Research On Geodesic Offset For Curves On Triangular Mesh

Offset of curves is an important tool in computer aided design. It is widely used in industry, such as robot path planning, solid modeling, generation of tool paths for NC machining etc. Usually, an offset curve refers to the offset points set for a plan curve, and the offset point is defined as the point obtained by moving a point of the progenitor curve by a specified offset distance along its normal direction. In this paper, geodesic offset of curves lying on 3D surfaces are considered. Geodesic offset of a curve on a surface is described as a set of points on the surface, which are at a constant geodesic distance form the curve. Geodesic offset is an essential tool in generating the composite ply shapes for the design of laminated composite parts. For the triangular mesh is widely used in industry and computer graphics systems, present research work is done for the geodesic offset approach for curves on triangular mesh surface, and the curve on the triangular mesh is defined by a sequence of points with every point lying on the mesh.A new offset method for curves on triangular mesh is presented. Firstly, all line segments on the original cuive are divided into more new line segments. Secondly, the offset point of each vertex of new line segments is exactly computed along geodesic direction, and the offset "arc segment" for the point with tangential discontinuity is generated, and at the same time, local self-intersections are detected and removed. Thirdly, all offset points are joined into the raw offset curve. Finally, the offset curve is obtained by deleting global self-intersections.The following is research work described in this thesis.1. The offset curve of a line segment on the triangular mesh is a 3D line segment chain on the triangular mesh. The relationship between the segment chain and its related triangular faces in the mesh is found, then a fast offset method for a line segment is proposed. Firstly, the progenitor line segment is divided into several new line segments according to a specified rule first, and the offset curve of the line segment is obtained by connecting offset points for each end point of the divided line segments.2. To ensure the offset distance from the point on the progenitor curve to the offset curve along the geodesic is a constant, the end point (it splits two offset curves of a pair neighbor line segments) of line segment on the curve should be offset into a "arc segment". The offset "arc segment" is generated by offsetting the end point along several geodesic directions which are found by a self-adaptive sampling method.3. The relationship between the convexity-concavity of vertices on the progenitor curve and the local invalid loops are explored. The inference relationship between line segments on the progenitor curve is defined. Utilizing the continuity of local invalid loops and the inference relationship between line segments, a fast detect and remove method for local self-intersections and local invalid loops is achieved by incremental searching start from corresponding vertex on the progenitor curve.4. The algorithm proposed by us for polygonal chains intersection based on the sweep line algorithm is extended, and it is used to detect and delete global self-intersections in the rough offset curve.In the end,the proposed method has been implemented and tested with open and closed curves in the environment of VC++ 6.0 and ACIS. Empirical tests show the method is robust, and fast.