Topologically reliable approximation of composite Bézier curves

Cho, Wonjoon; Maekawa, Takashi; Patrikalakis, Nicholas M.

doi:10.1016/0167-8396(95)00042-9

Cited by 19 publications

(23 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Even if the quality of the triangular elements is assumed to be sufficiently high in the parametric space, it can easily deteriorate through the mapping process due to distortion and/or stretching. To overcome this adverse effect, we introduced an auxiliary planar domain of triangulation for trimmed surface patches [8,10]. Approximate locally isometric mapping between the input trimmed surface patch and its planar triangulation domain is achieved by minimizing a locally isometric mapping error function E. Our algorithm also guarantees a homeomorphism between the triangulation domain and parametric space (and hence given surface patch) by robustly removing the possibility of self-intersection of the approximately developed surface patch.…”

Section: Construction Of Triangulation Domainmentioning

confidence: 99%

“…Approximate locally isometric mapping between the input trimmed surface patch and its planar triangulation domain is achieved by minimizing a locally isometric mapping error function E. Our algorithm also guarantees a homeomorphism between the triangulation domain and parametric space (and hence given surface patch) by robustly removing the possibility of self-intersection of the approximately developed surface patch. In case the minimized mapping error min(E) is unsatisfactory, we bisect the input surface patch and repeat the developing process until min(E) is within a certain threshold, as detailed in [8,10]. As an example, we consider a composite surface which consists of two trimmed bi-cubic Bézier surface patches, as shown in Fig.…”

Section: Construction Of Triangulation Domainmentioning

confidence: 99%

“…They play a role as the most appropriate initial nodes in case fully automatic surface tessellation process is required. Moreover, they permit quite optimal coarse approximation especially when the approximation tolerance is relatively loose or input geometry has a highly nonuniform distribution of intrinsic features [7].…”

Section: Introductionmentioning

confidence: 99%

“…The second problem, possible topological inconsistency between the input geometry and its approximation should be carefully identified and eliminated since such a topological inconsistency immediately results in system failure, e.g., during solution of computational mechanics problems in finite element analysis, or in the solid free-form fabrication (SFF) methods. For simple composite surfaces which do not have any inappropriate intersection, corresponding approximants also should not have any improperly intersecting elements to achieve a homeomorphism [2] between the exact input geometry and its approximation [7]. Therefore, a robust intersection test relying on RIA and the ability to perform local refinement of the approximation is required to avoid inappropriate intersections.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Topologically Reliable Approximation of Trimmed Polynomial Surface Patches

Cho

Maekawa

Patrikalakis

et al. 1999

Graphical Models and Image Processing

Self Cite

View full text Add to dashboard Cite

Section: Construction Of Triangulation Domainmentioning

confidence: 99%

Section: Construction Of Triangulation Domainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Topologically Reliable Approximation of Trimmed Polynomial Surface Patches

Cho

Maekawa

Patrikalakis

et al. 1999

Graphical Models and Image Processing

Self Cite

View full text Add to dashboard Cite

“…we have that the curvature critical set is identified with the zero set of g 0 , an expression given earlier in [2] in coordinate functions of the original plane curve…”

Section: Curvature Critical Set Of a Plane Curvementioning

confidence: 99%

Degree Reduction for NURBS Symbolic Computation on Curves

Chen

Riesenfeld

Cohen

IEEE International Conference on Shape Modeling and Applications 2006 (SMI'06)

View full text Add to dashboard Cite

BackgroundSymbolic computation of NURBS [16,5,4,26] refers to algebraic operations on one or more than one NURBS, resulting in another NURBS. The operations typically include primitive ones such as sum/difference, multiplication/division , differentiation, and composition, and derived ones such as dot product, cross product (of 2 B-splines) and triple-scalar product (of 3 B-splines) in R 3, and generalized cross product (of n − 1 B-splines) and scalar product (of n B-splines) in R n , etc. These operations are essential to the construction of complex NURBS models from simple ones, and more importantly to curve/surface interrogation [5,8,21,9,25]. NURBS symbolic computation has also been used in various other areas such as computing bisectors [10,11,12], blending surfaces [19] and offsetting * This work was supported in part by NSF ( CCR0310705 ) . All opinions, findings, conclusions or recommendations expressed in this document are those of the authors and do not necessarily reflect the views of the sponsoring agencies.† contact author: xchen@cs.utah.edu curves [7]. Actually, with the help of rational constraint solvers [28,13], NURBS symbolic computation plays a fundamental role virtually in every area of geometric computation on free-form curves/surfaces. However, there is one critical problem that severely restricts the power of NURBS symbolic computation -the rapidly increasing degree of the derived NURBS, which is especially true when rational B-splines are involved. Let us first review briefly symbolic computation on polynomial B-splines. To symbolically add/subtract two polynomial B-splines, both degree elevation and knot vector refinement have to be done so that both operands have the same degree and knot vector, and then addition/subtraction is simply applied point-wise to their control polygons. Dividing a polynomial B-spline by another polynomial scalar B-spline basically follows the same procedure, except that the final step is division and the derived B-spline is rational. Differentiation of a polynomial B-spline is simple and has the favorable property of decreasing the degree by one. Multiplication of two B-splines, though, is complicated with different approaches [23,5,29]; what is of concern to this paper is that polynomial B-spline multiplication results in another polynomial B-spline with a degree that is the sum of the degrees of the two operands.In contrast, rational NURBS symbolic computation is a rather different situation. Every operation on a rational B-spline is derived rather than primitive, and is implemented typically as more than one primitive operations on the denominator and numerator. As a result, addition/subtraction/division raises the degree just as multiplication does, and differentiation doubles the degree 1 . Because any operation on two rationals either adds together or doubles the degrees, any nontrivial symbolic computation on rational B-splines, especially when higher order differentiation is involved, would quite likely become impractical because of the huge d...

show abstract

Approximating Curves via Alpha Shapes

Sakkalis

Charitos

1999

Graphical Models and Image Processing

View full text Add to dashboard Cite

Topologically reliable approximation of composite Bézier curves

Cited by 19 publications

References 25 publications

Topologically Reliable Approximation of Trimmed Polynomial Surface Patches

Topologically Reliable Approximation of Trimmed Polynomial Surface Patches

Degree Reduction for NURBS Symbolic Computation on Curves

Approximating Curves via Alpha Shapes

Contact Info

Product

Resources

About