Ellipsform

Ellipsform is a term used in geometry and design to denote a planar ellipse-like figure that results from applying a non-singular linear transformation to a circle. The name emphasizes the form as a distortion of a circle rather than an independent construction; in many contexts it is synonymous with the conventional ellipse.

Mathematically, let L be a 2×2 invertible matrix. The ellipsform consists of points x = L u where u ranges over the unit circle (that is, ||u|| = 1). Equivalently, it is the set of x ∈ R^2 satisfying x^T (L L^T)^{-1} x = 1. A is positive definite; The parametrization is x(t) = L [cos t, sin t]^T, t ∈ [0, 2π).

The ellipse's center is at the origin in this formulation; its semiaxes are given by the singular values of L, and its orientation by the columns of L. The area is π |det L|, and the eccentricity is derived from the ratio of the semiaxes. Any rotation or scaling of the transformation preserves the ellipsform as an ellipse; a pure rotation combined with uniform scaling yields a circle as a special case.

In practical applications, ellipsforms appear in computer graphics, CAD, and architectural design as a compact representation of elliptical outlines, bounding shapes, or projection distortions. They serve as the 2D projection of circles under affine transformations and illustrate how linear distortions affect curvature and area.

See also: ellipse, affine transformation, conic section.

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