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Prove algebraically that the stereographic projection of a circle

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Prove algebraically that the stereographic projection of a circle (C) lying in a sphere (S) is either a circle or a straight line.

Definition:
Stereographic projection is done from the north pole of the sphere onto a plane tangent to the sphere at its south pole.

Hint:
A circle on the sphere is contained in a plane (P), so that C = P/S. A plane can be defined by equation P = {(x, y, z): a x + b y + c z = d, where a, b, c, and d are constants and (x, y, z) are Cartesian coordinates in R^3

The hint suggests an algebraic proof (those interested in a geometric proof may find one by Yana Zilberberg Mohany at http://math.ucsd.edu/~mohanty/nopix1.html#Eves2).

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Solution Summary

The expert proves algebraically that the stereographic projection of a circle.

Solution Preview

The following proof consists of 4 parts:
1. We set up a system of coordinates and notations
2. We find a relation between the coordinates of a point on a sphere and the coordinates of its projection
3. We prove that if the stereographic projection of a curve is a circle, this curve is a circle too
4. We prove that the stereographic projection of a circle is a circle or a straight line

1. Notations
Let us retain (x, y, z) as the coordinates of a point on a circle, let (u, v, 0) be the coordinates of the projection of this point, and let D be the diameter of the sphere, and let its north pole have coordinates (0, 0, D). A circle on the XY plane can be parameterized as u + iv = (s + it) + re^iφ, with φ running from 0 to 2π while s = const and t = const.

2. Projectee-Projection relation
Let us start from a simple case of y=0 and respectively v=0. It is good to make a drawing at this point, and to compare two triangles: ...

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