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January 1996
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436 Reads
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1,188 Citations
Mathematics of Computation
A classic reference and text, this book introduces the foundations used to create an accurate computer screen image using mathematical tools. This comprehensive guide is a handbook for students and practitioners and includes an extensive bibliography for further study.
April 1995
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11 Reads
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7 Citations
Computers & Mathematics with Applications
Free form volumes in rational Bézier representation are derived via homogeneous coordinates. Some properties and constructions are presented and two applications of free form volumes are discussed: definition of solid primitives and curve and surface modelling by the way of volume deformation.
January 1993
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10 Reads
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4 Citations
The composition of Bézier curves and tensor product Bézier surfaces, polynomial as well as rational, is applied to exactly and explicitly represent trim curves of tensor product Bézier surfaces. Trimming curves are assumed to be defined as Bézier curves in surface parameter domain. A Bézier spline approximation of lower polynomial degree is built up as well, which is based on the exact trim curve representation in coordinate space.
June 1992
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14 Reads
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6 Citations
Computers & Mathematics with Applications
We present a B-spline-Bézier representation of geometrically continuous quartic and quintic spline curves based on the Bézier representation of these curves described in a foregoing paper of the two authors. The influence of the design parameters is illustrated, and local support basis functions are given. Geometrically continuous curves include as special cases visually continuous curves as well as minimizing spline curves due to Nielson [1] and Hagen [2].
January 1992
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17 Reads
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13 Citations
January 1992
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4 Reads
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151 Citations
January 1992
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19 Reads
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8 Citations
Trimming of surfaces and volumes, curve and surface modeling via Bézier’s idea of destortion, segmentation, reparametrization, geometric continuity are examples of applications of functional composition. This paper shows how to compose polynomial and rational tensor product Bézier representations. The problem of composing Bézier splines and B-spline representations will also be addressed in this paper.
January 1992
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20 Reads
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151 Citations
January 1991
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8 Reads
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1 Citation
Die vom CAGD bekannten Techniken für rationale Freiform Bézier- Kurven und -Flächen werden auf rationale trivariate Darstellungen erweitert. Rationale Bézier-Volumina gestatten die Beschreibung von Freiform-Volumina als auch die exakte Darstellung von 3D-Primitiven, wie Kugel, Kegel, Torus, etc. Techniken zur Visualisierung rationaler Bézier-Volumina werden angesprochen.
November 1990
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4 Reads
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3 Citations
An algorithm that creates planar and arbitrarily curved sections of free-form volumes is presented. The definition of free-form volumes generalizes techniques from free-form curves and surfaces to trivariate representation. The definition is given for volumes in the Bernstein-Bezier representation. The author illustrates an intersection algorithm that can be used to perform intersection operations on free-form volumes. Some calculated examples are given. The algorithm can be used as a subroutine for algorithms which are able to perform more general intersections of free-form volumes, e.g. Boolean operations on two free-form volumes
... Here n and m define the polynomial degree of the Bernstein polynomials along the parameters t and s. This results in OEn + 1 m + 1 Bézier points bij, which define a bidirectional net of control points [9][10][11]. Fig. 3 shows the The Bernstein polynomials Bin(t) form a basis system of linearly independent functions, which are calculated according to Eq. 4. ...
January 1992
... The algorithm can be extended for C 0-continuous curves and possibly for open curves. Another potential application for using the theories that have been developed in this paper could be on blending of curves (see Chapter 14 in [29]). The application areas are being explored at present. ...
... The second step consists in extending our construction to the fourth degree of smoothness by means of the geometric continuity of order four in the parametric setting. This final result is based on the detailed description of geometric B-splines given in [2]. ...
June 1992
Computers & Mathematics with Applications
... Bézier volumes are particular cases with the special coefficients c i, j,k = ( m i )( n j )( l k ) m m n n l l [6]. Definition 1 is equivalent to the traditional definition introduced in much of the literature [25][26][27]. When the lattice polytope is m,n,l , the toric surface will be a 3D Bézier volume. ...
April 1995
Computers & Mathematics with Applications
... Functions with global support are not equal to zero outside a certain radius, but cover the whole interpolation space which leads to dense matrix systems. Some classical RBF's with global support provide a good approximation behaviour and are widely used in engineering problems, as by example the Duchon's thin plate splines (TPS n ), the Hardy's multiquadric biharmonic (MQB) and the inverse multiquadric biharmonics (IMQB) [12,63]. ...
January 1992
... This gives a wide ranging class of already well known methods but also certain new combinations offairness criteria, constraints and representations, characterizing new methods not developed yet. It can be seen that among others the conventional e(2) cubic splines (with minimal strain energy), the change of curvature minimal splines [13], the segment weighted cubic splines [6], the point weighted cubic splines (v-splines, [14]), the geometric splines [9] and the quintic ,-splines [8] are comprised as special cases proving the great flexibility of this approach. ...
June 1990
Computing
... Also, it implies that b(t) has zero reach (shortest distance to the cut locus) at S, so no smooth (trimmed) tube or offset can be constructed using b(t) as the generator [10]. To compute the potential self-intersection of an integral cubic, we could resort to numerical algorithms for Bézier curves of an arbitrary degree, such as those by Lasser [11] or Farouki [12], based on subdivision. However, for the specific cubic case, simple closed-form formulas are well known. ...
July 1989
Computers in Industry
... On the other hand, the drawbacks can be summarized according to Patrikalakis and Maekawa [221] as follows: (i) they are only able to isolate zero-dimensional solutions, (ii) there is no certainty that each root has been extracted, (iii) the number of roots in the remaining subdomains is typically not provided, and (iv) there is no explicit information about root multiplicities without additional computations. Last but not least, (v) the method is not efficient in case of high-precision or higher order evaluations [181,219]. ...
May 1986
Computer-Aided Design
... Ideas and concepts have been reformulated and worked on later using blossoming and functional composition principles,5678. TPB representations are the subject of [9]. Combinations of diverse shapes are trickier. ...
January 1992
... Trimming curves are discretized with a good approximation relying on a faithful mapping to a set of 3D Bézier curves. Based on the fact that a 2D Bézier curve can be mapped into a 3D Bézier curve faithfully if the curve lies on a tensor-product Bézier surface, we follow the methodology presented by Lasser et al. [LB95]. To summarize, the Bézier equivalents of trimming curves are divided by the knot spans of the decomposed Bézier surfaces of the NURBS surface such that each Bézier curve segment is restricted within one Bézier surface. ...
January 1993
