locusons

A locuson is a mathematical construct used to describe a family of loci that vary with a parameter, providing a compact way to study how a set of points evolves as that parameter changes. The term is used in speculative or theoretical discussions of geometry and computer graphics to capture the idea of a higher-order locus.

Formal definition and representation: Suppose a parameter t ranges over an interval T and, for each t, a curve Ct in the plane is defined by a relation F(x, y, t) = 0. The locuson L is the subset of three-dimensional space consisting of all triples (x, y, t) that satisfy F(x, y, t) = 0. The projection of L onto the (x, y) plane yields the curve Ct corresponding to a fixed t. Conversely, solving F(x, y, t) = 0 for t describes how a point (x, y) lies on the evolving locus as t varies.

Construction and properties: A locuson can be viewed as a surface or higher-dimensional object in (x, y, t) space. It can be described implicitly by F(x, y, t) = 0 or with parametric forms that incorporate t. Eliminating t from the defining relations yields envelopes, discriminants, or specializations of the locus as t changes. Singularities of the locuson often coincide with topological changes in Ct across parameter values.

Examples: If Ct is a circle of radius t centered at the origin, F(x, y, t) = x^2 + y^2 − t^2, and the locuson is the double cone x^2 + y^2 = t^2 in (x, y, t) space. If Ct is the horizontal line y = t, then F(x, y, t) = y − t, and the locuson is a plane y = t in (x, y, t) coordinates.

Applications: Locusons appear in theoretical geometry, computer graphics, animation, and robotics for modeling how contours, paths, or constraints evolve under parameter variation. See also locus, family of curves, envelopes, and discriminants.