Bert Jüttler

THB-splines: The truncated basis for hierarchical splines

The construction of classical hierarchical B-splines can be suitably modified in order to define locally supported basis functions that form a partition of unity. We will show that this property can be obtained by reducing the support of basis functions defined on coarse grids, according to finer levels in the hierarchy of splines. This truncation not only decreases the overlapping of supports related to basis functions arising from different hierarchical levels, but it also improves the numerical properties of the corresponding hierarchical basis - which is denoted as truncated hierarchical B-spline (THB-spline) basis. Several computed examples will illustrate the adaptive approximation behavior obtained by using a refinement algorithm based on THB-splines.

An algebraic approach to curves and surfaces on the sphere and on other quadrics

Abstract An explicit representation for any irreducible rational Bezier curve and Bezier surface patch on the unit sphere is given. The extension to general quadrics (ellipsoids, hyperboloids, paraboloids) is outlined. The construction is based on an algebraic result concerning Pythagorean quadruples in polynomial rings and can be additionally interpreted as a generalized stereographic projection onto the sphere. This projection is shown to be the composition of a hyperbolic projection (a special net projection) with a stereographic projection. The investigation of its properties leads to new results for the biquadratic Bezier patch on the sphere. Further attention is payed to the interpolation of a given point set with a spherical rational curve. The results are extended to rational B-spline curves and tensor product B-spline surfaces.

Computation of rotation minimizing frames

Due to its minimal twist, the rotation minimizing frame (RMF) is widely used in computer graphics, including sweep or blending surface modeling, motion design and control in computer animation and robotics, streamline visualization, and tool path planning in CAD/CAM. We present a novel simple and efficient method for accurate and stable computation of RMF of a curve in 3D. This method, called the double reflection method, uses two reflections to compute each frame from its preceding one to yield a sequence of frames to approximate an exact RMF. The double reflection method has the fourth order global approximation error, thus it is much more accurate than the two currently prevailing methods with the second order approximation error—the projection method by Klok and the rotation method by Bloomenthal, while all these methods have nearly the same per-frame computational cost. Furthermore, the double reflection method is much simpler and faster than using the standard fourth order Runge-Kutta method to integrate the defining ODE of the RMF, though they have the same accuracy. We also investigate further properties and extensions of the double reflection method, and discuss the variational principles in design moving frames with boundary conditions, based on RMF.

Visualization of moving objects using dual quaternion curves

Abstract The interpolation of some positions (= point + orientation) of a moving object is examined with help of dual quaternion curves. In order to apply the powerful methods of computer-aided geometric design, an interpolating motion whose trajectories are rational Bezier curves is constructed. The interpolation problem is discussed from a mechanical and a geometrical viewpoint. A representation formula for rational motions of fixed order is presented. Finally, the construction of rational spline motions is outlined. Dual quaternions prove to be very useful in computer graphics.

Swept Volume Parameterization for Isogeometric Analysis

Isogeometric Analysis uses NURBS representations of the domain for performing numerical simulations. The first part of this paper presents a variational framework for generating NURBS parameterizations of swept volumes. The class of these volumes covers a number of interesting free-form shapes, such as blades of turbines and propellers, ship hulls or wings of airplanes. The second part of the paper reports the results of isogeometric analysis which were obtained with the help of the generated NURBS volume parameterizations. In particular we discuss the influence of the chosen parameterization and the incorporation of boundary conditions.

Least-squares fitting of algebraic spline surfaces ∗

We present an algorithm for fitting implicitly defined algebraic spline surfaces to given scattered data. By simultaneously approximating points and associated normal vectors, we obtain a method which is computationally simple, as the result is obtained by solving a system of linear equations. In addition, the result is geometrically invariant, as no artificial normalization is introduced. The potential applications of the algorithm include the reconstruction of free-form surfaces in reverse engineering. The paper also addresses the generation of exact error bounds, directly from the coefficients of the implicit representation.

Strongly stable bases for adaptively refined multilevel spline spaces

The problem of constructing a normalized hierarchical basis for adaptively refined spline spaces is addressed. Multilevel representations are defined in terms of a hierarchy of basis functions, reflecting different levels of refinement. When the hierarchical model is constructed by considering an underlying sequence of bases {Γℓ}ℓ=0,…,N−1

Hermite interpolation by pythagorean hodograph curves of degree seven

\{\Gamma ^{\ell }\}_{\ell =0,\ldots ,N-1}

Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling

with properties analogous to classical tensor-product B-splines, we can define a set of locally supported basis functions that form a partition of unity and possess the property of coefficient preservation, i.e., they preserve the coefficients of functions represented with respect to one of the bases Γℓ

Medial axis computation for planar free-form shapes

\Gamma ^{\ell }