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Unformatted text preview: Homework II Stephen Taylor September 27, 2005 Page 10 : 4. Let Λ be a circle lying in S . Then there is a unique plane P in R 3 such that P ∩ S = Λ. Recall from analytic geometry that P = { ( x 1 ,x 2 ,x 3 ) : x 1 β 1 + x 2 β 2 + x 3 β 3 = l } where ( β 1 ,β 2 ,β 3 ) is a vector orthogonal to P and l is some real number. It can be assumed that β 2 1 + β 2 2 + β 2 3 = 1. Use this information to show that if Λ contains the point N then its stereographic projection on C is a straight line. Otherwise, Λ projects onto a circle in C . Proof: If Λ contains the north pole, without loss of generality we may assume that Λ is entirely contained in the x 1 x 2plane. Since x 2 = 0, it is clear that the stereographic projection of a circle containing the north pole is indeed a line which may be interpreted as a circle of infinite radius. To show the projection preserves circles, we first note that a circle on a unit sphere is given by the intersection of the set { ( x 1 ,x 2 ,x 3 )  x 2 1 + x 2 2 + x 2 3 = 1 } and a prescribed plane in R 3 β 1 x 1 + β 2 x 2 + β 3 x 3 = l Let N = (0 , , 1) be the north pole, Z = ( x 1 ,x 2 ,x 3 ) be a point on the unit sphere, and Z = ( x 1 ,x...
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This note was uploaded on 10/20/2009 for the course MATH 814 taught by Professor Cong during the Three '09 term at University of Adelaide.
 Three '09
 Cong
 Geometry

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