Methods to recover constant radius rolling ball blends in reverse engineering

Reverse engineering of geometric shape is the process of converting large amounts of measured data points into concise and consistent computer representations of geometry. Applications include the reproduction of engineering parts with no available documentation or CAD definition for such purposes as redesign, analysis and visualisation. Other applications include the creation of various mating surfaces for parts of the human body, for example. As described in a recent survey [18], reverse engineering typically consists of four phases: data acquisition, preprocessing, segmentation and geometric model creation. Each of these four phases is quite complicated and they are strongly interrelated. Depending on the basic assumptions made and the quality and quantity of the measured data, there is a great variety of applied algorithms and computer models, as reflected in many recent publications which investigate various aspects of reverse engineering of shape. The topic of this paper, recovering constant radius rolling ball blends, is mainly of interest when reconstructing mechanical engineering parts. From a geometric point of view, we assume that these objects are bounded by trimmed primary surfaces, which determine the basic shape of the object. These are relatively large in comparison to smaller blending surfaces which provide smooth transitions between the primary surfaces. Blends strongly depend on the primary surfaces, and are often used for reasons of aesthetics, manufacturability, stress reduction etc. As discussed in [21], there are many methods to create blends and represent them in various mathematical forms. Here we restrict our interest to the most widely used class of constant-radius rolling ball blends due to their simplicity and intuitive behaviour. Such blends are nominally generated by sweeping a rolling ball moving in contact with two adjacent primary surfaces. We assume that edge blends are small enough that they do not interfere with each other, except where they run together at the vertices; here vertex blend faces may need to be inserted to complete the object. The purpose of the current investigation is to present and compare algorithms for recovering constant radius rolling ball blends. This is a particular subproblem of the final model building phase of reverse engineering. We do not go into details concerning the various steps of preprocessing—algorithms to perform filtering, triangulation and decimation are described elsewhere [10, 17], but we assume that