A Variational Framework for Computing Geodesic Paths on Sweep Surfaces. (November 2021) - Meng, Wenlong; Xin, Shiqing; Zhao, Jinhui; Chen, Shuangmin; Tu, Changhe; He, Ying
Abstract: 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 ∈ [ 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 Φ = Φ ( t, θ ), where Φ ( [ t, t + d t ], θ ) defines the sectorial generatrix curve segment at the angle of θ while r t ( θ ) = Φ ( t, θ ), θ ∈ [ 0, 2 π ], 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 ( θ ) : (1) r t ( θ ) is a constant, independent of t and θ, (2) r t ( θ ) depends on only t, independent of θ, and (3) r t ( θ ) depends on both t and θ . We validate the effectiveness and high performance of our method through extensive experimental results. Graphical abstract: Highlights: We propose a variational framework to compute geodesics on sweep surfaces. We optimize the energy functional in a super-linear convergence rate. We establish a relationship between the number of inserted points and the error.