ellipseens

Ellipseen are closed, convex plane curves that generalize the circle by allowing different scaling along distinct directions. In many sources the term ellipseen is used interchangeably with ellipses; the standard English plural is ellipses. Mathematically, an ellipseen can be obtained as the image of a circle under a non-singular affine transformation, or as a level set of a positive definite quadratic form.

Form and axes: An ellipseen can be described by the equation (x − x0)^T A (x − x0) = 1, where A is a 2x2 symmetric positive definite matrix and x0 is the center. The eigenvectors of A determine the directions of the principal axes, and the corresponding semi-axes lengths are a = 1/√λ1 and b = 1/√λ2, with λ1, λ2 the eigenvalues. The ellipseen can be rotated by the angle determined by the eigenvectors relative to the coordinate axes.

Area, perimeter and eccentricity: The area of an ellipseen is A = πab. The perimeter has no simple closed form, but can be approximated (for example by Ramanujan’s formula). The eccentricity e = sqrt(1 − (min(a,b)^2 / max(a,b)^2)) lies in [0,1], with e = 0 for a circle and e approaching 1 for a highly elongated shape.

Properties and relations: Ellipseen exhibit central symmetry and two orthogonal axes of symmetry. Under affine transformations, some classical focal properties of a standard ellipse may not be preserved, though the overall convex, smooth nature remains.

Applications and occurrences: Ellipseen appear in data fitting, computer graphics, optical design, and various physical and engineering contexts as convenient models for elongated, smoothly bounded shapes.

See also: Ellipse, Conic sections, Affine transformation.