10.7 Geodesic offsets

In this section we focus on geodesic offsets which are different from the classical offset definition. Geodesic offsets or geodesic parallels are well known in classical differential geometry. Let us consider an arbitrary curve on a surface. The locus of points at a constant distance measured from curve along the geodesic curve drawn orthogonal to is called geodesic offset (see Fig. 10.12). Patrikalakis and Bardis [296] provide an algorithm to construct such geodesic offsets on NURBS surfaces. The equations of the geodesics consist of four first order nonlinear ordinary differential equations (10.17) to (10.20) which are solved as an initial value problem.

Let us consider a progenitor curve lying on a parametric surface given by and an arc length parametrized geodesic curve orthogonal to . We select points on the progenitor curve , and compute a geodesic path for each point by a distance equal to as an IVP. The initial direction = = can be determined by the condition that the tangent vector along the progenitor curve and the unit tangent vector of the geodesic curve are orthogonal (see (10.68))

and the normalization condition

leading to

where and . The positive and negative signs in (10.73) and (10.74) correspond to the two possible directions of the geodesic path relative to the progenitor curve.

The terminal points of the geodesic paths, departing orthogonally from selected points of the progenitor curve on the surface, are interpolated in the surface patch parameter space by a B-spline curve assuring that the offset curve lies entirely on the surface.

Wolter and his associates [340] compute medial curves on a surface, which is the locus of points which are equidistant from two given curves on the surface, utilizing the geodesic offset function. Their method is also applicable to the plane curve case. Also Wolter and his associates [214] applied the above method to compute a Voronoi diagram on a parametric surface instead of the Voronoi diagram in Euclidean space.

Traditionally the spacing between adjacent tool paths, which is referred to as side-step or pick-feed, has been kept constant in either the Euclidean space or in the parameter space. Recently geodesic offset curves are used to generate tool paths on a part for zig-zag finishing using 3-axis NC machining with ball-end cutter so that the scallop-height, which is the cusp height of the material removed by the cutter, will become constant [417,366]. This leads to a significant reduction in size of the cutter location data and hence in the machining time.

Figure 10.12: Geodesic offset curve on a Bézier surface. Thick solid line represents the progenitor curve and the thick dotted line represents the geodesic offset curve (adapted from [250])