Sweeping solids on manifolds

Abstract

This work describes a new method for modeling of sweep solids on manifolds, considering various geometric and functional constrains. The proposed method is applied in semi-automatic computer aided design of ventilation tubes for hearing aid devices. The sweeping procedure begins with definition of a trajectory. Besides smoothness and minimal length, other requirements may be considered. Therefore, it is convenient to formulate the optimal trajectory problem as a geodesic computing over Riemannian manifold. The trajectory defined on the manifold is ofsetted, in order to make the sweep solid tangent to the manifold. The offset curve shape is iteratively smoothed while preserving minimal distance from the manifold. Then, a frame field is defined over the offset curve and the cross section contour is transformed according to this field. The major problem is how to construct the frame field such that the resulting sweep solid will be smooth and free of self-intersections. It is well known, that Frenet frame imposes restrictions on the trajectory and may create undesirable twist. In order to overcome these obstacles, an efficient procedure is proposed to compute the discrete minimal rotation frame. Finally, a new approach to the self-intersection problem of sweep solids is proposed. The key idea is to weaken the orthogonality requirement between the cross section plane and the trajectory curve, in order to avoid self-intersections. The described method was implemented and tested in real production environment, where it was proved robust and efficient. The proposed techniques can be utilized in many related applications where sweep surface modeling and manipulation is involved.