TITLE: The straightest possible path embeddable in a path of polygons
QUESTION [6 upvotes]: 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.

REPLY [4 votes]: The natural generalization of your "straightest-possible" constraint is a curvature-constrained path.
I know this is not what you asked, but I wanted to mention that even finding a path with maximum
curvature 1 inside a convex polygon is already difficult.  This was studied by
Agarwal     et al. in a paper,
"Curvature-Constrained Shortest Paths in a Convex Polygon,"
(SIAM Journal on Computing, Volume 31, Issue 6, 2002)
Here is one of their figures, which hints at why it is difficult to find a path between initial $I$ and final $F$
positions/orientations:

Returning to the question (or questions) you did ask,
I think looking at the paper
"Polygonal path simplification with angle constraints," by Chen et al.
(Computational Geometry, Volume 32, Issue 3, November 2005, Pages 173-187), might help.
They explicitly consider the "tunnel" version of your revised question, phrased in terms of an
error tolerance $\epsilon$.  Perhaps most usefully for your purposes, they describe all the related
work in this domain, which has considered many variations.
Usually those variations start with a path $P$, and then try to find another path (a "simplification")
with certain properties: (1) The min-# problem: Find a path that remains within $\epsilon$ of $P$ but has the fewest
segments; (2) The min-$\epsilon$ problem: Given a fixed number $m$ of segments, find a path using
no more than $m$ segments that minimizes $\epsilon$.  This specific paper solves the min-# problem
with the addition of angular constraints, which seems close to your initial formulation (as clarified in the comments).
Edit. Here is a preliminary version of the "Simplification" paper: simplification.pdf.