A Variational Framework for Computing Geodesic Paths on Sweep Surfaces

Sweep is a natural, intuitive, and convenient 3D modeling method in computer-aided design. Sweep surface can be obtained by extruding a 2D cross-sectional profile along a guide curve x = x(t), t is an element of [a, b]. A small segment of the sweep volume can also be understood by rotating a 2D sectorial generatrix curve around the guide curve. We assume that sweep surfaces have a parametric form Phi = Phi(t, theta), where Phi ([t, t dt], theta) defines the sectorial generatrix curve segment at the angle of theta while r(t) (theta) = Phi(t, theta), theta is an element of [0, 2 pi], defines the circumferential closed curve. Geodesic computation on sweep surfaces is a fundamental geometric operation in many scenarios like the manufacturing process of filament winding. In order to compute a geodesic path between two points on sweep surfaces, we propose a variational framework that works on the 2D parametric domain, without the step of discretizing the surface into a polygonal mesh. The solution to the objective function is a polyline curve of n equally spaced vertices that approximates the real geodesic path, where n is a user-specified parameter for accuracy control. We prove that the polyline approaches the real geodesic in quadratic order. Furthermore, it can be easily extended to compute N-round geodesic helix curves. We also discuss various configurations of r(t) (theta): (1) r(t) (theta) is a constant, independent of t and theta, (2) r(t) (theta) depends on only t, independent of theta, and (3) r(t) (theta) depends on both t and theta. We validate the effectiveness and high performance of our method through extensive experimental results. (C) 2021 Elsevier Ltd. All rights reserved.