Shapea

Shapea is a term used in geometry to describe a family of two‑dimensional shapes defined by a simple parametric construction that yields dihedral symmetry. It is often presented in didactic contexts to illustrate how varying a single parameter changes polygon structure while preserving central symmetry. The concept can also be used in computer graphics as a controllable test shape for rendering and recognition tasks.

For a given integer n ≥ 3, a shapea_n is the polygon formed by joining the vertices (x_k, y_k) for k = 0 to 2n−1, where the polar angle θ_k equals kπ/n and the radial distance r_k alternates between R+Δ and R−Δ. Specifically, r_k = R+Δ when k is even, and r_k = R−Δ when k is odd. The Cartesian coordinates of the vertices are x_k = r_k cos θ_k and y_k = r_k sin θ_k. Consecutive vertices are connected by straight edges, forming a simple closed boundary with central angles of π/n between neighbors.

The construction yields symmetry described by the dihedral group of order 2n: rotations by 2π/n and reflections across n axes. The shapes are convex when Δ ≤ R; increasing Δ beyond R can produce nonconvex boundaries. Area and perimeter vary smoothly with R, Δ, and n, and polygon‑area formulas can be applied to compute precise values.

Variations include reducing to a regular 2n‑gon when Δ = 0, or exploring other alternation patterns and nonuniform angular spacing to achieve different symmetry classes. Shapea serves as an instructional example for teaching symmetry, tiling potential, and algorithmic shape handling in graphics and vision applications.

See also: polygon, symmetry, dihedral group, tiling.