Gudrun Albrecht

Construction of C 2 Pythagorean-hodograph interpolating splines by the homotopy method

The complex representation of polynomial Pythagorean-hodograph (PH) curves allows the problem of constructing aC2 PH quintic “spline” that interpolates a given sequence of pointsp0,p1,...,pN and end-derivativesd0 anddN to be reduced to solving a “tridiagonal” system ofN quadratic equations inN complex unknowns. The system can also be easily modified to incorporate PH-splineend conditions that bypass the need to specify end-derivatives. Homotopy methods have been employed to compute all solutions of this system, and hence to construct a total of 2N+1 distinct interpolants for each of several different data sets. We observe empirically that all but one of these interpolants exhibits undesirable “looping” behavior (which may be quantified in terms of theelastic bending energy, i.e., the integral of the square of the curvature with respect to arc length). The remaining “good” interpolant, however, is invariably afairer curve-having a smaller energy and a more even curvature distribution over its extent-than the corresponding “ordinary”C2 cubic spline. Moreover, the PH spline has the advantage that its offsets arerational curves and its arc length is apolynomial function of the curve parameter.

Construction of Be´zier rectangles and triangles on the symmetric Dupin horn cyclide by means of inversion

Abstract Dupin cyclides may be obtained as offsets of a special Dupin cyclide, the so-called symmetric Dupin horn cyclide. A novel approach based on the concept of inversion is presented for generating rational Bezier patches on the symmetric Dupin horn cyclide. This leads to a new formulation for rational rectangular biquadratic cyclide Bezier patches, and to a rational Bezier representation of triangular patches of degree 4 on the symmetric Dupin horn cyclide.

A note on Farin points for rational triangular Be´zier patches

A geometric way of defining the weights of a rational triangular Bezier patch is presented by introducing a new kind of auxiliary points for the patch. These points are situated along spiral-shaped polygons of control points of the patch.

Determination and classification of triangular quadric patches

Abstract A method for determining, if a given rational triangular Bezier patch of degree 2 lies on a quadric surface, and if so, for establishing the quadrics affine type, is presented. First, the question whether the patch is a quadric patch is solved by means of the related Veronese surface in five-dimensional projective space. Once established that the patch lies on a quadric the Gaussian curvature in one of the corner points of the patch is used for a rough classification yielding the projective type of the quadric. Then, the quadrics affine type is obtained by means of the quadrics intersection with the plane at infinity. An easy algorithm for the method is finally presented, together with several examples.

On the approximation order of tangent estimators

A classic problem in geometric modelling is curve interpolation to data points. Some of the existing interpolation schemes only require point data, whereas others, require higher order information, such as tangents or curvature values, in the data points. Since measured data usually lack this information, estimation of these quantities becomes necessary. Several tangent estimation methods for planar data points exist, usually yielding different results for the same given point data. The present paper thoroughly analyses some of these methods with respect to their approximation order. Among the considered methods are the classical schemes FMILL, Bessel, and Akima as well as a recently presented conic precision tangent estimator. The approximation order for each of the methods is theoretically derived by distinguishing purely convex point configurations and configurations with inflections. The approximation orders vary between one and four for the different methods. Numerical examples illustrate the theoretical results.

Eine Bemerkung zum Satz von Ivory

In 1809 J. IVORY found a distance relation between any two confocal ellipsoids in Euclidean space E3. After that, the “theorem of IVORY” has been formulated for any two nondegenerate confocal quadrics of the same type. In this paper we succeed in completing these results and in formulating and proving them uniformly for any two quadrics of the same type belonging to a general system of confocal quadrics.

Invariante Gütekriterien im Kurvendesign — Einige neuere Entwicklungen

This survey deals with invariance aspects for fairness criteria in the design of curves. It focuses on the developments of the past few years regarding both, direct or pointwise and indirect fairness criteria. Several techniques are reviewed that aim at controling the effect of parameter transformations and scalings of the curve and at achieving parameter invariance and/or scale invariance.

Applications of a Fruitful Relation between a Dupin Cyclide and a Right Circular Cone

A powerful way of handling a Dupin cyclide is presented. It is based on the concept of inversion, which yields a fruitful relation between the symmetric Dupin horn cyclide, from which all other Dupin cyclides may be obtained by offsetting, and a right circular cone. This relation has two important applications. First, it is used for constructing rational rectangular and triangular Bezier patches on the cyclide. Second, it allows to establish an approximative isometry between cyclide and cone patches, a useful result e.g. for scattered data interpolation techniques on Dupin cyclides.