Minkowski Sum
Animate the Minkowski sum of two polygons.
About this tool
A + B contains all sums a+b with a in the filled polygon A and b in the filled polygon B. Here both polygons must be simple and convex. Their full result is computed from the convex hull of pairwise vertex sums. Nonconvex inputs are rejected, not silently filled in.
The yellow polygon is one translation A+b. Select b along the boundary of B with the position slider. A boundary tour alone does not generally fill the whole sum; the green boundary is computed independently in advance. Play makes one four-second tour, Pause preserves the current position and hidden pages pause playback. The projection remains fixed during a tour.
Custom mode: one x y pair per line; whitespace or semicolons separate coordinates, and a decimal comma is accepted. Use 3–24 distinct vertices in boundary order, clockwise or counterclockwise. A repeated closing vertex is optional; collinear boundary vertices are allowed. Coordinates range −10000…10000 in a common arbitrary unit; area uses its square. The hexagon/pentagon example uses a polygon, not an approximated disk.
After the first calculation, valid input changes update the sum automatically. Reset stops playback and returns the position to 25%. Drag the orange marker or choose a point on B to move the translation; the slider is the keyboard alternative. The colored checkboxes show or hide A, B, A+b and A+B. Coordinates and the result vertices are in Measurements. Only raw polygon inputs and the selected example persist, including invalid drafts; reload and navigation wait for Calculate. CSV contains A, B and the result vertices. SVG/PNG preserve the current visible layers and translation with full result metadata. Export pauses playback.
Predicates use stored binary coordinates; summed coordinates, measures and drawing use floating-point approximations. Very small features may be invisible at the shared scale. Unsupported positive measures produce an error.