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Stereographic Projection Of Plane Algebraic Curves Onto The Sphere


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S. Welke


Stereographic 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