A Variational Framework for Computing Geodesic Paths on Sweep Surfaces

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+dt],θ) defines the sectorial generatrix curve segment at the angle of θ while rt(θ)=Φ(t,θ),θ∈[0,2π], 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 rt(θ): (1) rt(θ) is a constant, independent of t and θ, (2) rt(θ) depends on only t, independent of θ, and (3) rt(θ) depends on both t and θ. We validate the effectiveness and high performance of our method through extensive experimental results.