Computational geometry

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

A novel model-based approach to 3D hand tracking from monocular video is presented. The 3D hand pose, the hand texture, and the illuminant are dynamically estimated through minimization of an objective function. Derived from an inverse problem formulation, the objective function enables explicit use of temporal texture continuity and shading information while handling important self-occlusions and time-varying illumination. The minimization is done efficiently using a quasi-Newton method, for which we provide a rigorous derivation of the objective function gradient. Particular attention is given to terms related to the change of visibility near self-occlusion boundaries that are neglected in existing formulations. To this end, we introduce new occlusion forces and show that using all gradient terms greatly improves the performance of the method. Qualitative and quantitative experimental results demonstrate the potential of the approach.

/pdf/model-based-3d-hand-pose-estimation-from-monocular-video-2b1afr8vel.pdf

Model-Based 3D Hand Pose Estimation from Monocular Video

https://hal.inria.fr/inria-00070350/document

Subdivision methods for solving polynomial equations

We report on the effects of a filter based on modular arithmetic that has been introduced recently into the EXACUS library. Our experiments with planar arrangements for curves up to degree four show that the exact construction and comparison of real algebraic numbers are some of the most time consuming operations when solving intersection problems for curved objects. In our experiments the modular filter accelerated the computation for arrangements of cubic curves by a factor of about 6.

/pdf/effective-computational-geometry-for-curves-and-surfaces-12shwzo0sp.pdf

Effective computational geometry for curves and surfaces

An algorithm is presented for the geometric analysis of an algebraic curve f(x, y) = 0 in the real affine plane. It computes a cylindrical algebraic decomposition (CAD) of the plane, augmented with adjacency information. The adjacency information describes the curve's topology by a topologically equivalent planar graph. The numerical data in the CAD gives an embedding of the graph.The algorithm is designed to provide the exact result for all inputs but to perform only few symbolic operations for the sake of efficiency. In particular, the roots of f(∝, y) at a critical x-coordinate .The algorithm is implemented as C++ library AlciX in the EXACUS project. Running time comparisons with top by Gonzalez-Vega and Necula (2002), and with cad2d by Brown demonstrate its efficiency.

https://domino.mpi-inf.mpg.de/intranet/ag1/ag1publ.nsf/20a0a8aecef76f33c12569e30040d63d/6b96a81443c11220c125733200398e3c/$FILE/ekw-fast-2007-authprep.pdf

Fast and exact geometric analysis of real algebraic plane curves

We give a unified ("basis free") framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes' rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = Eni=0aiXi with integer coefficients |ai|

https://domino.mpi-inf.mpg.de/intranet/ag1/ag1publ.nsf/d8ab60b7b27d8142c12569e30040d645/6e09cb7fbe62ea64c12571b20040a5d4/$file/esy-desctree-issac06-authprep.pdf

Almost tight recursion tree bounds for the Descartes method

We give an exact geometry kernel for conic arcs, algorithms for exact computation with low-degree algebraic numbers, and an algorithm for computing the arrangement of conic arcs that immediately leads to a realization of regularized boolean operations on conic polygons. A conic polygon, or polygon for short, is anything that can be obtained from linear or conic halfspaces (= the set of points where a linear or quadratic function is non-negative) by regularized boolean operations. The algorithm and its implementation are complete (they can handle all cases), exact (they give the mathematically correct result), and efficient (they can handle inputs with several hundred primitives).

A computational basis for conic arcs and boolean operations on conic polygons

The Descartes method is an algorithm for isolating the
 real roots of square-free polynomials with real coefficients. We assume
 that coefficients are given as (potentially infinite) bit-streams. In other
 words, coefficients can be approximated to any desired accuracy, but are not
 known exactly. We show that
 a variant of the Descartes algorithm can cope with bit-stream
 coefficients. To isolate the real roots of a
 square-free real polynomial $q(x) = q_nx^n+ldots+q_0$ with root
 separation $
ho$, coefficients $abs{q_n}ge1$ and $abs{q_i} le 2^	au$,
 it needs coefficient approximations to $O(n(log(1/
ho) + 	au))$
 bits after the binary point and has an expected cost of
 $O(n^4 (log(1/
ho) + 	au)^2)$ bit operations.

A Descartes Algorithms for Polynomials with Bit-Stream Coefficients

We show how to compute the planar arrangement induced by segments of arbitrary algebraic curves with the Bentley-Ottmann sweep-line algorithm. The necessary geometric primitives reduce to cylindrical algebraic decompositions of the plane for one or two curves. We compute them by a new and efficient method that combines adaptive-precision root finding (the Bitstream Descartes method of Eigenwillig et al., 2005) with a small number of symbolic computations, and that delivers the exact result in all cases. Thus we obtain an algorithm which produces the mathematically true arrangement, undistorted by rounding error, for any set of input segments. Our algorithm is implemented in the EXACUS library AlciX. We report on experiments; they indicate the efficiency of our approach.

https://domino.mpi-inf.mpg.de/intranet/ag1/ag1publ.nsf/d6dd8219a4a2a18cc12569e30040d655/30cb4a34db378e4ec12573700054479d/$file/ek-arrangements-2008-authprep.pdf

Arno Eigenwillig

Papers

Fast and exact geometric analysis of real algebraic plane curves

Almost tight recursion tree bounds for the Descartes method

A computational basis for conic arcs and boolean operations on conic polygons

A Descartes Algorithms for Polynomials with Bit-Stream Coefficients

Exact and efficient 2D-arrangements of arbitrary algebraic curves