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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 nsphere 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 α ( θ ) = (cos θ ) p + (sin θ ) q for 0 ≤ θ ≤ π/ 2 , (1) belongs to S n (why?, you can check that k α ( θ ) k = 1 by computing α ( θ ) · α ( θ ) and remembering that p · q = 0) with α (0) = p and α ( π/ 2) = q . Note that α will trace out a great circle on S n if θ is allowed to range from 0 to 2 π . (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 you’ll see). Let’s 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 α ( θ ) = (cos θ ) p + (sin θ ) e and note that α ( θ ) ∈ S n for each θ ∈ R (you’ve already checked this in part (a)) and that α (0) = p and α ( π ) = 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 α ( θ ) = (cos θ ) p + (sin θ ) e . The great circle traced out by...
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 Spring '09
 Vector Space, Dot Product, Cos, ... ..., Rn+1

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