I got an interesting idea today.<\/p>\n
As you may already know, I’ve been making models<\/a> of Klein bottles an’ stuff; heretofore they’ve all been in the form of bent rods, but where possible I’d prefer a continuous surface. (A hollow body must have holes so that unused powder can be shaken out; but not all of my designs have enclosed spaces.) How to place a minimum number of vertices so that deviations from the abstract shape are within the resolution of the process? That’s less obvious with more degrees of freedom.<\/p>\n So, today’s idea. Start with an arbitrary set of sample nodes (in the abstract space of the parametric variables, rather than on the target surface itself), and their Delaunay triangulation<\/a>. Along each edge of the triangulation, measure the deviation of the surface from a straight line; this gives the edge a weight. Move each node to the weighted average of its neighbors (with a bit of noise); thus, an edge whose image is strongly curved gets shorter.<\/p>\n After the movement phase, each edge ought to be checked, whether it’s still a Delaunay edge or needs to be replaced by the other diagonal of the quadrilateral formed by its two triangles. I don’t yet have criteria for adding nodes where existing nodes are too far apart, or merging them if they become redundant.<\/p>\n","protected":false},"excerpt":{"rendered":" I got an interesting idea today. As you may already know, I’ve been making models of Klein bottles an’ stuff; heretofore they’ve all been in the form of bent rods, but where possible I’d prefer a continuous surface. (A hollow … Continue reading