Computational Geometry

Note: This post has been updated to work with HiGHS. A fun geometry problem to think about is: given two polygons, do they intersect? That is, do they touch on the border or overlap? Does one reside entirely within the other? While this question has obvious applications in computer graphics (see: arcade games of the 1980s), it’s also important in areas such as cutting and packing problems. There are a number of way to answer this. In computer graphics, the problem is often approached using a clipping algorithm. This post examines a couple of simpler techniques using linear inequalities and properties of convexity. To simplify the presentation, we assume we’re only interested in convex polygons in two dimensions. We also assume that rotation is not an issue. That is, if one of the polygons is rotated, we can simply re-test to see if they overlap. ...

Note: This post was written before Gurobi supported nonlinear optimization. It has been updated to work with Python 3. A common problem in handling geometric data is determining the center of a given polygon. This is not quite so easy as it sounds as there is not a single definition of center that makes sense in all cases. For instance, sometimes computing the center of a polygon’s bounding box may be sufficient. In some instances this may give a point on an edge (consider a right triangle). If the given polygon is non-convex, that point may not even be inside or on its boundary. ...