expansionshape

Expansionshape is a geometric construct describing the result of enlarging a given shape by a fixed distance in all directions. It is defined as the Minkowski sum of the base shape with a closed disk of radius r, meaning the set of all points within distance r of the original shape. In practice, expansionshape is also called an offset or dilation in morphological operations.

For a polygon or polyshape in the plane, the expansion adds edges parallel to the original ones and creates circular arcs of radius r at the vertices. If the base shape is convex, its perimeter increases by 2πr and its area increases according to the Steiner formula: A(r) = A0 + P0 r + π r^2, where A0 and P0 are the original area and perimeter. For non-convex shapes, the area change depends on the specific geometry and may involve filling of concavities as r grows.

Computation and practical considerations involve various offset algorithms. Vector-based methods produce the new boundary by offsetting each edge and inserting rounded corners, while raster or voxel approaches simulate dilation with a structuring element. Challenges include handling self-intersections, topology changes, and numerical stability for small or large radii.

Applications of expansionshape span multiple fields. In robotics and path planning, offsets create safe clearance around obstacles. In geographic information systems, they are used to generate buffer zones. In computer graphics and computer-aided design, expansionshape supports tolerance analysis and quality control. Simple examples include expanding a square to a rounded square and expanding a circle to a larger concentric circle.