[alogo] Stereographic projection

This is a map of the points of the circle onto the points of a line constructed by means of lines and intersection points. Usually one takes a point A on the circle and projects radially its points on the tangent to the diametral point O of A.
Below is illustrated such a case. The circle has its center on the y-axis and is tangent to the x-axis. The stereographic projection corresponds to every point B different from A the intersection point C of AB with the x-axis. Denoting by t the ordinate of C, we see easily that t = (2rx)/(2r-y), r denoting the radius of the circle.
For the inverse calculation, of (x,y) in terms of t, use is made of the equation of the circle: x2+(y-r)2=r2 <==> x2-2ry+y2 = 0. The inverse of the stereographic projection is given by:

[0_0] [0_1]
[1_0] [1_1]

Considering the projectification of the euclidean plane and the standard projective coordinate system [x,y,z], the homogeneous coordinates of the inverse of the stereographic projection are given by (replacing in the above formulas x with x/z and y with y/z:


Using again the projectification of the line (x-axis) and replacing t with u/v, the equations become (after multiplication of the tripple by v2):


The use of projective coordinates allows the extension of the stereographic projection to the excepted point A. This corresponds to the point at infinity of the projective line, which is represented by coordinate pairs of the form (u,0).
The stereographic projection and its inverse is the prototype of the so called good parametrizations of conics (through the points of projective lines), and their inverses, which are rational parametrizations of conics, discussed in GoodParametrization.html .

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