Finding a curve that lies on the arbitrary sphere given $\alpha(0)$ and $\alpha'(0)$.

Question

Find a curve $\alpha : (−ε,ε) → \Sigma$ on the sphere which has $\alpha(0) = (1,0,0)$ and $\alpha′(0) = (0, 5, 6)$.

I'm unsure how to approach this. I know the parametarization of a sphere, and obviously the bases of the tangent space, but I don't know if this will help me?

Answer 1

In general, for two orthogonal unit vectors $p, Z \in \mathbb{R}^3$, the map $$ \gamma: s \mapsto p\cos s + Z\sin s $$ parametrizes a unit speed curve on the unit circle with $\gamma(0) = p$ and $\dot\gamma(0) = Z.$ Can you adapt this to your case?

Answer 2

Let us place here the conditions for future referencing :

$$V(0)=(1,0,0)^T \ (a) \ \ \ \text{and} \ \ \ V'(0)=(0,5,6)^T \ (b) \tag{1}$$

There is an obvious candidate

• for the trajectory : a great circle of the sphere, obtained here by a rotation of the equator by an angle $a$ around the $x$ axis.

• in a second step, its cinematics : how must this trajectory be parametrized ? It suffices clearly, thinking to a uniform angular speed) to take $kt$ instead of $t$ ; all this gives :

$$V(t)=\begin{pmatrix}1&0& \ \ 0\\ 0&\cos(a)&-\sin(a)\\ 0&\sin(a)&\ \ \cos(a)\end{pmatrix}\begin{pmatrix}\cos(kt)\\ \sin(kt)\\ 0\end{pmatrix}=\begin{pmatrix}\cos(kt)\\ \cos(a)\sin(kt)\\ \sin(a)\sin(kt)\end{pmatrix} \tag{2}$$

We have thus to determine constants $a$ and $k$ in order that conditions (1) are fulfilled.

The first condition is clearly fullfilled.

It remains to comply to 2nd condition (b).

As :

$$V'(t)=(-k \sin(t), k \cos(a) \cos(kt), k \sin(a)\cos(kt))^T.$$

Taking $t=0$ yields, by identification with the values of the 2nd condition $(0,5,6)^T$ :

$$k \cos(a)=5 \ \ (a) \ \ \text{and} \ \ k \sin(a)=6 \ \ (b). \tag{3}$$

Taking the quotient of (b) by (a) gives $\tan(a)=6/5$ : Thus $a$ can be taken as $a_0=\mathrm{atan}(6/5)$.

Taking the square of (a) and (b) and adding them, one gets $k_0= \sqrt{61}\approx 7.8$ (taking $k=-\sqrt{61}$ would make going the other way).

Thus the solution is given by (2) with $k=k_0$ and $a=a_0$.

Last thing : The interval of variation for parameter $t$ ? You must not pass twice on the same point, thus, take $t_{lim}$ (your $\varepsilon$) such that $\sqrt{61}t_{lim}<\pi \ \iff \ t_{lim}<\pi/\sqrt{61}$, for example $t_{lim}=0.4$.