Formulas to find the end point of an arc given the start point and midpoint on the unit sphere?

Question

So the radius and the center of the arc are $1$ and $(0, 0, 0)$ respectively because the points and the rac is on the unit sphere $S^2$. Given the start and the midpoint of the arc, is there any formula to deduce the endpoint? I know that normalizing the midpoint on the straight line between the start and end returns the midpoint of the arc. Thanks in advance.

Answer 1

Let $O$ be the center of the sphere, $S$ and $E$ be the endpoints of the arc, and $M$ be the midpoint. These all lie within the plane though $O, S, M$, so this becomes a two-dimensional problem in a circle instead of a three dimensional problem in a sphere.

Let $N$ be the midpoint of the chord $\overline{SE}$, which is also the intersection of $\overline{SE}$ and $\overline{OM}$. Because $N$ is the midpoint, the chord and radius intersect in a right angle. Let $\theta$ be the angle $\angle SOM$ and $h = ON$, the distance from the center to $N$. Then $h = \cos \theta$.

If we make $O$ the origin, and consider all the points to be vectors $$\mathbf s = S\\\mathbf m = M\\\mathbf n = N\\\mathbf e = E$$ Then $$\mathbf n = \frac{\mathbf s + \mathbf e}2\\\mathbf n = (\cos\theta)\mathbf m\\\cos\theta = \mathbf s \cdot \mathbf m$$ so $$\mathbf e = 2(\mathbf s\cdot \mathbf m)\mathbf m - \mathbf s$$ Where $\mathbf s\cdot\mathbf m = s_1m_1 + s_2m_2 + s_3m_3$ is the dot product.