Convex and Star Polygons
Explore convex polygons and star polygons derived from the same oriented half-spaces.
Open demonstration →Interactive demonstrations accompanying the 1999 work of Ergun Akleman and Jianer Chen
This site collects compact interactive examples built around the implicit shape-modeling operations introduced in our 1999 paper. The same basic operations are used in increasingly general constructions—from convex half-space models to simple polygons, set differences, and sampled B-spline curves. The goal is to make the behavior of the representations visible and interactive rather than to repeat the mathematics of the paper.
The examples are intentionally compact. Together they reveal advantages and limitations of the different constructions without turning this page into a formal comparison. Our more recent Hölder–Minkowski work extends the approximate-operation direction and will be presented on a separate site.
Explore convex polygons and star polygons derived from the same oriented half-spaces.
Open demonstration →Build more complex regions as unions of multiple convex components.
Open demonstration →Combine additive and subtractive convex components to construct holes and related shapes.
Open demonstration →Draw a simple polygon, compute its monotone Boolean representation, then explore the functional operations.
Open demonstration →Edit a general simple polygon while its monotone representation and implicit field are recomputed continuously.
Open demonstration →Use two dynamically represented simple polygons, A and B, to explore implicit set difference.
Open demonstration →Sample a closed B-spline and use the resulting polygonal monotone representation as an implicit model.
Open demonstration →Represent an open curve by a thin offset polygon with controllable thickness and smoothed normals.
Open demonstration →Begin with convex and star polygons, then explore unions and set differences of convex components. The monotone-formula examples extend the construction to general simple polygons. The B-spline examples show how the same framework can be applied to sampled parametric curves, including both closed and open curves.