Variantsshape

Variantsshape is a term used in geometry and computer graphics to denote a family of related shapes generated by applying a parameterized deformation to a base shape. The concept emphasizes the collection of variants produced by varying one or more parameters, rather than a single object, and is often used to study the shape space accessible to a given base form.

Mathematically, a variantsshape is the image of a base shape S0 under a deformation map f_theta, where theta lies in a parameter space Theta. The base shape can be any manifold embedded in Euclidean space, commonly a polygonal mesh in 3D or a contour in 2D. Desired properties include continuity of the map with respect to theta and, for smooth models, differentiability of f_theta. Distances between variants can be defined using metrics such as the Hausdorff distance or sequence-based metrics on point clouds or meshes.

Common deformation classes include affine, projective, non-linear warp fields, and stochastic perturbations. In practice, variantsshapes are used to explore design spaces, generate training data for shape recognition, or create morphable models for animation and manufacturing. Researchers may study invariants that survive within a deformation class, as well as the topology changes that occur under large deformations.

Variantsshape is related to the broader ideas of shape space and morphable models, and it is sometimes used interchangeably with terms like shape family or deformation family in informal discussions. No standardized formal definition exists across disciplines, and the term is most often used in exploratory or theoretical contexts.