By the extraction of ideas from computer vision, geometrical design and complexity analysis, a structure called visibility emerges. The paper describes a way in which a 3D workpiece is mapped onto the unit sphere, and its visibility is determined. For applications, manufacturing machines are classified by their degrees of freedom into point, line and surface visible processes. Algorithms for optimal workpiece orientation are then formulated as simple intersections on the sphere. Convexity is a notion that facilitates geometrical computation. An entity is said to be convex if the line segment pq that joins two arbitrary points p and q from a set lies entirely in the set. However, entities such as curves and surfaces are not always convex. The notion of visibility promises to extend beyond convexity. Take the art-gallery formulation 1, for example. Given a nonconvex polygon that resembles the floor plan of an art gallery, one asks how many guards are needed such that the entire interior is visible to the guards. Intuition suggests that the lines of sight, or point-to-point visibility, partition a nonconvex polygon into convex (visible) regions, hence bringing convexity to an entity that was nonconvex. This paper takes analytically complex entities, and maps them onto a unit sphere into analytically simple entities called visibility maps that turn out to be convex. Problems that are combinatorially complex are then addressed by a set of simple spherical algorithms that invoke the intersection between the visibility map and a point, a great circle or a spherical rectangle; these are also convex. The establishment of a line of sight between a point on a workpiece and a point on an effector can be quite Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109-2117, USA Paper received." 10 November 1992. Revised." 26 April 1993 useful in manufacturing. In numerical-control (NC) machining, assembly with robots, or inspection by coordinate-measurement machines (CMM), the computation for a tool path often relies on an intersection-free line of sight. Though the geometries of the effector of the manufacturing processes differ, the question about visibility in its most basic form can be stated as follows. Basic enquiry about visibility." Given the geometries of the workpiece and the effector,find a number of orientations of the workpiece such that all the points of interest on the workpiece are visible to the effector. That a minimum of orientations is crucial to productivity in manufacturing is evident. In an automotive application of the laser inspection of cylinder heads, the scanning time is approximately 20 min at 70 point/line/s. The setup time for fixturing and calibrating the workpiece is about 6 h. Since a reorientation requires the dismounting and setting up of the workpiece, the total number of orientations (to achieve complete visibility) should be minimized. To be sure, there are manufacturing operations in which only a single setup is affordable or permitted. The use of the stereolithography apparatus (SLA) in layered manufacturing is one such instance. (In layered manufacturing, a facsimile of a 3D computer representation is fabricated layer by layer, stacking along the z axis. The material, which is a powder or a liquid, is selectively solidified by sintering or photocuring in the xy plane. For sheet material, a contour in the xy plane is created, and unwanted portions are discarded. However, regardless of the process, there is only one setup, and hence single implied orientation is permitted.) The choice of an orientation in layered manufacturing can make a difference. In an experiment in which the housing of a mobile phone was fabricated, two orientations showed a difference in fabrication time. In the orientation in which the antenna pointed upwards, it took over 22 h to fabricate the housing on an SLA 250. In the orientation 0010-4485/94/010006-11
Woo introduced the concept of spherical visibility maps and applied them to various manufacturing problems like optimal workpiece orientation REF .
Visibility maps and spherical algorithms
Comput. Aided Des.
Comput. Aided Des.
Computer Science Mathematics