Papers by Chanderjit Bajaj
On the duality of intersection and closest points
Geometric optimization and computational complexity
Our purpose here is to study problems involving geometric optimization, namely, questions of the ... more Our purpose here is to study problems involving geometric optimization, namely, questions of the type: Is there at least a minimum or at most a maximum number of certain geometric figures, that are within certain distances of other figures (objects). We are also concerned with the optimization of the size of these geometric figures. These problems arise as geometric reductions from various classes of location-allocation optimization problems and are inherently not pure combinatorial. Our primary aim, then, is to discover techniques of dealing with such geometric optimization problems, while adapting to these problems the older combinatorial design and analysis methods. The task of classifying problems accurately in the polynomial hierarchy is one of increasing importance. To solve an optimization problem deterministically it seems that one must solve both an $NP$ and a $Co-NP$ problem. The significance of the classes $NP$ and $Co-NP$ are that none of the problems they include is known to have a polynomial time solution. We show that if $NP \neq Co-NP$ then there are interesting natural geometric optimization problems (location-allocation problems under minsum) in $\Delta^{P}_{2}$ that are in neither $NP$ nor $Co-NP$. Hence, all these problems are shown to belong properly to $\Delta^{P}_{2}$, the second level of the polynomial hierarchy. We also show that if $NP \neq Co-NP$ then there are again some interesting geometric optimization problems (location-allocation problems under minmaz) properly in $\Delta^{P}_{2}$ and furthermore they are complete for a class $D^{P}$ (which is contained in $\Delta^{P}_{2}$ and contains $NP \bigcup Co-NP$). Also considered are the above geometric location-allocation optimization problems for the case when the allocation is predetermined. Both efficient algorithms and worst-case lower bounds are derived. Necessary conditions for the existence of mazima and minima in optimization problems are generally tied to the question of solvability of an equation or a system of equations. In calculus these equations are algebraic. By generating the minimal polynomial whose root over the field of rational numbers is the solution of the geometric optimization problem on the real (Euclidean) plane, we are able to prove the non-solvability of certain geometric optimization problems by radicals. The algebraic degree of the optimizing solution, which is the degree of the irreducible minimal polynomial for the problem, correlates with the inherent difficulty of constructing the solution and provides an algebraic complexity measure for these geometric optimization problems.
Computer-Aided Design, 1987
A Igorithms that can obtain rational and special parametric equations for degree three algebraic ... more A Igorithms that can obtain rational and special parametric equations for degree three algebraic curves (cubics) and degree three algebraic surfaces (cubicoids), given their implicit equations are described. These algorithms have been implemented on a VAX8600 using VAXIMA.
ACM Transactions on Graphics, 1989
For an irreducible algebraic space curve C that is implicitly defined as the intersection of two ... more For an irreducible algebraic space curve C that is implicitly defined as the intersection of two algebraic surfaces, f ( x , y , z ) = 0 and g ( x , y , z ) = 0, there always exists a birational correspondence between the points of C and the points of an irreducible plane curve P , whose genus is the same as that of C . Thus C is rational if the genus of P is zero. Given an irreducible space curve C = ( f ∩ g ), with f and g not tangent along C , we present a method of obtaining a projected irreducible plane curve P together with birational maps between the points of P and C . Together with [4], this method yields an algorithm to compute the genus of C , and if the genus is zero, the rational parametric equations for C . As a biproduct, this method also yields the implicit and parametric equations of a rational surface S containing the space curve C . The birational mappings of implicitly defined space curves find numerous applications in geometric modeling and computer graphics sin...
Data visualization techiques
Visualization Paradigms (C. Bajaj) Efficient Techniques for Volume Rendering of Scalar Fields (R.... more Visualization Paradigms (C. Bajaj) Efficient Techniques for Volume Rendering of Scalar Fields (R. Yagel) Accelerated Isocontouring of Scalar Fields (C. Bajaj, et al.) Surface Interrogation: Visualization Techniques for Surface Analysis (S. Hahmann) Vector Field Visualization Techniques (R. Crawfis & N. Max) Applications of Texture Mapping to Volume and Flow Visualization (N. Max, et al.) Continuous Bayesian Tissue Classification of Visualization (D. Laidlaw) References.
Proceedings of the fifth annual symposium on Computational geometry - SCG '89
We present a simple characterization of the lowest degree, implicitly defined, real algebraic sur... more We present a simple characterization of the lowest degree, implicitly defined, real algebraic surfaces, which smoothly contain any given number of points and algebraic space curves, of arbitrary degree. The characterization is constructive, yielding efficient algorithms for generating families of such algebraic surfaces. Smooth containment of space curves yields Cl-continuous surface fitting, and is a generalization of standard Hermite interpolation applied to fitting curves through point data, equating derivatives at those points. We deal with the containment and matching of "normals" (vectors orthogonal to tangents), possibly varying along the entire span of the space curves. Such Hermite interpolated surfaces prove useful as "blending" or "joining" surfaces for solid models as well as "fleshing" surfaces for curved wireframe models.
mut Alt (FU Berlin), Bernard Chazelle (Princeton University) and Emo Welzl (FU Berlin). The 31 pa... more mut Alt (FU Berlin), Bernard Chazelle (Princeton University) and Emo Welzl (FU Berlin). The 31 participants came from 8 countries, 12 of them came from North America and Israel. 29 lectures were given at the seminar, covering quite a number of topics in computational geometry. Unlike last year, there was no special concentration on any subject. In fact, there were talks on graph algorithms, parallel algorithms, motion planning, application-oriented problems, numerical robustness, similarity and congruence, randomized algorithms, dynamic algorithms, and a talk on implementations. As last year, an open problem session was held on Monday evening, chaired by Micha Sharir. It was stated that most of the problem discussed in last year's session had been solved (or at least some progress had been made). Let us hope that this yearr s session (reported here by Micha Sharir) will prove as fruitful. Berichterstatter: Otfried Schwarzkopf Participants: Helmut Alt, Freie
Proceedings. 1986 IEEE International Conference on Robotics and Automation
We describe an efficient parallel solution for the problem of jinding the shortest Euclidean path... more We describe an efficient parallel solution for the problem of jinding the shortest Euclidean path between two points in three dimensional space in the presence of polyhedral obstacles. We consider the important case where the order in which the obstacles are encountered in this shortest path is known. Inparticular for this case we describe an eficient parallel numerical iterative method on a concurrent-read exclusive-write synchronous shared-memory model. The iterations are essentially convergent non-linear block Gauss-Seidel. For special relative orientations of the, say n , polyhedral obstacles, we further describe a direct method that gives the exact solution in O (log n) time using n processors.
Curves and Surfaces in Computer Vision and Graphics II, 1992
We present two algorithms to construct C1-smooth models of skeletal structures from CT/NMR voxel ... more We present two algorithms to construct C1-smooth models of skeletal structures from CT/NMR voxel data. The boundary of the reconstructed models consist of a C1-continuous mesh of triangular algebraic surface patches. One algorithm first constructs Cl-continuous piecewise conic contours on each of the CT/NMR data slices and then uses piecewise triangular algebraic surface patches. to C 1 interpolate the contours on adjacent slices. The other algorithm works directly in voxel space and replaces an initial CO triangular facet approximation of the model with a highly compressed C1-continuous mesh of triangular algebraic surface patches. Both schemes are adaptive, yielding a higher density of patches in regions of higher curvature.
Proceedings of the fifth ACM symposium on Symbolic and algebraic computation - SYMSAC '86, 1986
We use simple argwnents from Galois theory to prove the impossibility of exact algorilhms for pro... more We use simple argwnents from Galois theory to prove the impossibility of exact algorilhms for problems under various models of computation. In particular we show that lhere exist applied computational problems for which there are no closed form solutions over models such as Q(+,", .,/, v), Q (+, _, '10, /, tv), and Q(+,-, '1<,/, k...J, q(x», where Q is the field of rationals and q(x)eQ[x] are polynomials with non-solvable Galois groups.
Rational hypersurface display
ACM SIGGRAPH Computer Graphics, 1990
Algorithms are presented for polygonalizing implicitly defined, quadric and cubic hypersurfaces i... more Algorithms are presented for polygonalizing implicitly defined, quadric and cubic hypersurfaces in n ≥ 3 dimensional space and furthermore displaying their projections in 3D. The method relies on initially constructing the rational parametric equations of the implicitly defined hypersurfaces, and then polygonalizing these hypersurfaces by an adaptive generalized curvature dependent scheme. The number of hyperpolygons used are optimal, in that they are the order of the minimum number required for a smooth Gouraud like shading of the hypersurfaces. Such hypersurface projection displays should prove useful in scientific visualization applications. The curvature dependent polygonal meshes produced, should also prove very useful in finite difference and finite element analysis programs for multi-dimensional domains.
Design and Implementation of Symbolic Computation Systems, 1990
The GANITH algebraic geometry toolkit manipulates arbitrary degree polynomials and power series. ... more The GANITH algebraic geometry toolkit manipulates arbitrary degree polynomials and power series. It can be used to solve a system of algebraic equations and visualize its multiple solutions. Example applications of this for geometric modeling and computer graphics are curve and surface display, curve-curve intersections, surface-surface intersections, global and local parameterizations, implicitizations, and inversions. It also incorporates techniques for multivariate interpolation and least-squares approximation to an arbitrary collection of points and curves.
Journal of Symbolic Computation, 1986
We explain how factoring polynomials modulo primes can be used in proving that for certain geomet... more We explain how factoring polynomials modulo primes can be used in proving that for certain geometric optimisation problems there exists no exact algorithm under models of computation where the root of an algebraic equation is obtained using arithmetic operations and the extraction of kth roots. This leaves only numerical or symbolic approximations to the solution of these problems under these models. This letter describes work which is described in more detail in Bajaj (1984)-here we concentrate on the use of computer algebra, in particular factoring polynomials over the rationals using the MACSVMA system. Consider the following geometric problem which is of fundamental importance with an equally long and interesting history. Simply stated one wishes to obtain the optimum location of a single source point in the plane, so that the sum of the Euclidean distances to n fixed destination points is a minimum. Given n fixed destination points in the plane with integer coordinates (at, bt), determine the opt#hum location (x, y) of a sfngle source point, that is minimisex.,f(x, y) = ~ x/(x-al)Z +(y-b~) 2. i=l .,.n Weber (1937) was probably the first who formulated this problem in light of the location of a plant, with the objective of minimising the sum of transportation costs from the plant to two sources of raw materials and a market centre. Hence this problem for n points has also come to be known as the Generalised Weber problem. In the decision version of this problem we ask if there exists (x, y) such that for given integer L, x/(x-a~)2 +(y-b,) 2 <~ L? i= l ,..n This problem is not even known to be in NP. Since on guessing a solution one then attempts to verify/f 2 L?, l=l...n in time polynomial in the number of bits needed to express certain rational numbers cl "" c,, and L. However, no such polynomial time algorithm is known (Graham, 1984; Odlyzko, 1985). Such a decision problem is fundamental in that it also occurs in numerous other geometric optimisation problems such as in finding the minimum length Euclidean Travelling Salesman Tour and the minimum length Euclidean Steiner Tree.
The International Journal of Robotics Research, 1988
We consider the problem of determining shortest paths in the presence of polyhe dral obstacles be... more We consider the problem of determining shortest paths in the presence of polyhe dral obstacles between two points in Euclidean three-space. For the special case when paths are constrained to the surfaces of three-dimensional objects, simple planar unfoldings are used to obtain the shortest path. For the general case when paths are not constrained to lie on any surface, we describe general ized unfoldings wherein the shortest path in three-space again becomes a straight line. These unfoldings of consist of multiple rotations about the edges of the polyhedral obstacles.
Information Processing Letters, 1987
Suppose we are gIven a set S of n (possibly intersecting) simple objects in the plane, such that ... more Suppose we are gIven a set S of n (possibly intersecting) simple objects in the plane, such that for every pair of objects in S, the intersection of the boundaries of these two objects has at most a connected components. The integer a is independent of n, Le. a.=O (1). \Ve consider the problem of detennining whether there exists a straight line that goes through every object in S. We give an 0 (n logn)'(n)) time algorithm for this problem, where y(n) is a very slowly growing function of n. If a<3 then our algorithm runs in 0 (n logn) time. Previously, only special cases of this problem were considered: In [6] the case when every object is a straight-line segment, in [2] the case when the objects are equal-radius circles and in [5] the case when objects all maintain the same orientation. All these cases follow from our general approach, which places no constraints on the size and/or configuration of the objects in S.
Discrete & Computational Geometry, 1989
In this paper we show a number of natural geometric optimization problems in the plane to be comp... more In this paper we show a number of natural geometric optimization problems in the plane to be complete for a class D p. The class D p contains both NP and Co-NP and is contained in A p = pNP. Completeness in D p is exhibited under many-one and positive reductions. Further an OptP(O(log n)) result is also obtained for some of these optimization problems.
Discrete & Computational Geometry, 1988
In this paper we apply Galois methods to certain fundamental geometric optimization problems whos... more In this paper we apply Galois methods to certain fundamental geometric optimization problems whose exact computational complexity has been an open problem for a long time. In particular we show that the classic Weber problem, along with the line-restricted Weber problem and its three-dimensional version are in general not solvable by radicals over the field of rationals. One direct consequence of these results is that for these geometric optimization problems there exists no exact algorithm under models of computation where the root of an algebraic equation is obtained using arithmetic operations and the extraction of kth roots. This leaves only numerical or symbolic approximations to the solutions, where the complexity of the approximations ia shown to be primarily a function of the algebraic degree of the optimum solution point.
This paper presents a scan curve algorithm for evaluating the face area of solids in constructive... more This paper presents a scan curve algorithm for evaluating the face area of solids in constructive solid geometry (CSG) representation. Compared to previous methods, the algorithm is more accurate and computationally faster. The applicable domain is limited to solids bounded by three classes of surfaces: all quadric surfaces, cylindrical surfaces, and surfaces of revolution which are algebraic surfaces with rational parametric equations for their generating curves. The algorithm has been implemented in FORTRAN 77 on a VAX 11/780 machine. The extensions of this algorithm may also be applied to the solution of the following three problems: (1) Boundary representation (BREP) evaluation from CSGj (2) Face area evaluation for solids in BREPj and (3) Triangulation of the faces of solids in eSG or BREP.
This paper uses some well known theorems of algebraic geometry to characterize polynomial Hermite... more This paper uses some well known theorems of algebraic geometry to characterize polynomial Hermite interpolation in any dimension. Efficient numerical algorithms are presented for interpolatory curve fits through points in the plane, surface fits through points and curves in space, and in general, hypersuface fits through. points, curves, surfaces, and sub-varieties in n dimensional space. These interpolatory fits may also be made to match derivative information at the data points.
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Papers by Chanderjit Bajaj