Prosenjit Gupta

On some geometric optimization problems in layered manufacturing

Abstract Efficient geometric algorithms are given for optimization problems arising in layered manufacturing, where a 3D object is built by slicing its CAD model into layers and manufacturing the layers successively. The problems considered include minimizing the stair-step error on the surfaces of the manufactured object under various formulations, minimizing the volume of the so-called support structures used, and minimizing the contact area between the supports and the manufactured object—all of which are factors that affect the speed and accuracy of the process. The stair-step minimization algorithm is valid for any polyhedron, while the support minimization algorithms are applicable only to convex polyhedra. The techniques used to obtain these results include construction and searching of certain arrangements on the sphere, 3D convex hulls, halfplane range searching, and constrained optimization.

Further results on generalized intersection searching problems: counting, reporting, and dynamization

In a generalized intersection searching problem, a set, S, of colored geometric objects is to be preprocessed so that given some query object, q, the distinct colors of the objects intersected by q can be reported or counted efficiently. In the dynamic setting, colored objects can be inserted into or deleted from S. These problems generalize the well-studied standard intersection searching problems and are rich in applications. Unfortunately, the techniques known for the standard problems do not yield efficient solutions for the generalized problems. Moreover, previous work on generalized problems applies only to the reporting problems and that too mainly to the static case. In this paper, a uniform framework is presented to solve efficiently the counting/reporting/dynamic versions of a variety of generalized intersection searching problems, including: 1-, 2-, and 3-dimensional range searching, quadrant searching, and 2-dimensional point enclosure searching. Several other related results are also mentioned.

Efficient geometric algorithms for workpiece orientation in 4- and 5-axis NC machining

Abstract In 4- and 5-axis NC machines, the time to dismount, recalibrate, and remount the workpiece after each set of accessible faces of the workpiece has been machined can be considerable in comparison to the actual machining time. Unfortunately, the problem of minimizing the number of setups is NP-hard. In this paper, efficient algorithms are given for a greedy heuristic, where the goal is to find an orientation for the workpiece which maximizes the number of faces that can be machined in a single setup—using either a ball-end or a filletend cutter. The algorithms are based on geometric duality, topological sweep, interesting new properties concerning intersection and covering on the unit-sphere, and on techniques for efficiently constructing and searching an arrangement of polygons on the unit-sphere. The results imply that the optimal number of set-ups can be approximated to within a logarithmic factor. Evidence is also provided that it may not be possible to improve substantially on the proposed algorithms.

Minimizing support structures and trapped area in two-dimensional layered manufacturing

Abstract Algorithms are given for the two-dimensional versions of optimization problems arising in layered manufacturing, where a polygonal object is built by slicing its CAD model and manufacturing the slices successively. The problems considered are minimizing (i) the contact-length between the supports and the manufactured object, (ii) the area of the support structures used, and (iii) the area of the so-called trapped regions—factors that affect the cost and quality of the process.

Computing a flattest, undercut free parting line for a convex polyhedron, with application to mold design

Abstract A parting line for a polyhedron is a closed curve on its surface, which identifies the two halves of the polyhedron for which mold-boxes must be made. A parting line is undercut-free if the two halves that it generates do not contain facets that obstruct the de-molding of the polyhedron. Computing an undercut-free parting line that is as “flat” as possible is an important problem in mold design. In this paper, algorithms are presented to compute such a parting line for a convex polyhedron, based on different flatness criteria.

Fast algorithms for collision and proximity problems involving moving geometric objects

Consider a set of geometric objects, such as points or axes-parallel hyperrectangles in ℝd, that move with constant but possibly different velocities along linear trajectories. Efficient algorithms are presented for several problems defined on such objects, such as determining whether any two objects ever collide and computing the minimum inter-point separation or minimum diameter that ever occurs. In particular, two open problems from the literature are solved: Deciding in o(n2) time if there is a collision in a set of n moving points in ℝ2, where the points move at constant but possibly different velocities, and the analogous problem for detecting a red-blue collision between sets of red and blue moving points. The strategy used involves reducing the given problem on moving objects to a different problem on a set of static objects, and then solving the latter problem using techniques based on sweeping, orthogonal range searching, simplex composition, and parametric search.

Efficient algorithms for generalized intersection searching on non-iso-oriented objects

Generalized intersection searching problems are a class of geometric query-retrieval problems where the questions of interest concern the intersection of a query object with aggregates of geometric objects (rather than with individual objects.) This class contains, as a special case, the well-studied class of standard intersection searching problems and is rich in applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented (i.e., axes-parallel) or where the aggregates satisfy additional properties (e.g., connectedness). In this paper, efficient algorithms are given for several generalized problems involving non-iso-oriented objects. These problems include: generalized halfspace range searching, segment intersection searching, triangle stabbing, and triangle range searching. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search.

A technique for adding range restrictions to generalized searching problems

Abstract In a generalized searching problem, a set S of n colored geometric objects has to be stored in a data structure, such that for any given query object q, the distinct colors of the objects of S intersected by q can be reported efficiently. In this paper, a general technique is presented for adding a range restriction to such a problem. The technique is applied to the problem of querying a set of colored points (respectively fat triangles) with a fat triangle (respectively point). For both problems, a data structure is obtained having size O(n1 + e) and query time O((log n)2 + C). Here, C denotes the number of colors reported by the query, and e is an arbitrarily small positive constant.

Computing a Flattest, Undercut-Free Parting Line for a Convex Polyhedron, with Application to Mold Design

A parting line for a convex polyhedron, \(\mathcal{P}\), is a closed curve on the surface of \(\mathcal{P}\). It defines the two pieces of \(\mathcal{P}\) for which mold-halves must be made. An undercut-free parting line is one which does not create recesses or projections in \(\mathcal{P}\) and thus allows easy de-molding of \(\mathcal{P}\). Computing an undercut-free parting line that is as flat as possible is an important problem in mold design. In this paper, an O(n2)-time algorithm is presented to compute such a line, according to a prescribed flatness criterion, where n is the number of vertices in \(\mathcal{P}\).

On Some Geometric Optimization Problems in Layered Manufacturing

Efficient geometric algorithms are given for optimization problems arising in layered manufacturing, where a 3D object is built by slicing its CAD model into layers and manufacturing the layers successively. The problems considered include minimizing the degree of stair-stepping on the surfaces of the manufactured object, minimizing the volume of the so-called support structures used, and minimizing the contact area between the supports and the manufactured object-all of which are factors that affect the speed and accuracy of the process. The stair-step minimization algorithm is valid for any polyhedron, while the support minimization algorithms are applicable to convex polyhedra only. Algorithms are also given for optimizing supports for non-convex, simple polygons. The techniques used include construction and searching of certain arrangements on the sphere, 3D convex hulls, halfplane range searching, ray-shooting, visibility, and constrained optimization.