Envelopten

Envelopten is a term encountered in theoretical geometry to denote the boundary locus formed by a one-parameter family of curves or surfaces generated by a common process. While closely related to the classical notion of an envelope, the envelopten is used in some discussions to emphasize the boundary object itself as a distinct geometric construct, potentially with its own singularities and parametrization.

Definition and construction: Let {S_t} be a one-parameter family of surfaces in three-dimensional space defined by F(x, y, z, t) = 0. The envelopten E, when it exists, is the set of points for which F(x, y, z, t) = 0 and ∂F/∂t(x, y, z, t) = 0 for some t. For planar families of curves defined by F(x, y, t) = 0, the same two-equation system defines the planar envelopten. Solving these equations yields an implicit or parametric description of E.

Properties: Envelopten curves or surfaces describe the limiting boundary of the moving family and may be smooth in regular regions or acquire singularities at tangency points. The existence and shape of an envelopten depend on the regularity of F with respect to t and on the chosen parametrization; different parameterizations can, in principle, yield different envelopes.

Examples: In planar geometry, a family of lines tangent to a fixed convex curve has an envelopten that coincides with that curve. In a scaling family, copies of a shape scaled about a point can produce an envelopten that outlines the maximal extent of the family.

Applications: The concept appears in geometric modeling, computer-aided design, architectural form-finding, and optics, where envelope-like boundaries describe limiting behavior of families of shapes or surfaces.

See also: envelope (geometry), boundary, locus, parameterization.