regularruled

Regularruled is a term used in some discussions of differential geometry to denote a subclass of ruled surfaces with an additional regularity condition imposed on the director curve of the ruling lines. The term is not widely standardized in standard texts, but is sometimes employed to emphasize smooth variation of ruling directions.

Definition: A ruled surface can be written as r(u,v) = c(u) + v d(u), where c(u) is the directrix and d(u) is a unit direction vector of the ruling at parameter u. The surface is regular if r_u and r_v are linearly independent for all (u,v) in the domain. A regularruled surface further requires that the director curve d(u) be regular, with d and d' not colinear, equivalently the map u -> d(u) on the unit sphere S^2 satisfies |d × d'| ≠ 0.

Implications: Regularruled excludes degenerate rulings where the direction stalls or reverses, and excludes developable surfaces where the ruling directions are constant. The Gaussian curvature of a regularruled surface can vary, and such surfaces include examples like the hyperbolic paraboloid and the helicoid when endowed with the stated regularity on the ruling directions.

Applications: In geometric modeling and CAD, regularruled surfaces provide smooth, easily parameterizable forms with smoothly varying rulings, useful for architectural shells and surface shading. In theory, they form a natural bridge between general ruled surfaces and developable surfaces.

See also: Ruled surface, developable surface, helicoid, hyperbolic paraboloid, differential geometry.