mathematics//computational geometry

Computational geometry is the branch of mathematics and computer science that designs algorithms and data structures for shapes (points, lines, polygons, surfaces, volumes), and it is what lets a computer store a part, a building or a terrain, test whether two objects collide, plan a path around obstacles, or turn a scan into a surface. Every CAD tool, game engine, robot motion planner, slicer for a 3D printer and finite-element mesher runs on it.

Computational geometry is the branch of mathematics and computer science that designs algorithms and data structures for shapes (points, lines, polygons, surfaces, volumes), and it is what lets a computer store a part, a building or a terrain, test whether two objects collide, plan a path around obstacles, or turn a scan into a surface. Every CAD tool, game engine, robot motion planner, slicer for a 3D printer and finite-element mesher runs on it.

The family shares one problem: a continuous shape has to live in a finite machine. Each member chooses a representation and pays for it. A polygon mesh stores the surface as flat faces glued along edges, compact and fast to draw, the lingua franca between tools. An implicit representation stores a function over space whose zero set is the surface, which makes inside and outside trivial to test and blends shapes smoothly; its surface is recovered as an isosurface. A point cloud, the raw output of a lidar or a depth camera, stores samples of the surface and nothing about how they connect. Converting between them (point cloud to mesh, field to mesh, mesh to field) is where most practical work and most artefacts live.

The formulas are easy; robustness is the hard part.

The geometry of a triangle is school mathematics; deciding with floating-point numbers whether a point lies exactly on an edge, without contradicting a decision made a microsecond earlier, is what makes geometric code crash or leak holes.

For a robot or a drone the field appears as collision checking and free-space computation: an occupancy grid or a signed distance field around the vehicle answers how far is the nearest obstacle, which a planner queries thousands of times a second (SLAM builds the map these queries run on).

For simulation it appears as meshing: a finite-element solver needs the part cut into small elements whose shape quality decides the accuracy and the stability of the solution.

Generative models now produce geometry directly (3D generation), and their output is judged with the tools of this field: is the mesh closed, manifold, free of self-intersections.