In recreational mathematics, polytans are polygons formed by edge-connecting isosceles right triangles. Order-n polytans are those constructed from n such triangles. My question is this:
Given a set of n angles and an N x N grid, how do I draw a graph using a set of angles? I am trying to find if the angles form the perimeter of a polytan (i.e., given a starting point on the plane, does progressively iterating through the angles get me back to the point of origin?). For example, given these three sets: {{270, 315, 315}, {315, 270, 315}, {315, 315, 270}}, the output would show two isosceles right triangles: {270, 315, 315} and {315, 315, 270}, and one non-polygon: {315, 270, 315}, as seen in the image below. In the image, the initial reference line is {{0,0}->{0,1} (on the y-axis), and the point of origin is {0,1}.
I need the constructions to be graphs so that I can check whether two polytans of the same order are identical, but just in different orientations (like the two triangles in the image). Vertex count, vertex degree, edge-connectedness, etc., would also be useful.
It appears that you are also constraining the side length to be $1$ if the side is horizontal or vertical and $\sqrt 2$ if it is diagonal. If so, you can just keep track of the lattice points and see if the polygon closes. For the first, you have points $(0,0), (0,1), (1,1), (0,0)$. You can note that the last point is the same as the first to see that the polygon closes. To check if two are the same, you can put the two $270$ degree turns at the origin, rotate one polygon around (there are only four choices) and see if the other vertices match.