sweptsurface

A swept surface, or sweep surface, is a two-dimensional surface created by moving a fixed curve, called the profile, along a spatial path, called the trajectory or directrix, using a rigid body motion. As the profile translates and possibly rotates, every position traces a copy of the profile, and the union of these copies forms the surface. The concept generalizes extrusion, rotation, and other rigid motions.

Common instances include sweeping a line along a curve produces a ruled surface; sweeping a circle along a straight line yields a circular cylinder; sweeping a circle along a curved path yields a tube around the path; sweeping a line along a straight line can give a plane or general cylinder; if the trajectory is a closed loop and the profile remains perpendicular, you may obtain a toroidal or other tube-like surface. The orientation of the profile (whether it remains perpendicular to the trajectory or rotates) affects the geometry.

Mathematically, if the profile is C(s) and the rigid motion at parameter t is g(t) (comprising a rotation R(t) and a translation d(t)), the swept surface is S(t, s) = g(t)(C(s)) = R(t)C(s) + d(t). In special cases R(t) is identity and d(t) is linear in t, S is an extrusion; if C is a circle and d(t) follows a line with a constant offset, S is a cylinder; if R(t) changes with t, the cross-section rotates along the path.

Applications include computer-aided design, computer graphics, and manufacturing, where swept surfaces model ducts, cables, pipes, or extruded profiles. Practical considerations include potential self-intersections, cusps, and distortions, and often one studies properties such as developability, curvature, and smoothness of the sweep.