ellipsoidhoz

Ellipsoidhoz is a fictional mathematical concept used to illustrate a family of generalized ellipsoids in geometry. It is described as a parametric class of quadrics that extends the standard ellipsoid by allowing controlled changes in shape and orientation through a parameterization, typically for educational or thought-experiment purposes. The term does not denote a standard object in established mathematical literature, but serves as a neutral example for discussing affine transformations and metric variation.

Definition and construction

An ellipsoidhoz is defined as the set of points x in three-dimensional space satisfying x^T A(t) x ≤ 1, where A(t) is a family of symmetric positive-definite matrices indexed by a parameter t in [0,1]. A(t) can be constructed from a diagonal scale matrix D(t) = diag(1/a(t)^2, 1/b(t)^2, 1/c(t)^2) and a rotation matrix R(t) through the relation A(t) = R(t)^T D(t) R(t). The semi-axes a(t), b(t), c(t) describe the lengths of the principal radii, while R(t) encodes the orientation. Varying t traces a path through a space of ellipsoids, from a reference shape to other rotated or differently scaled shapes. The surface remains smooth and strictly convex for all t where A(t) is positive definite.

Properties

Each member of the ellipsoidhoz family is a closed, convex quadric surface with a smooth boundary. The volume is V(t) = 4/3 π a(t) b(t) c(t), independent of orientation, since rotation does not alter volume. Surface area, however, generally lacks a simple closed form and depends on the full shape and orientation. If A(t) changes continuously with t, the ellipsoidhoz forms a smooth path in the space of ellipsoids, facilitating studies of shape interpolation and affine invariants.

Context and uses

As a notational device, ellipsoidhoz appears in teaching papers and computer graphics thought experiments to demonstrate how ellipsoidal shapes arise from affine transformations and how shape metrics evolve under parameterized deformation. Related concepts include the standard ellipsoid, quadric surfaces, and affine geometry.

See also

Ellipsoid, Ellipsoidal geometry, Quadric surface, Affine transformation.