ShapesI

shapesI is a term used in geometry to denote a family of invariant-based shape representations. It refers to a hypothetical framework for describing planar shapes by a vector of invariant features that do not change under similarity transformations (translation, rotation, scaling). The I stands for invariants, and the framework aims to provide a compact, comparison-friendly description of shape geometry.

A shapesI object consists of either a closed polygonal chain or a smooth parametric curve in the plane together with an associated invariant vector I. I is computed from the boundary and may include area (normalized by the convex hull area), normalized perimeter, curvature statistics, and frequency-domain descriptors such as Fourier or Zernike moments, designed to be invariant to location, orientation, and size.

The typical pipeline involves contour extraction, normalization to a canonical scale, and computation of invariant components. The resulting I-vector enables simple distance measures between shapes and supports clustering, classification, and retrieval tasks. ShapesI is agnostic to the exact polygonal representation, focusing on invariant shape content rather than absolute coordinates.

Examples and limitations: Regular shapes like circles, squares, and equilateral triangles map to compact, low-variance I-vectors, while highly irregular contours yield higher variance in I. The framework is sensitive to segmentation quality and noise; preprocessing steps are critical for reliable invariants.

Relation and usage: ShapesI aligns with broader shape-descriptor research, sharing goals with Fourier descriptors, moment invariants, and shape contexts, but emphasizes a unified, invariant vector for efficient comparison in computational geometry, computer vision, and pattern recognition.