Reflections on a sphere

Solution 1:

It seems that “Computing a Point of Reflection on a Sphere” was done by David Eberly and it needs to compute roots of a quartic polynomial.

Solution 2:

Find the bisector of $\angle{CSP}$, then point X is the intersection of this bisector with the circle/sphere.

Note: Above simple answer only works when point C and P are symmetric to center S.

In general, the vector $\vec{SX}$ bisects $\angle{CXP}$ (not $\angle{CSP}$). So, to find point X, represented as $X=S+(rcost,rsint)$, we have to solve for the following equation:

$\frac{\overrightarrow{XC}}{|\overrightarrow{XC}|}\cdot \vec{n} = \frac{\overrightarrow{XP}}{|\overrightarrow{XP}|}\cdot \vec{n} $

, where $\vec{n}=\frac{\overrightarrow{SX}}{|\overrightarrow{SX}|}$

It is not easy to solve for point X from this equation analytically. But you can do that numerically.