I'm studying a problem involving the sets of discrete curves that can be embedded in a non-trivial polygon, from a source to a target point, as shown below.

Initially my interest was limited to the set containing discrete curves with the possible smallest segment count and within this set the discrete curve with the shortest length. I've been able to find this discrete curve (not shown in the above graphic).

Now I'm considering the problem of the straightest-possible discrete curve that can be embedded in the polygon, and I suspect, or rather I'm hoping, that both curves, the one with the smallest segment count and the straightest one are identical. I'd appreciate help showing that is or isn't the case.

Note: I'm aware of the concept of geodesics as shortest possible paths. However, the curve I'm studying isn't the shortest path but the path with the smallest segment count. In the example provided above, the shortest path would cut around corners as it seeks to minimize the path length. Interestingly, this curving of the path increases its segment count.

**Clarification of the question** (*courtesy of arex*)

Suppose a simple polygon is given, along with two points, s and t, within the polygon. Is there a polygonal path from s to t that simultaneously minimizes the number of segments and amount of turning. Simple means homeomorphic to a disk; a polygonal path is a piecewise-linear curve; and turning is measured by the sum of the absolute values of the turning angles.

I'm making the following restatement of the initial problem, now solved, because it describes the sort of path that I'm investigating. Also the description may trigger insight into the question of bounds on straightness:

*The polygon can be considered as a tunnel in which there is no line of sight between two points, which we can imagine to be the source and destination of a light beam. We seek to install the smallest number of light beam relays required to forward a signal as quickly as possible though the tunnel from source to the destination. The locations of the relays are the equivalent to the intermediate vertices of the shortest discrete curve with the minimum possible segment count embeddable in a non-trivial/non-convex polygon.*

35more comments