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Let $S$ be a surface. What conditions can we place on $S$ so that for any two points $a$, $b$ on $S$, the minimal geodesic from $a$ to $b$ is the shortest path on $S$ from $a$ to $b$ ?

Why I am asking

Balasubramanian, Polimeni & Schwartz (2009) "Exact Geodesics and Shortest Paths on Polyhedral Surfaces" makes the following claim: Let $S$ be the boundary of a convex polyhedron in Euclidean $\mathbb{R}^3$. For any vertices $a$, $b$ of the polyhedron that underpins $S$, the minimal geodesic between $a$ and $b$ is the shortest path between $a$ and $b$. [Section IV, para 1 with abridging of definitions from Section III, para 1].

The claim is supported by a reference to Mitchell, Mount & Papadimitriou (1987) "The Discrete Geodesic Problem", but I can't find the corresponding text in the article.

Edit

I suspect that Balabsubramanian et al. were actually thinking of the following lemma from Hershberger & Suri (1998) "Practical methods for approximating shortest paths on a convex polytope in $\mathbb{R}^3$" that is in turn attributed to Sharir & Schorr (1986) "On Shortest Paths in Polyhedral Spaces":

Lemma 3.7 (Unfolding rule). Let $P$ be a polyhedron in 3-space, and let $p$, $q$ be two points on the surface of $P$. Let $f_p, f_1, f_2, \dots f_k, f_q$ be the sequence of faces of $P$ crossed by the shortest path $\pi(p,q)$. If we unfold the faces $f_1, \dots f_k, f_q$ in sequence until they all become coplanar with $f_p$ then the shortest path $\pi(p,q)$ unfolds to a straight line.

(Hershberger & Suri's Lemma 3.7 is (I think) derived from Sharir & Schorr's Lemma 3.1 that [paraphrasing] amounts to "On a convex polyhedron in 3-dimensions, when a shortest path passes through an edge on a surface, then the angle of entry and exit are the same.")

What I have tried

After reviewing this question and this question, this discussion, and the example below, I can see that convexity of $S$ is necessary. Is it sufficient?

Figure 5 from Balasubramanian, Polimeni & Schwartz (2009) "Exact Geodesics and Shortest Paths on Polyhedral Surfaces"

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Even for a $\mathbb R^2$ simple cone surface there is a infinite set of geodesics satisfying zero geodesic curvature by differential geometric condition for smooth surfaces.

For convex polyhedral surfaces incident and emerging straight rays should make equal angles at the polyhedral edge or fold before flattening it out on development. The equality should be respected even if a choice of edge split is made.

A very rough hand trace for geodesic tracking on a dodecahedron:

12hedrGeod

All possible minimum geodesic distances in the neighborhood should be computed and sorted by increasing geodesic lengths. The one nearest to LOS in $\mathbb R^3$ should be selected as that is the global minimum.

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  • $\begingroup$ Thank you @Narasimham, I think I'm starting to see. I needed to revisit my understanding of 'geodesic' and 'minimal geodesic' - the answer at math.stackexchange.com/questions/1432985/… helped. Thanks again. $\endgroup$ – Patrick Hew Jul 13 '18 at 12:31
  • $\begingroup$ For geodesic tracking on polyhedra ( at least Platnic ones) flatly laid out cardboard models give I believe better starting insight than on a computer. $\endgroup$ – Narasimham Jul 14 '18 at 22:16
  • $\begingroup$ @Narasinham It's likely that I have missed something as there are two things that I'm not understanding: 1) What is it about surfaces of convex polyhedra that guarantees that a minimum geodesic is a shortest path? and 2) How can there be minimum geodesics of different length? Thanks again. $\endgroup$ – Patrick Hew Jul 15 '18 at 6:33
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    $\begingroup$ 1) In case of Platonic regular convex polyhedrals derived from a sphere the shortest path is unique and can be found by projection of great circle onto faces from the sphere . 2) Geodesics with vanishing $\kappa_g$ need not necessarily be of minimum length.Non zero minimum perpendicular distances to geodesics can be found. At present I cannot define generalization criteria. $\endgroup$ – Narasimham Jul 15 '18 at 7:02

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