sizeshape

Sizeshape is a concept used in geometry and related disciplines to describe how the size and shape of an object relate to one another. In this framework, size refers to a linear scale factor that maps a reference or canonical shape to a target instance, while shape refers to the geometry of that instance independent of scale, captured by a dimensionless descriptor. A sizeshape representation thus consists of either a paired description (size, shape) or a single composite metric that combines these components.

Formally, if R is a reference shape and s > 0 is a scale factor, the target shape is sR. The shape descriptor S is invariant to scaling and encodes the geometry of R. By pairing s with S, or by composing them into a sizeshape vector, one can compare objects across different sizes while preserving information about their form.

Applications appear in computer vision, computer-aided design, and morphometrics, where it is useful to classify or optimize shapes that appear at multiple scales. For example, the ellipse with major axis a and minor axis b has size parameter a (or area) and a shape descriptor given by eccentricity e = sqrt(1 - (b^2/a^2)). More complex shapes use Fourier descriptors, Zernike moments, or principal component analysis to produce S.

Limitations include dependence on the chosen shape descriptor, sensitivity to noise, and potential ambiguity when multiple shapes share similar descriptors at different scales. Related ideas include scale invariance, similarity transformations, and morphometrics.