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Functional Composition Algorithms via Blossoming
, 1993
"... In view of the fundamental role that functional composition plays in mathematics, it is not surprising that a variety of problems in geometric modeling can be viewed as instances of the following composition problem: given representations for two functions F and G, compute a representation of the fu ..."
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In view of the fundamental role that functional composition plays in mathematics, it is not surprising that a variety of problems in geometric modeling can be viewed as instances of the following composition problem: given representations for two functions F and G, compute a representation of the function H = F ffi G: We examine this problem in detail for the case when F and G are given in either B'ezier or Bspline form. Blossoming techniques are used to gain theoretical insight into the structure of the solution which is then used to develop efficient, tightly codable algorithms. From a practical point of view, if the composition algorithms are implemented as library routines, a number of geometric modeling problems can be solved with a small amount of additional software. This paper was published in TOG, April 1993, pg 113135 Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling  curve, surface, and object representations; J.6 [...
From Degenerate Patches to Triangular and Trimmed Patches
 CURVES AND SURFACES
, 1997
"... CAD systems are usually based on a tensor product representation of free form surfaces. In this case, trimmed patches are used for modeling non rectangular zones. Trimmed patches provide a reasonable solution for the representation of general topologies, provided that the gap between equivalent trim ..."
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CAD systems are usually based on a tensor product representation of free form surfaces. In this case, trimmed patches are used for modeling non rectangular zones. Trimmed patches provide a reasonable solution for the representation of general topologies, provided that the gap between equivalent trimming curves in the euclidean space is small enough. Several commercial CAD systems, however, represent certain non rectangular surface regions through degenerate rectangular patches. Degenerate patches produce rendering artifacts and can lead to malfunctions in the subsequent geometric operations. In the present paper, two algorithms for converting degenerate tensorproduct patches into triangular and trimmed rectangular patches are presented. The algorithms are based on specific degree reduction algorithms for B'ezier curves. In both algorithms, the final surface approximates the initial one in a quadratic sense while inheriting its boundary curves. In the second one, " \Gamma G 1 cont...
n with control points Ti,j,k is defined by
, 2000
"... This paper presents an explicit formula that converts a triangular Bézier patch of degree n to a degenerate rectangular Bézier patch of degree n × n by reparametrization. Based on this formula, we develop a method for approximating a degenerate rectangular Bézier patch by three nondegenerate Bézier ..."
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This paper presents an explicit formula that converts a triangular Bézier patch of degree n to a degenerate rectangular Bézier patch of degree n × n by reparametrization. Based on this formula, we develop a method for approximating a degenerate rectangular Bézier patch by three nondegenerate Bézier patches; more patches can be introduced by subdivision to meet a userspecified error