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Does the shortest path from an initial point to a goal point, in an environment with obstacles, also gives the minimum sum of turning angles?

It doesn't seem to be obvious, but I couldn't come up with a counter example! If it is indeed true, is there a proof which states that?

Note: the shortest path means minimum sum of Euclidean distance. Additional context: I was using A* on a visibility graph to find the optimal path. Two different cost functions were used. 1. Just the Euclidean distance and 2. Euclidean distance + theta, turning angle, multiplied by some constant.

Shortest path, which also gives the minimum sum of turning angles

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    $\begingroup$ I don't think so. I was able to generate a setting in which the turning angles through the shortest path was 180 degrees, but through the longest path was less. I'll try to make an image and show you $\endgroup$ – Michael Stachowsky Dec 9 '16 at 19:06
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The following image, constructed by the original poster, demonstrates that the claim is false. The shortest path (green) has a larger total turning angle than the longer path (red).

enter image description here

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  • $\begingroup$ Please see this link imgur.com/a/HXR27 The green line appears to be the shortest path and it also gives the minimum sum of turning angles. Am I missing something? Also, the shortest path is aways through the vertices of the obstacle. $\endgroup$ – Saurav Dec 9 '16 at 19:20
  • $\begingroup$ I made the image imprecisely. If you move the square down so that it touches the rectangle, then I believe my turning angles are still correct. I know that the shortest path should be through the vertices, but I lack the paint skills to do it :-P $\endgroup$ – Michael Stachowsky Dec 9 '16 at 19:23
  • $\begingroup$ Great! I think that works. I shall double check and get back. $\endgroup$ – Saurav Dec 9 '16 at 19:27
  • $\begingroup$ I've edited the figure to demonstrate $\endgroup$ – Michael Stachowsky Dec 9 '16 at 19:27
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    $\begingroup$ You can use this imgur.com/a/2Zdx7 image for a clearer answer and I shall accept it. $\endgroup$ – Saurav Dec 9 '16 at 19:51

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