Shapeencompassing

Shapeencompassing is a concept in geometry and related fields describing the task of identifying or constructing a single shape that contains a set of given shapes or geometric primitives. The goal is to produce a bounding envelope that covers all input elements with minimal redundancy, subject to chosen constraints such as convexity or alignment. In practice, shape encompassing is used to simplify complex configurations, enable fast spatial queries, or provide a compact geometric summary.

Common enveloping shapes include the convex hull, the minimum enclosing circle, and the minimum bounding box (axis-aligned or oriented). More flexible envelopes are formed by alpha shapes, which can create non-convex boundaries that still enclose the data based on a parameter. Minkowski sums can be used to inflate a shape to cover others, while unions or hulls of multiple shapes can produce more complex envelopes.

For points, convex hulls can be computed with Graham scan or Quickhull; smallest enclosing circle with Welzl's algorithm; minimum bounding box by rotating calipers; alpha shapes via Delaunay triangulations.

Shape encompassing is applied in collision detection, computer graphics, geographic information systems, robotics, clustering visualization, and shape matching. Envelopes are not unique; convex versus non-convex choices yield different results. There is often a trade-off between tightness of fit and computational cost, and sensitivity to outliers can influence the envelope produced. See also related concepts such as convex hull, bounding volume hierarchies, alpha shapes, and bounding boxes.