Wolfgang Boehm

Inserting new knots into B-spline curves

Abstract For some applications, further subdivision of a segment of a B-spline curve or B-spline surface is desirable. This paper provides an algorithm for this. The structure is similar to de Boors algorithm for the calculation of a point on a curve. An application of the subdivision is illustrated.

Bezier and B-Spline Techniques

1 Geometric fundamentals.- 2 Bezier representation.- 3 Bezier techniques.- 4 Interpolation and approximation.- 5 B-spline representation.- 6 B-spline techniques.- 7 Smooth curves.- 8 Uniform subdivision.- 9 Tensor product surfaces.- 10 Bezier representation of triangular patches.- 11 Bezier techniques for triangular patches.- 12 Interpolation.- 13 Constructing smooth surfaces.- 14 Gk-constructions.- 15 Stationary subdivision for regular nets.- 16 Stationary subdivision for arbitrary nets.- 17 Box splines.- 18 Simplex splines.- 19 Multivariate splines.- References.

Geometric concepts for geometric design

This book is a comprehensive tool both for self-study and for use as a text in classical geometry. It explains the concepts that form the basis for computer-aided geometric design.

The insertion algorithm

Abstract A complete formulation is given of a multiple knot insertion algorithm. The presented algorithm compares favourably with other existing multiple knot insertion algorithms.

On cyclides in geometric modeling

Abstract Dupins cyclides are useful in blending conventional solids in solid modeling. Most of the properties of cyclides can easily be derived from a construction using an ellipse and a string as given 125 years ago by J. Clerk Maxwell. It covers the construction of the Bezier presentation of a simple basic patch as well as the double-cyclid blend of two conics, a solution of the Cranfield problem and the blend of a tripod.

On de Casteljau's algorithm

Within the last 20 years de Casteljaus algorithms became a fundamental tool in CAGD. His idea of control points and his geometric view of polar forms gave an immediate insight to how these tools work and control points are so effective in their applications in car and ship design, in aircraft industry as well as in medical and geological representations. This paper is meant to give some historical remarks and a short introduction to de Casteljaus powerful and worldwide used method. It also includes simple proofs.

On the efficiency of knot insertion algorithms

The principal differences of knot insertion algorithms are discussed. Their efficiencies are compared.

Calculating with box splines

Box splines are multivariate splines over regular grids. Two recursion formulas for box splines are developed: (1) a Mansfield-de Boor-like expression of box splines as linear combinations of box splines of lower degree and (2) a deBoor-like reduction of the net of box spline control points. The ideas follow those from the paper by deBoor in 1972. The proofs are geometrical and simple.

On de Boor-like algorithms and blossoming

Abstract Algorithms that use repeated linear interpolation of points, such as the de Boor algorithm but also the Aiken-Neville algorithm, will be considered. Their common geometric structure will be discussed from a more general viewpoint and its relationship with the so-called blossoming will be described.

Be´zier presentation of airfoils

Abstract Bezier presentations and cubic spline presentations of NACA-4-digit airfoils are developed and their errors are discussed. The given representations may help to use the Bernstein-Bezier technique in the design process of wings.