25 April 2014
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Stereographic projection of plane algebraic curves onto the sphere

Author(s): S. Welke

Abstract:
raphic projection of plane algebraic curves onto the sphere S.

Welke, 7n der Spwelke@aol com

1 Introduction:
Stereographic projection is a conformal map from the x — y— plane to the unit sphere S* C R^.

The geometric definition is: Definition 1 Let P = (x, y, 0) be a point in the x — y— plane, let N = (0, 0, 1) be the North Pole of the unit sphere in R^, and let lp be the unique straight line lp through P and N .

The intersection with the unit sphere is a point P' ^ N .

The correspondence p : P \ — > P* establishes a one-to-one map p : R^ — > S*\{N} called Stereographic projection. We frequently identify (x,y) R% with (x,y, 0) G R^.

Note that there is no point P in the entire plane with p(P) = N. Given a point P, the line lp is the set {(0,0,1) +t(x,y,-l) \t R} = {(tx,ty,t — 1) \t G R}.

Because P' belongs to the unit sphere, its coor- dinates satisfy the following quadratic equation: (tx)* -f (ty)* + (1 — t)* = 1 with ...

Pages: 8
Size: 791 kb
Paper DOI: 10.2495/IMS970631

 

 

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Innovation in Mathematics

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