This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 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 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...
View Full
Document
 Three '09
 Cong
 Geometry

Click to edit the document details