Arno Eigenwillig

Fast and exact geometric analysis of real algebraic plane curves

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 curves 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.

A Computational Basis for Conic Arcs and Boolean Operations on Conic Polygons

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).

EXACUS: efficient and exact algorithms for curves and surfaces

We present the first release of the Exacus C++ libraries. We aim for systematic support of non-linear geometry in software libraries. Our goals are efficiency, correctness, completeness, clarity of the design, modularity, flexibility, and ease of use. We present the generic design and structure of the libraries, which currently compute arrangements of curves and curve segments of low algebraic degree, and boolean operations on polygons bounded by such segments.

Complete, exact, and efficient computations with cubic curves

The Bentley-Ottmann sweep-line method can be used to compute thearrangement of planar curves provided a number of geometricprimitives operating on the curves are available. We discuss themathematics of the primitives for planar algebraic curves of degreethree or less and derive efficient realizations. As a result, weobtain a complete, exact, and efficient algorithm for computingarrangements of cubic curves. Conics and cubic splines are specialcases of cubic curves. The algorithm is complete in that it handles all possibledegeneracies including singularities. It is exact in that itprovides the mathematically correct result. It is efficient in thatit can handle hundreds of curves with a quarter million of segmentsin the final arrangement.

Exact, efficient, and complete arrangement computation for cubic curves

The Bentley-Ottmann sweep-line method can compute the arrangement of planar curves, provided a number of geometric primitives operating on the curves are available. We discuss the reduction of the primitives to the analysis of curves and curve pairs, and describe efficient realizations of these analyses for planar algebraic curves of degree three or less. We obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Special cases of cubic curves are conics as well as implicitized cubic splines and Bezier curves. The algorithm is complete in that it handles all possible degeneracies such as tangential intersections and singularities. It is exact in that it provides the mathematically correct result. It is efficient in that it can handle hundreds of curves with a quarter million of segments in the final arrangement. The algorithm has been implemented in C++ as an Exacus library called CubiX.

Snap rounding of Bézier curves

We present an extension of snap roundingfrom straight-line segments (see Guibas and Marimont, 1998)to Bézier curves of arbitrary degree, and thus the first method for geometric roundingof curvilinear arrangements.Our algorithm takes a set of intersecting Bézier curvesand directly computes a geometric rounding of their true arrangement, without the need of representing the true arrangement exactly.The algorithms output is a deformation of the true arrangementthat has all Bézier control points at integer pointsand comes with the same geometric guarantees as instraight-line snap rounding: during rounding, objects do not movefurther than the radius of a pixel, and features of thearrangement may collapse but do not invert.

Multiplikation langer Zahlen (schneller als in der Schule)

Multiplizieren haben wir alle schon in der Grundschule gelernt. Um zwei ganze Zahlen a und b miteinander zu multiplizieren, multipliziert man a mit jeder Ziffer von b und arrangiert diese Teilprodukte in einem Stufenschema. Dann addiert man die Teilprodukte spaltenweise.

Multiplication of Long Integers - Faster than Long Multiplication

In this chapter the authors present an algorithm for fast multiplication that is much more efficient than the standard grade-school method, especially if one wants to multiply large numbers consisting of many digits. The authors present and analyze the efficiency of Karatsuba’s method – named after its inventor, he came up with the idea in the 1960s. The method exploits recursion, a fundamental technique in computer science, and it also involves the trick of dividing the problem into three subproblems of half the size.