Algorithmic Motion Planning and Related Geometric Problems on Parallel Machines (Dissertation Proposal)

The problem of algorithmic motion planning is one that has received considerable attention in recent years. The automatic planning of motion for a mobile object moving amongst obstacles is a fundamentally important problem with numerous applications in computer graphics and robotics. Numerous approximate techniques (AI-based, heuristics-based, potential field methods, for example) for motion planning have long been in existence, and have resulted in the design of experimental systems that work reasonably well under various special conditions [7, 29, 30]. Our interest in this problem, however, is in the use of algorithmic techniques for motion planning, with provable worst case performance guarantees. The study of algorithmic motion planning has been spurred by recent research that has established the mathematical depth of motion planning. Classical geometry, algebra, algebraic geometry and combinatorics are some of the fields of mathematics that have been used to prove various results that have provided better insight into the issues involved in motion planning [49]. In particular, the design and analysis of geometric algorithms has proved to be very useful for numerous important special cases. In the remainder of this proposal we will substitute the more precise term of algorithmic motion planning by just motion planning

Gap Processing for Adaptive Maximal Poisson-Disk Sampling

In this paper, we study the generation of maximal Poisson-disk sets with varying radii. First, we present a geometric analysis of gaps in such disk sets. This analysis is the basis for maximal and adaptive sampling in Euclidean space and on manifolds. Second, we propose efficient algorithms and data structures to detect gaps and update gaps when disks are inserted, deleted, moved, or have their radius changed. We build on the concepts of the regular triangulation and the power diagram. Third, we will show how our analysis can make a contribution to the state-of-the-art in surface remeshing.Comment: 16 pages. ACM Transactions on Graphics, 201

Optimal Mesh Algorithms for the Voronoi Diagram of Line Segments, Visibility Graphs and Motion Planning in the Plane

The motion planning problem for an object with two degrees of freedom moving in the plane can be stated as follows: Given a set of polygonal obstacles in the plane, and a two-dimensional mobile object B with two degrees of freedom, determine if it is possible to move B from a start position to a final position while avoiding the obstacles. If so, plan a path for such a motion. Techniques from computational geometry have been used to develop exact algorithms for this fundamental case of motion planning. In this paper we obtain optimal mesh implementations of two different methods for planning motion in the plane. We do this by first presenting optimal mesh algorithms for some geometric problems that, in addition to being important substeps in motion planning, have numerous independent applications in computational geometry. In particular, we first show that the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane can be constructed in O(√n) time on a √n x √n mesh, which is optimal for the mesh. Consequently, we obtain an optimal mesh implementation of the sequential motion planning algorithm described in [14]; in other words, given a disc B and a polygonal obstacle set of size n, we can plan a path (if it exists) for the motion of B from a start position to a final position in O (√n) time on a mesh of size n. Next we show that given a set of n line segments and a point p, the set of segment endpoints that are visible from p can be computed in O (√n) mesh-optimal time on a √n x √n mesh. As a result, the visibility graph of a set of n line segments can be computed in O(n) time on an n x n mesh. This result leads to an O(n) algorithm on an n x n mesh for planning the shortest path motion between a start position and a final position for a convex object B (of constant size) moving among convex polygonal obstacles of total size n

Real-Time Cave Destruction Using 3D Voronoi

Arvutimängudes kasutatakse keskkonna muutmiseks enamasti lihtsaid meetodeid, nagu maailma kujutamist vokslitena, või eelkalkuleeritud hävitamist. Käesolevas töös uuritakse, kuidas muuta seda reaalsemaks, kasutades Voronoi diagramme. Selles lähenemises kujutatakse kogu maailma ühe 3D Voronoi diagrammina, millesse lisatud koopad on saadud Voronoi rakkude eemaldamise teel. Töö eesmärgiks on leida sobivad algoritmid sellise koopa genereerimiseks, võrrelda nende sobivust ja luua prototüüprakendus Unity mängumootoris, millega testida, kas selline lähenemine on mõistlik. Selles simulatsioonis saab kasutaja mõjutada koobast, lõigates sealt tükke välja ning seeläbi suurendades Voronoi diagrammi reaalajas. Töös uuritakse ka erinevaid lähenemisi juba olemasolevast geomeetriast tükkide välja lõikamiseks ja vaadeldakse erinevaid algoritme geomeetria manipuleerimiseks.Environment modification in video games are often done by using simple methods like voxels or pre-calculated destruction. The aim of this thesis is to study different ways of making it more realistic by generating the environment destruction in real time using Voronoi diagrams. This approach represents the world as a 3D Voronoi diagram where the cave is represented as a region where some of the Voronoi cells have been removed. The goal of this thesis is to find the suitable algorithms for such cave generation, compare them and implement a proof of concept simulation in Unity game engine. In this simulation the user can modify the cave by cutting out more pieces, thus expanding the Voronoi diagram in real-time. To cut off pieces of already fixed geometry different approaches for geometry manipulation are also compared