Intersection of a ruled surface with a free-form surface

Abstract

This paper presents a simple method for computing the intersection curve of a ruled surface and a free-form surface. The basic idea is to reduce the problem of surface intersection to the one of projecting an appropriate curve such as a directrix of the ruled surface, along its indicatrix curve (direction vector field of its generating lines), onto the free-form surface; the projection curve is just the intersection curve. With techniques in classical differential geometry, we derive the differential equations of the intersection curve in the cases of parametrically and implicitly defined free-form surfaces. The intersection curve naturally inherits the parameter of the chosen directrix. Moreover, it is independent of the base surface geometry and its parameterization, and is obtained by numerically solving the initial-value problem for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for parametric case or in 3D space for implicit case. Some experimental examples are also given to demonstrate that the presented method is effective and potentially useful in computer aided design and computer graphics.

3D Mass Customization Toolkits Design, Part I: Survey and an Evaluation Model

“…Other strategies use a mesh description of the surface and use an optimizer algorithm to determine the optimal angle orientation in each finite element of the mesh. One of this algorithm is used in [35], [56], [58], [69] and [72]. Then methods such in [77] or the use of streamlines in [49] can be used to compute a path based on the angle found previously.…”

“…Three other methods are presented in [23] to compute parallel curves on a NURBS surface. The first method named section curves is similar to the ones presented in [58], [60], [69], [77]. The other two consist of generating orthogonal curves to the reference by either taking vector-field curves or geodesic curves.…”

Integration of Virtual Reality in a Knowledge-based Engineering System for Preliminary Configuration and Quotation of Assembly Lines

“…Therefore, the error increases by taking a further offset curve (in the case of wider roller), and in the case of highly curved surface. A more accurate method is presented in [58], [60], [69], [77], by taking the intersection between the plane perpendicular to the curve and the mold surface. To do so, a numerical approach presented in [49] is used to determine the resulting curve.…”