Grundlagen der geometrischen Datenverarbeitung

Abstract

Das Werk einschlief3lich aller seiner Teile ist urheberrechtlich geschiitzt. Jede Verwertung auf3erhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlages unzuliissig und stratbar. Das gilt besonders fiir Vervielfiiltigungen, Obersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen.

An adaptive hierarchical approximation method on the sphere using axisymmetric locally supported basis functions

“…~) : 12 ~ R, ~ ~ F The reader acquainted with scattered data methods will recognize that the above method is similar to the Shepard method used in meteorology and geology (see e.g. [14]). In contrast to Shepard's algorithm the above method is developed on the sphere and does not aim at interpolating the data but at approximation.…”

Rational Ruled Surfaces and Their Offsets

“…In the following, a(u) and g(u) are always assumed to be rational in u. Denote by n(u, v) the unit normal of R S ϭ ((r 0 ϫ r 1 ) 2 , (r 0 , r 1 , r 1 )r 0 ϩ (r 0 , r 0 , r 1 )r 1 (9) at r(u, v), i.e., ϩ (͗r 0 , r 1 ͘ Ϫ ͗r 1 , r 0 ͘)(r 0 ϫ r 1 )). Here, (a, b, c) denotes the determinant of the three vectors.…”

“…degree reductions occur if the six univariate polynomials Are there computational advantages of the form (6) over in R(t) have a common zero (over ‫. )ރ‬ This happens if at the standard CAGD representation of a ruled surface as least one of the t-lines can be degree reduced or if the a rational tensor product surface [7][8][9], t-lines s ϭ 0 and s ϭ 1 possess a common point (over ‫;ރ‬ then all t-lines pass through this point).…”

Transfinite interpolation on the medians of a triangle and best L1-approximation

“…Their main application is in the construction of smooth surfaces over triangulations or rectangulations in terms of so-called Gordon-Coons patches, see [6,Chapter 8], which is a widely used method in Computer Aided Geometric Design. There, transfinite interpolation is performed along the perimeter of the basic elements, a method which in case of triangles was established by Barnhill, Birkhoff and Gordon in [1], see also Example 3 in [4].…”