Scaleshape

Scaleshape is a term used in geometry and related fields to describe the family of shapes obtained by uniformly scaling a base object. For a base shape S in Euclidean space and a positive scale factor s, the scaled shape S_s is formed by applying the transformation x -> s x (or, more generally, scaling about a center c: x -> c + s (x - c)). The collection {S_s : s > 0} constitutes a scale family or a homothetic progression of S. Since uniform scaling is a similarity transformation, every S_s is similar to S; the shape itself is preserved while the size changes.

Key properties include that lengths scale by s, areas by s^2, and curvatures scale by 1/s. Angles and overall geometric structure are preserved under isotropic scaling. Convexity, smoothness, and topology are generally preserved as well, making scaleshape a useful concept for analyzing shapes across different sizes without altering their essential form.

Variants and related ideas:

- Isotropic (uniform) scaling yields true scaleshape families; nonuniform (anisotropic) scaling alters shape more broadly and is typically treated separately under affine transformations.

- In applications such as computer graphics and computer vision, scaleshape concepts underpin multi-scale representations and scale-space analysis, enabling shape comparison and feature detection across varying sizes.

- In mathematics, the idea connects to homothety and similarity, foundational notions for studying how figures relate under size changes.

See also: similarity, homothety, scale-space, affine transformation, geometric invariants.