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.