11138955
Computing an Almost Minimum Set of Spanning Line Segments of a Polyhedron
A set of spanning line segments (SLS) is a subset of the edges of a finite polyhedron in E3 such that an arbitrary plane intersects the polyhedron if and only if the plane intersects at least one of the line segments of the SLS. In this paper an algorithm is presented for computing an almost minimum set of spanning line segments for an arbitrary polyhedron . When the number of extreme vertices of is odd, the computed SLS is minimum; when the number of extreme vertices of is even, the size of the computed SLS is at most the minimum size plus one. The algorithm has linear-time complexity for a convex polyhedron, hence is optimal in this case; its time complexity is Θ(m log m) for an arbitrary polyhedron, where m is the number of vertices of the polyhedron.
2992545
Constrained 3D shape reconstruction using a combination of surface fitting and registration
We investigate 3D shape reconstruction from measurement data in the presence of constraints. The constraints may fix the surface type or set geometric relations between parts of an object's surface, such as orthogonality, parallelity and others. It is proposed to use a combination of surface fitting and registration within the geometric optimization framework of squared distance minimization (SDM). In this way, we obtain a quasi-Newton like optimization algorithm, which in each iteration simultaneously registers the data set with a rigid motion to the fitting surface and adapts the shape of the fitting surface. We present examples to show the applicability of our method to constrained 3D shape fitting for reverse engineering of CAD models and to high accuracy fitting with kinematic surfaces, which include surfaces of revolution (reconstructed from fragments of archeological pottery) and spiral surfaces, which are fitted to 3D measurement data of shells. Our optimization algorithm can combine registration of multiple scans of an object and model fitting into a single optimization process which is shown to be superior to the traditional procedure, which first registers the data and then fits a model to it.
14471402
Computing quadric surface intersections based on an analysis of plane cubic curves
Computing the intersection curve of two quadrics is a fundamental problem in computer graphics and solid modeling. We present an algebraic method for classifying and parameterizing the intersection curve of two quadric surfaces. The method is based on the observation that the intersection curve of two quadrics is birationally related to a plane cubic curve. In the method this plane cubic curve is computed first and the intersection curve of the two quadrics is then found by transforming the cubic curve by a rational quadratic mapping. Topological classification and parameterization of the intersection curve are achieved by invoking results from algebraic geometry on plane cubic curves. 2003 Elsevier Science (USA). All rights reserved.
45794206
Rational Quadratic Parameterizations of Quadrics
Every irreducible quadric in E3 has infinitely many different rational quadratic parameterizations. These parameterizations and the relationships between them are investigated. It is shown that every faithful rational quadratic parameterization of a quadric can be generated by a stereographic projection from a point on the quadric, called the center of projection (COP). Two such parameterizations for the same quadric are related by a rational linear reparameterization if they have the same COP; otherwise they are related by a rational quadratic reparameterization. We also consider unfaithful parameterizations for which, in general, a one-to-one correspondence between points on the surface and parameters in the plane does not exist. It is shown that all unfaithful rational quadratic parameterizations of a properly degenerate quadric can be characterized by a simple canonical form, and there exist no unfaithful rational quadratic parameterizations for a nondegenerate quadric. In addition, given a faithful rational quadratic parameterization of a quadric, a new technique is presented to compute its base points and inversion formula. These results are applied to solve the problems of parameterizing the intersection of two quadrics and reparameterizing a given quadric parameterization with respect to a different COP without implicitization.
7101460
On the Difference Method for Drawing Conic Arcs
Abstract The difference method is an efficient way to generate an inscribed polygon of a conic section. We address the implementation issues of the difference method, that is, how to choose the difference parameter and initial points so that the inscribed polygon has a minimal number of line segments and is within a given tolerance of the conic arc. We also discuss how the difference method can be used to draw conic arcs represented by rational quadratic Bezier curves.
16675497
Revisiting the [mu]-basis of a rational ruled surface
The µ-basis of a rational ruled surface P(s,t) = P0(s) + tP1(s) is defined in Chen et al. (Comput. Aided Geom. Design 18 (2001) 61) to consist of two polynomials p(x, y, z, s) and q(x, y, z, s) that are linear in x, y, z. It is shown there that the resultant of p and q with respect to s gives the implicit equation of the rational ruled surface; however, the parametric equation P(s, t) of the rational ruled surface cannot be recovered from p and q. Furthermore, the µ-basis thus defined for a rational ruled surface does not possess many nice properties that hold for the µ-basis of a rational planar curve (Comput. Aided Geom. Design 18 (1998) 803). In this paper, we introduce another polynomial r(x, y, z, s, t) that is linear in x, y, z and t such that p, q, r can be used to recover the parametric equation P(s, t) of the rational ruled surface; hence, we redefine the µ-basis to consist of the three polynomials p, q, r. We present an efficient algorithm for computing the newly-defined µ-basis, and derive some of its properties. In particular, we show that the new µ-basis serves as a basis for both the moving plane module and the moving plane ideal corresponding to the rational ruled surface.
41898252
Vertex-pancyclicity of edge-face-total graphs
The edge-face-total graph r(G) of a plane graph G is the graph defined on the vertex set E(G)∪F(G) so that two vertices in r(G) are joined by an edge if and only if they were adjacent or incident in G. In this paper we prove that (1) the edge-face-total graph of a 2-connected plane graph is vertex-pancyclic and there exists a connected plane graph G with cut vertices such that r(G) is non-Hamiltonian; (2) the line graph of a 2-connected plane graph with at most one face of degree ⩾4 is pancyclic.
15064123
A new proof of Melnikov's conjecture on the edge-face coloring of plane graphs
In 1975, Melnikov conjectured that the edges and faces of each plane graph G may be colored with Δ(G)+3 colors so that any two adjacent or incident elements receive different colors, where Δ(G) is the maximum degree of G. Two similar, yet independent, proofs of this conjecture have been published recently by Waller (J. Combin. Theory Ser. B 69 (1997) 219) and Sanders and Zhao (Combinatorica 17 (1997) 441). Both proofs made use of the Four-Color Theorem. This paper presents a new proof of Melnikov's conjecture independent of the Four-Color Theorem.
9952702
Equitable Total Coloring of Graphs with Maximum Degree 3
Abstract. The equitable total chromatic number χrdquo;e(G) of a graph G is the smallest integer k for which G has a total k-coloring such that the number of vertices and edges in any two color classes differ by at most one. We prove in this paper that χrdquo;e(G)≤5 if G is a multigraph with maximum degree at most 3.
5167032
AGILE: a general approach to detect transitions in evolving data streams
In many applications such as e-commerce, system diagnosis and telecommunication services, data arrives in streams at a high speed. It is common that the underlying process generating the stream may change over time, either as a result of the fundamental evolution or in response to some external stimulus. Detecting these changes is a very challenging problem of great practical importance. The overall volume of the stream usually far exceeds the available main memory and access to the data stream is typically performed via a linear scan in ascending order of the indices of the records. In this paper, we propose a novel approach, AGILE, to monitor streaming data and to detect distinguishable transitions of the underlying processes. AGILE has many advantages over the traditional Hidden Markov Model, e.g., AGILE only requires one scan of the data.
16100691
Design multicast protocols for non-cooperative networks
Conventionally, most network protocols assume that the network entities that participate in the network activities will always behave as instructed. However, in practice, most network entities will try to maximize their own benefits instead of altruistically contribute to the network by following the prescribed protocols, which is known as selfish. Thus, new protocols should be designed for the non-cooperative network, which is composed of selfish entities. In this paper, we specifically show how to design strategyproof multicast protocols for non-cooperative networks such that these selfish entities will follow the protocols out of their own interests. By assuming that a group of receivers is willing to pay to receive the multicast service, we specifically give a general framework to decide whether it is possible, and how if possible to transform an existing multicast protocol to a strategyproof multicast protocol. We then show how the payments to those relay entities are shared fairly among all receivers so that it encourages collaboration among receivers. As a running example, we show how to design the strategyproof multicast protocol for the currently used core-based multicast structure. We also conduct extensive simulations to study the relations between payment and cost of the multicast structure.