chap5 - Chapter 5 Hints and(partial Solutions Exercise 5.1...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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...
View Full Document

Page1 / 3

chap5 - Chapter 5 Hints and(partial Solutions Exercise 5.1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online