Directional offset of a 3D curve

ABSTRACT

Being a fundamental operation in geometric modeling, there have been a number of researches on offsetting of 2D curves and 3D surfaces. However, there is no commonly accepted definition of 3D curve offset. In this paper, we propose a 3D curve offset method, named directional offset, motivated from the observation of the needs in many engineering design practices. Since the normal vector of a 3D curve at a point is not unique, a 3D curve offset definition is about how to select the offset direction vector on the normal plane of the curve. A previous research on this issue specifies the offset direction vector with a constant angle from the principal normal vector. In directional offset, the offset direction vector on the normal plane is chosen to be perpendicular to the user-specified projection direction vector k. Each point on the original curve is then moved along the offset direction by given offset distance. The directional offset has the following characteristics: (a) directional offset is a natural extension of 2D curve offset, in the sense that they produce the same result when applied to 2D planar curve, with k being normal to the plane, and (b) when k is parallel to Z-axis, the directional offset of a 3D curve is similar to 2D curve offset on XY plane projected image, while inheriting the Z-axis ordinate from the original curve. These properties make it useful in many engineering design applications such as the flange of a sheet metal part, the overflow area design of a forging die, and the cutting blade design of a trimming die for a stamped part. An overall procedure to compute a directional offset for a position-continuous NURBS curve is described with an emphasis on avoiding self-intersection loop.

References

1. Coquillart S, "Computing offsets of B-spline curves:, Computer-Aided Design, 1987, v19(6), pp.305--309
2. Elber G, Lee I-K, Kim M-S, "Comparing Offset Curve Approximation Methods", IEEE computer graphics and applications, 1997, v17(3), pp.62--71
3. Hoschek J, Wissel N, "Optimal approximate conversion of spline curves and spline approximation of offset curves", Computer-Aided Design, 1988, v20(8), pp.475--483
4. Hoschek J, "Spline approximation of offset curves", Computer Aided Geometric Design, 1988, v5(1), pp.33--40
5. Kim D-S, "Hodograph Approach to a Geometric Characterization of Parametric Cubic Curves", Computer Aided Design, 1993, v25(10), pp.644--654
6. Lipschultz M, Differential Geometry, 1969, McGraw Hill
7. Maekawa T, "An overview of offset curves and surfaces", Computer-Aided Design, 1999, v31, pp.165--173
8. Park SC, Choi BK, "Uncut free pocketing tool-paths generation using pair-wise offset algorithm", Computer-Aided Design, 2001, v33(10), pp.739--746
9. Park SC, Shin H, "Polygonal Chain Intersection", Computers & Graphics, in press
10. Pham B, "Offset approximation of uniform B-splines", Computer-Aided Design, 1988, v20(8), pp.471--474
11. Pham B, "Offset curves and surfaces: a brief survey", Computer-Aided Design, 1992, v24(4), pp.223--229
12. Piegl L, Tiller W, The NURBS book, 2nd ed, New York: Springer, 1997
13. Peigl L, Tiller W, "Computing offsets of NURBS curves and surfaces", Computer-Aided Design, 1999, v31(2), pp.147--156
14. Peternell M, Pottmann H, "A Laguerre geometric approach to rational offsets", Computer Aided Geometric Design, 1998, v15, pp.223--249
15. Tiller W, Hanson EG "Offsets of two-dimensional profiles", IEEE Computer Graphics and Applications, 1984, v.4 (9), pp.61--69
16. Wallner J, Sakkalis T, Maekawa T, Pottmann H, Yu G, "Self intersections of offset curves and surfaces", Intl. Journal of Shape Modeling, 2001, v1, pp.1--22

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

      • Parametric curve and surface models

      • Volumetric models