I need to find a parametrisation in terms of $t$ for a half circle on a sphere with radius $R$. The circle goes from $(R,0,0)$ to $(-R,0,0)$ and is going through the point $(0, R/\sqrt{2},R/\sqrt{2})$.
I was thinking about putting the sphere in terms of spherical coordinates, but can't seem to figure out how to get a good parametric equation from there: Something of the form: $$\mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j}+z(t)\mathbf{k}, \, a\leq t \leq b.$$ For some $a$ and $b$.
Can somebody give me a push in the right direction?
Thanks in advance.
Find the equation of the plane determined by the three points. Then use spherical coordinates.