematic corner cases?