No description available.

No description available.

Mechanical CAD/CAM (computer aided design/manufacturing) as a field research concerns itself with the algorithms and the mathematics necessary to simulate mechanical parts of the computer, that is to produce a computer model. Solid modelling is a subdiscipline in which the computer model accurately simulates volumetric, i.e. 'solid', properties of mechanical parts. This dissertation deals with a particular type of free-form surface, the blending surface, which is particularly well-suited for solid modelling. A blending surface is one which replaces creases and kinks in the original model with smooth surfaces. A fillet surface is a simple example. We introduce an intuitive paradigm for devising different types of blending forms. Using the paradigm, three forms are derived: the circular, the rolling-ball, and the super-elliptic forms. Important mathematical properties are investigated for the blending surfaces, e.g. continuity, smoothness, containment etc. Blending on blends is introduced as a notion which both extends the flexibility of blending surfaces and allows the blending of multiple surfaces. Blending on blends requires one to think about the way in which the defining functions act as a distance measure from a point in space to a surface. The function defining the super-elliptic blend is offered as an example or a poor distance measure. The zero surface of this function is then embedded within a function which provides an improved distance measure. Mathematical properties are derived for the new function. A weakness in the continuity properties of above blending form is rectified by defining another method to embed the super elliptic blend into a function with better distance properties. This is the displacement form. The concern with this form is its computational reliability which is, therefore, considered in more depth. In the process of integrating the blending surface geometry into a solid modelling environment so it was usable, it was discovered that three other formidable problems needed some type of resolution. These were the topological, the intersection and the display problems. We report on the problems, and solutions which we developed.

Given a set of circles C = {c₁, ..., cn}on the Euclidean plane with centers {(a₁, b₁), ..., (an, b_n)}and radii {r₁..., r<n},the smallest enclosing circle (of fixed circles) problem is to find the circle of minimum radius that encloses all circles in C. We survey four known approaches for this problem, including a second order cone reformulation, a subgradient approach, a quadratic programming scheme, and a randomized incremental algorithm. For the last algorithm we also give some implementation details. It turns out the quadratic programming scheme outperforms the other three in our computational experiment. / Singapore-MIT Alliance (SMA)

At SODA'10, Agarwal and Sharathkumar presented a streaming algorithm for approximating the minimum enclosing ball of a set of points in d-dimensional Euclidean space. Their algorithm requires one pass, uses O(d) space, and was shown to have approximation factor at most 1.3661. We prove that the same algorithm has approximation factor less than 1.22, which brings us much closer to a 1.207 lower bound given by Agarwal and Sharathkumar. We also apply this technique to the dynamic version of the minimum enclosing ball problem (in the non-streaming setting). We give an O(dn)-space data structure that can maintain a 1.22-approximate minimum enclosing ball in O(dlog n) expected amortized time per insertion/deletion. Finally, we prove that a 1+ϵ approximation to the problem can be found in (0.5+δ)/ϵ passes over the input, for an arbitrarily small constant δ, which is an improvement over the previous result that used 2/ϵ passes.

Thesis--Auburn University, 2010. / Abstract. Includes bibliographic references (p.28).

No description available.

At SODA'10, Agarwal and Sharathkumar presented a streaming algorithm for approximating the minimum enclosing ball of a set of points in d-dimensional Euclidean space. Their algorithm requires one pass, uses O(d) space, and was shown to have approximation factor at most 1.3661. We prove that the same algorithm has approximation factor less than 1.22, which brings us much closer to a 1.207 lower bound given by Agarwal and Sharathkumar. We also apply this technique to the dynamic version of the minimum enclosing ball problem (in the non-streaming setting). We give an O(dn)-space data structure that can maintain a 1.22-approximate minimum enclosing ball in O(dlog n) expected amortized time per insertion/deletion. Finally, we prove that a 1+ϵ approximation to the problem can be found in (0.5+δ)/ϵ passes over the input, for an arbitrarily small constant δ, which is an improvement over the previous result that used 2/ϵ passes.

Approximate convex decomposition (ACD) is a technique that partitions an input object into "approximately convex" components. Decomposition into approximately convex pieces is both more efficient to compute than exact convex decomposition and can also generate a more manageable number of components. It can be used as a basis of divide-and-conquer algorithms for applications such as collision detection, skeleton extraction and mesh generation. In this paper, we propose a new method called Fast Approximate Convex Decomposition (FACD) that improves the quality of the decomposition and reduces the cost of computing it for both 2D and 3D models. In particular, we propose a new strategy for evaluating potential cuts that aims to reduce the relative concavity, rather than absolute concavity. As shown in our results, this leads to more natural and smaller decompositions that include components for small but important features such as toes or fingers while not decomposing larger components, such as the torso that may have concavities due to surface texture. Second, instead of decomposing a component into two pieces at each step, as in the original ACD, we propose a new strategy that uses a dynamic programming approach to select a set of n_c non-crossing (independent) cuts that can be simultaneously applied to decompose the component into n_c + 1 components. This reduces the depth of recursion and, together with a more efficient method for computing the concavity measure, leads to significant gains in efficiency. We provide comparative results for 2D and 3D models illustrating the improvements obtained by FACD over ACD and we compare with the segmentation methods given in the Princeton Shape Benchmark.

In the area of motion planning, considerable work has been done on guarding problems, where "guards", modelled as points, must guard a polygonal space from "intruders". Different variants of this problem involve varying a number of factors. The guards performing the search may vary in terms of their number, their mobility, and their range of vision. The model of intruders may or may not allow them to move. The polygon being searched may have a specified starting point, a specified ending point, or neither of these. The typical question asked about one of these problems is whether or not certain polygons can be searched under a particular guarding paradigm defined by the types of guards and intruders. In this thesis, we focus on two cases of a chain of guards searching a room (polygon with a specific starting point) for mobile intruders. The intruders must never be allowed to escape through the door undetected. In the case of the two guard problem, the guards must start at the door point and move in opposite directions along the boundary of the polygon, never crossing the door point. At all times, the guards must be able to see each other. The search is complete once both guards occupy the same spot elsewhere on the polygon. In the case of a chain of three guards, consecutive guards in the chain must always be visible. Again, the search starts at the door point, and the outer guards of the chain must move from the door in opposite directions. These outer guards must always remain on the boundary of the polygon. The search is complete once the chain lies entirely on a portion of the polygon boundary not containing the door point. Determining whether a polygon can be searched is a problem in the area of visibility in polygons; further to that, our work is related to the area of planning algorithms. We look for ways to find optimal schedules that minimize the distance or time required to complete the search. This is done by finding shortest paths in visibility diagrams that indicate valid positions for the guards. In the case of the two-guard room search, we are able to find the shortest distance schedule and the quickest schedule. The shortest distance schedule is found in O(n^2) time by solving an L_1 shortest path problem among curved obstacles in two dimensions. The quickest search schedule is found in O(n^4) time by solving an L_infinity shortest path problem among curved obstacles in two dimensions. For the chain of three guards, a search schedule minimizing the total distance travelled by the outer guards is found in O(n^6) time by solving an L_1 shortest path problem among curved obstacles in two dimensions.