I'm studying a problem involving the sets of discrete curves that can be embedded in a non-trivial path or sequence of polygonspolygon, from a source to a target point, as shown below.
Now I'm considering the problem of the straightest-possible discrete curve through the path of polygons from the source to that can be embedded in the target pointpolygon, 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, what the curve I'm looking for studying isn't just the shortest path but the straightest-possible path with the smallest segment count. In this casethe example provided above, the geodesic shortest path would cut around corners as it seeks to minimize the path distancelength. The Interestingly, this curving of the path would increase the increases its segment countand probably .
Clarification of the question (not entirely sure courtesy of thisarex)increase
Suppose a simple polygon is given, along with two points, s and t, within the total polygon. Is there a polygonal path from s to t that simultaneously minimizes the number of segments and amount of turningangle . Simple means homeomorphic to a disk; a polygonal path is a piecewise-linear curve; and turning is measured by the sum of the pathabsolute values of the turning angles.
I'm making the following restatement of the initial problemon , now solved, because it describes the off chance sort of path that it will provide additional I'm investigating. Also the description may trigger insight into the the new problemquestion of bounds on straightness:
The polygon path can be considered as a tunnel in which there is no line of sight between the source and target 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 polygonpath.