Rida T. Farouki

On the numerical condition of polynomials in Berstein form

Abstract The Bernstein-Bezier curve and surface forms have enjoyed considerable popularity in computer aided design applications, due to their elegant geometric properties and the simple recursive algorithms available for processing them. In this paper we describe a different and largely unexplored favorable aspect of Bernstein-Bezier methods: their inherent numerical stability under the influence of imprecise computer arithmetic or perturbed input data. We show that the condition numbers of the simple real roots of a polynomial on the unit interval are always smaller in the Bernstein basis than in the power basis. These arguments are then generalized to accommodate multiple roots and roots on an arbitrary interval. We further show that the standard Bernstein polynomial subdivision and degree elevation procedures induce a strictly monotonic decrease in root condition numbers. Finally, we prove that among a large family of polynomial bases characterized by certain simple properties, the Bernstein basis exhibits optimal root conditioning. Some well-known examples are employed to illustrate these results, and their practical implications for floating-point implementation of curve and surface intersection algorithms in a geometric modeling system are discussed. Important issues requiring further attention in the processing of polynomials in Bernstein form are also briefly indicated.

Algorithms for polynomials in Bernstein form

Abstract Aspects of the formulation and numerical stability of geometric modeling algorithms in the Bernstein polynomial basis are further explored. Bernstein forms for various basic polynomial procedures (the arithmetic operations, substitution of polynomials, elimination of variables, determination of greatest common divisors, etc.) required in such algorithms are developed, and are found to be of similar complexity to their customary power forms. This establishes the viability of systematic computation with the Bernstein form, avoiding the need for (numerically unstable) basis conversions. Procedures which nominally improve root condition numbers—the power-to-Bernstein conversion and Bernstein degree elevation and subdivision techniques—are examined in greater detail. These procedures are found to have a relatively poor inherent conditioning, which essentially cancels the expected improvement, and their explicit use does not therefore provide a means of enhancing numerical stability. We also examine the condition of computation in floating point arithmetic of the power and Bernstein formulations for the basic polynomial procedures. However, the detailed dependence of the computational errors on the input parameters, the algorithm structure, and the floating point system preclude comparisons as definitive and general as those pertaining to the root condition numbers in the power and Bernstein bases. Empirical evaluation of carefully selected test cases may provide firmer conclusions in this regard.

Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable

By virtue of their special algebraic structures, Pythagorean-hodograph (PH) curves offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. This book offers a comprehensive and self-contained treatment of the mathematical theory of PH curves, including algorithms for their construction and examples of their practical applications. Special features include an emphasis on the interplay of ideas from algebra and geometry and their historical origins, detailed algorithm descriptions, and many figures and worked examples. The book may appeal, in whole or in part, to mathematicians, computer scientists, and engineers.

The Bernstein polynomial basis: A centennial retrospective

One hundred years after the introduction of the Bernstein polynomial basis, we survey the historical development and current state of theory, algorithms, and applications associated with this remarkable method of representing polynomials over finite domains. Originally introduced by Sergei Natanovich Bernstein to facilitate a constructive proof of the Weierstrass approximation theorem, the leisurely convergence rate of Bernstein polynomial approximations to continuous functions caused them to languish in obscurity, pending the advent of digital computers. With the desire to exploit the power of computers for geometric design applications, however, the Bernstein form began to enjoy widespread use as a versatile means of intuitively constructing and manipulating geometric shapes, spurring further development of basic theory, simple and efficient recursive algorithms, recognition of its excellent numerical stability properties, and an increasing diversification of its repertoire of applications. This survey provides a brief historical perspective on the evolution of the Bernstein polynomial basis, and a synopsis of the current state of associated algorithms and applications.

Surface Analysis Methods

Parametric polynomial surfaces defined to satisfy given interpolatory or boundary constraints often suffer extraneous undesired features because of poor control over their many degrees of freedom. Current techniques for detecting and remedying such unexpected surface characteristics are primitive and inadequate. This article describes several surface analysis tools useful in detecting such anomalous surface features. These include standard techniques such as contouring and high-resolution shaded image displays based on direct ray tracing, and some novel methods such as maps of the principal curvatures, the integration of lines of curvature to show the variation ofthe principal directions, and the determination of geodesic paths on the surface.

The approximation of non-degenerate offset surfaces

Abstract This paper describes algorithms for approximating the offsets to general piecewise parametric surfaces by networks of bicubic patches. The surface to be offset may be composed of arbitrary parametric patches, provided precise tangent continuity obtains across all patch boundaries and the surface metric is everywhere non-singular (i.e., a unique surface normal is defined). The approximation scheme consists of three stages: (1) A differential surface analysis is performed to ascertain the extremum principal radii of curvature for each patch — this constrains the offset magnitude that may be specified without degeneration of the offset. (2) The parametric domain of each patch is sub-divided, and a bicubic approximant to the offset for each sub-domain is computed. (3) A tolerance analysis is performed on each patch of the offset approximation to evaluate its extremum deviations from the true offset. The accuracy of the offset approximation increases with the degree of subdivision of the parametric domain of each patch. Fractional errors in the range 10 −2 to 10 −3 are typical for two- or three- fold sub-division of each parametric variable (four- and nine- fold area division respectively), provided the surface is smooth and its smallest concave radius of curvature considerably exceeds the specified offset magnitude. A fully automatic tolerance-based surface offset capability may be developed by providing feedback between stages (3) and (2), successive degrees of parametric sub-division being determined by the errors from prior approximations until the desired accuracy is achieved.

Analytic properties of plane offset curves

Abstract We survey the principal geometric and topological features of plane offset curves. With appropriate sign conventions, the irregular points of the offset at distance d from a regular generator curve arise where the generator has curvature κ=− 1 d . Usually, this induces a cusp on the offset, but if κ is also a local extremum, we observe instead a tangent-continuous extraordinary point of infinite curvature. Such irregular points are intimately related to the evolute, or locus of centers of curvature, of the generator. Certain special regular points are then identified: those of horizontal or vertical tangent, and those where the curvature or its derivative vanish. A one-to-one correspondence (with due allowance for irregularities) is established between such characteristic points on the generator and its offsets at each distance d. In the absence of irregular points, simple relations between certain global properties of the generator and offset curves, such as their arc length, the area they bound, and their mean square curvature or “smoothness” may be derived. The self-intersections of offset curves, and the trimming of certain extraneous loops they delineate, are also addressed.

Exact Taylor series coefficients for variable-feedrate CNC curve interpolators

Abstract Recent papers [Yang DCH, Kong T. Computer Aided Design 1994;26:225–34; Yeh S-S, Hsu P-L. Computer Aided Design 1999;31:349–57] formulate real-time CNC interpolators for variable feedrates along parametric curves. These interpolators employ truncated Taylor series to compute successive reference-point parameter values, but in both papers an erroneous coefficient for the highest (quadratic) term is cited. The derivation of the proper coefficients is a straightforward, although somewhat convoluted, exercise in chain-rule differentiation. Compact recursive formulae are presented here to compute the correct coefficients, up to the cubic term, in cases where the feedrate depends on: (i) elapsed time; (ii) curve arc length; or (iii) local path curvature. The local and cumulative effects of truncation errors on the accuracy of such interpolator schemes are also assessed, and compared with the essentially exact interpolators for Pythagorean-hodograph curves.

On the optimal stability of the Bernstein basis

We show that the Bernstein polynomial basis on a given interval is optimally stable, in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval. This result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations. We further show, by means of some low-degree examples, that the Bernstein form is not uniquely optimal in this respect. However, it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval. These ideas are illustrated by comparing the stability properties of the power, Bernstein, and generalized Ball bases.

Algebraic properties of plane offset curves

Abstract We consider the offsets to a polynomial or rational parametric generator r(t) as algebraic curves, specified by implicit equations ƒ o (x, y)=0 . Efficient resultant formulations for computing these irreducible equations—which represent simultaneously the ‘exterior’ and ‘interior’ offsets—are presented, and certain special conditions, which cause the degree of ƒ o (x, y) to be diminished, are identified. These include the presence of cusps, tangencies to the line at infinity, and passages through the circular points at infinity. (In the case of a rational generator with cusps or tangencies to the line at infinity coincident with the circular points, we remedy an apparent defect in the classical degree formula due to Cayley.) For polynomial or rational generators, the theory of singular points of algebraic curves offers further insight into the nature of the ‘cusps’ and ‘extraordinary points’ of offset curves, which we have previously described from a purely analytic perspective. Algebraic methods based on polynomial resultant and real-root isolation procedures also furnish an algorithmic basis for identifying the self-intersections of offset curves, and hence computing the trimmed offsets which are desired in most practical applications.