Note: I originally posed this question to Mathematics, but it was recommended that I try here as well.

Edit: Since posting, I've since realised that my solution attempt is incorrect as the resulting traversal is not guaranteed to form a Eulerian circuit. Feel free to skip this section and the My Solution Attempt Issues section.

In section 3.4 of Accurate and Efficient Drawing Method for Laser Projection the paper describes using Hierholzer’s algorithm for finding an optimal Eulerian circuit with the amendment that during traversal of each vertex you select the unvisited edge along the angle closest to a straight line. One issue that occurs to me with this approach is that it is not clear to me that this always results in the absolute optimal circuit, only one that is probably more optimal than a naive construction without this added amendment.

Note: I originally posed this question to Mathematics, but it was recommended that I try here as well.

Edit: Since posting, I've since realised that my solution attempt is incorrect as the resulting traversal is not guaranteed to form an Eulerian circuit. Feel free to skip this section and the My Solution Attempt Issues section.

In section 3.4 of Abderyim, Halabi, Fujimoto, and Chiba - Accurate and Efficient Drawing Method for Laser Projection the paper describes using Hierholzer’s algorithm for finding an optimal Eulerian circuit with the amendment that during traversal of each vertex you select the unvisited edge along the angle closest to a straight line. One issue that occurs to me with this approach is that it is not clear to me that this always results in the absolute optimal circuit, only one that is probably nearer to optimal than a naïve construction without this added amendment.

My attempt at solving this was to first simplify the problem by looking at each vertex individually. We know that each vertex must have an even degree, and thus for each vertex there must be an optimal set of incoming/outgoing edge pairs (where each edge is used once) that results in a minimum accumulative angular distance. By minimum accumulative angular distance, I'm referring to the sum of the difference between the result of the difference between the angle of each incoming/outgoing edge pair and a straight line. For example, given the following vertex A and its neighbours B, C, D and E:

1. Is there an existing solution to the original Problem stated above? If so, is there somewhere I might read further on this?
2. If not, does my attempted solution sound like a reasonable approach? If so, do you have an idea of how I might represent the sub-graph for determining the set of edge pairs resulting in the minimum accumulative angular distance for each vertex?
3. If not, can you recommend an approach I might be able to take to make progress on solving the previously mentioned Problem?

Edit: Since posting, I've since realised that my solution attempt is incorrect as the resulting traversal is not guaranteed to form a Eulerian circuit. Feel free to skip this section and the My Solution Attempt Issues section.

My attempt at solving this was to first simplify the problem by looking at each vertex individually. We know that each vertex must have an even degree, and thus for each vertex there must be an optimal set of incoming/outgoing edge pairs (where each edge is used once) that results in a minimum accumulative angular distance. By minimum accumulative angular distance, I'm referring to the sum of the difference between the result of the difference between the angle of each incoming/outgoing edge pair and a straight line. For example, given the following vertex A and its neighbours B, C, D and E:

1. Is there an existing solution to the original Problem stated above? If so, is there somewhere I might read further on this?
2. If not, does my attempted solution sound like a reasonable approach? If so, do you have an idea of how I might represent the sub-graph for determining the set of edge pairs resulting in the minimum accumulative angular distance for each vertex? Approach determined to be invalid.
3. If not, can you recommend an approach I might be able to take to make progress on solving the previously mentioned Problem?