Home_2 - Homework II Stephen Taylor September 27, 2005 Page...

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Unformatted text preview: Homework II Stephen Taylor September 27, 2005 Page 10 : 4. Let be a circle lying in S . Then there is a unique plane P in R 3 such that P S = . Recall from analytic geometry that P = { ( x 1 ,x 2 ,x 3 ) : x 1 1 + x 2 2 + x 3 3 = l } where ( 1 , 2 , 3 ) is a vector orthogonal to P and l is some real number. It can be assumed that 2 1 + 2 2 + 2 3 = 1. Use this information to show that if contains the point N then its stereographic projection on C is a straight line. Otherwise, projects onto a circle in C . Proof: If contains the north pole, without loss of generality we may assume that is entirely contained in the x 1 x 2-plane. 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 1 x 1 + 2 x 2 + 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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Home_2 - Homework II Stephen Taylor September 27, 2005 Page...

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