The following is an excerpt from a 2005 paper on geodesics on triangular meshes, taken from section 3.5
In this case $p$ is a point on some arbitrary face in a mesh, $p'$ is a point on one of the 3 edges in the triangle and $D(p')$ is the geodesic distance to that point.
I am very confused as to how this minimization is achieved. The paper doesn't go into detail, so I assume it's trivial, but I am not seeing the solution of the top of my head.
Given an arbitrary interval (segment) the shortest point to it is either the orthogonal projection of the point $p$ onto the segment or one of the end points of the segment (depending on where $p$ is located with respect to the segment).
However, this is not necessarily the point that minimizes the expression $||p - p'|| + D(p')$ or in other words, $p'$ is not, in general, the orthogonal projection of $p$ onto the segment.
The only algorithm I have is to naively check epsilon offsets along the segment and picking the shortest one. There has to be a better way than that.
I have drawn a diagram on Geogebra to show the problem:
$\overline{BA}$ is the segment, $C$ is the point we are looking to minimize the distance from. $D$ is the frontier point, which is the orthogonal projection of the Geodesic source (the source point of all geodesics in the mesh), $E$ is the orthogonal projection of $C$ onto $\overline{BA}$. Clearly the optimal point is somewhere between $E$ and $D$ but I don't know how to find it.