I want to try and identify a geometric structure I thought up while doing some weird stuff with making things walk on the surface of a 3D model and trying to incorporate backface culling into the surface geometry itself. See, in computer graphics each side of a polygon or triangle are considered separate entities and so I specifically desired to capture this within the geometry I constructed. Below, I will describe the different properties I know of to see if anyone can identify it as anything previously studied.
The Structure Itself
Let us define a special triangular mesh. Let's just call it a "half-triangle mesh" since I don't know what else to call it. In this context we define a half-triangle mesh to be a collection of half-triangles and we define a half-triangle to be an ordered triplet of points. These points technically form the vertices of Euclidean triangle in space.
Now comes the somewhat weird part. We can say that a half-triangle only has one side. If we look at it from a geodesic perspective and a physics perspective, from the side in space where the points are in clockwise order, there is nothing on that side. Literal emptiness. The geodesics will behave as if that triangle isn't there. However, from the other side, the half-triangle does exist and geodesics extending onto that triangle will behave as if it is there. Think of it like a one way window.
Now I did technically say a bit about the geodesics, but let me be more rigorous. When triangles facing the right direction form a surface like in a triangular mesh or this image geodesics behave like you would expect. In fact, two half-triangles with the same three points but facing in opposite directions form a regular triangle.
The unusual case is if we had a shape like in the T-surface within the below thing I found on google images. If all the protrusions are formed by normal triangles, then under my system the blue line is a geodesic/straight line.
Whereas if the surface was formed by half-triangles and the unseen backsides did not have any half-triangles, then by extending the green lines according to the allowed rule set forth in Euclid's second postulate which states "lines may be extended infinitely in either direction" we get the following:
I hope the image makes sense. Basically, when there is no side from where the line is coming from, the line just ignores it and passes by it (hence the action of the blue line). Furthermore because there is no backside for the red line to wrap around onto, it just ends. These are just two examples.
Note, that this also means that by extending backwards along the blue line, the blue line can essentially grow a perpendicular segment that is part of the line. This essentially means that the lines in this system can split and branch wherever a lone half-triangle intersects a plane on the side that exists. I feel that this is in particular an important property of this geometric system.
Let's presume that this concept can be extended to 'surfaces' in general. Not just polygons. I dare not attempt it myself, but I'm sure the polygonal case is clear enough for people to get an idea of how that might extend. What sort of 'surface' geometry am I doing it and what is it classified under?