I am working on a project to automate a lot of my work, but I wanted a stronger mathematical background under my belt to make sure I tackle this properly. I am a drafter that will be writing a module to select and generate the proper geometry for our models given some input data before it is seen or touched by other people. To that end, I've been trying to pick up some useful terminology to help me search for more relevant material on the topic. Let me explain what I'm trying to do:

Pic related is a simplified example of what I wish to do. We can assume that the leftmost and and rightmost profile will always be present for a particular model. The true dimensions are not what is important, it is the relational constraints that are (aka, the left is always a vertical line perpendicular to a horizontal line, each of arbitrary length). Given those two constraining paths, I wish to "attach" a intermediate open profile between the two at their respective endpoints. Once again, the relational constraints of these profiles are all that must be preserved, and the dimensions are irrelevant.

I'm not a pure math buff, and am not currently familiar with the notation common to academic publications, but it is not out of my wheelhouse to pick it up very rapidly. I have a short list of potentially useful words to look into, and I am hoping you could give me further guidance. If you can point me to the right topics to research, I can teach myself the notation and dive deeply to learn what I need for my project. Here is the surface level first pass list of keywords I've compiled:
>affine geometry
>configuration
>Isomorphism
>homomorphism
>homeomorphism
>continuous graph
>simple path
>group theory
>topography