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Unformatted text preview: Chapter 5: Hints and (partial) Solutions Exercise 5.1 Start with distinct points p and q on the n-sphere S n . So p and q are unit vectors in R n +1 . There are three cases (well... only two really, the first is a special case of the third): (a) If p q (i.e., p q = 0) then the point ( ) = (cos ) p + (sin ) q for 0 / 2 , (1) belongs to S n (why?, you can check that k ( ) k = 1 by computing ( ) ( ) and remembering that p q = 0) with (0) = p and ( / 2) = q . Note that will trace out a great circle on S n if is allowed to range from 0 to 2 . (b) If q =- p , that is, if p and q are both on a line through the origin then intuitively any great circle containing either p or q should contain the other (sketch a picture and youll see). Lets check to make sure intuition is correct here: first choose a point e S n orthogonal to p (hence, orthogonal to q , ...could you construct such a point? the next case will give you a clue on how to do this by first selecting any point r S n not on the line through p and q ). Now proceed as in part (a) above, except with e in place of q : put ( ) = (cos ) p + (sin ) e and note that ( ) S n for each R (youve already checked this in part (a)) and that (0) = p and ( ) =- p = q . (c) If p and q are not on the same line then try this approach: replace q with another point e S n such that e p and e is in the same plane as the points p , q and . Then, again as in (a) put ( ) = (cos ) p + (sin ) e . The great circle traced out by...
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- Spring '09