EDB — 1P7

Exercises

1. [1P7] Note:Nice formula taken from [ 56 ] .

   Let \(S=S(0,1)⊆ ℝ^ n\) be the unit sphere \(S=\{ x: |x|=1\} \). Let \(v,w∈ S\) with \(v≠ w\) and \(v≠ -w\); let \(T = \arccos ( v⋅ w )\) so that \(T∈(0,𝜋)\); then the geodesic (that is, the arc-parameterized minimal length curve) \(𝛾(t):[0,T]→ S\) connecting \(v\) to \(w\) inside \(S\) is

   \[ 𝛾(t)=\frac{\sin \big(T-t\big) }{\sin (T)} v + \frac{\sin \big(t\big) }{\sin (T)} w\quad , \]

   and its length is \(T\).
   (You may assume that, when \(v⋅ w=0\) that is \(T=𝜋/2\), then the geodesic is \(𝛾(t) = v \cos (t) + w \sin (t)\)). 

   Solution 1

   [1P8]↺↻