EDB — 1P7

Exercises

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

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

   $$𝛾(t)=\frac{\sin (T-t)}{\sin (T)}v+\frac{\sin (t)}{\sin (T)}w,$$

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

   Solution 1

   [1P8]↺↻