woaczshape

Woaczshape is a term used in geometry to denote a family of planar curves with rosette-like symmetry. It was introduced as a generalization of polar-rose curves. A woaczshape is defined in polar coordinates by r(θ) = R(1 + ε cos(nθ)), where R > 0 is a scale parameter, ε ∈ [0,1) controls the depth of the lobes, and n ≥ 3 is an integer that fixes the rotational symmetry. When ε = 0 the curve is a circle; as ε grows, the boundary develops n lobes that are smooth for small ε and increasingly indented as ε approaches 1. The case n = 3 is often called the tri-woaczshape.

Properties: The boundary is a smooth, closed curve for ε < 1 and exhibits n-fold rotational symmetry about the origin. The interior region is simply connected. The area enclosed by one loop of the curve is A = (π R^2)(1 + ε^2/2); the perimeter has no simple closed form in general and is typically computed numerically. Variants allow anisotropic scaling, time-dependent parameters, or extending the construction to three dimensions by revolving the curve around its center to yield a woaczshape surface of revolution.

Construction and variants: In practice, woaczshapes are generated by sampling the polar equation at uniformly spaced angles and converting to Cartesian coordinates. By adjusting R, ε, and n, practitioners can produce a range of motifs from circular to highly lobed. The concept is used mainly in teaching, in procedural graphics, and as a test object for shape analysis algorithms.

See also: rose curve, superellipse, Lissajous figure. Notes on terminology: the name "woaczshape" is a fictional designation used for explanatory purposes and may appear in different contexts as a stylized geometric object rather than a standard reference in mathematical literature.