Rational surfaces with linear normals and their convolutions with rational surfaces

Sampoli, Maria Lucia; Peternell, Martín; Jüttler, Bert

doi:10.1016/j.cagd.2005.07.001

Cited by 41 publications

(12 citation statements)

References 14 publications

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“…Similarly, revolution surfaces with monotone slope profile curves have explicit reparameterization and can be computed efficiently [10]. Rational convolution surfaces can be obtained between linear normal surfaces and generic rational surfaces [11,12,13].…”

Section: Surface Convolution Minkowski Sums and Offsetmentioning

confidence: 99%

Computing Minkowski Sum of Periodic Surface Models

Wang

2009

Computer-Aided Design and Applications

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Recently we proposed a periodic surface model to assist geometric construction in computer-aided nano-design. This implicit surface model helps create super-porous nano structures parametrically and support crystal packing. In this paper, we study construction methods of Minkowski sums for periodic surfaces. A numerical approximation approach based on the Chebyshev polynomials is developed and can be applied in the formulations of surface normal direction matching and volume translations.

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Section: Surface Convolution Minkowski Sums and Offsetmentioning

confidence: 99%

Computing Minkowski Sum of Periodic Surface Models

Wang

2009

Computer-Aided Design and Applications

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“…We may note that the envelope operator E is a linear mapping and defines an isomorphism between the linear spaces C 1 (D, R) and its images, where the addition in the image space is given by the so called convolution of the surfaces (for more details see [15,9]). …”

Section: Preliminariesmentioning

confidence: 99%

“…Recently it was found out that quadratic triangular Bézier splines also belong to the class of surfaces with odd rational support functions and therefore it can be proved that they belong to the family of surfaces which can be equipped with a linear field of normal vectors. This nice feature allows the exact computation of a rational parameterization of offsets and convolution surfaces, [10,11,12,15]. A Bézier patch, by construction, is uniquely defined once its control points are assigned.…”

Section: Approximation Of Surfaces and Their Support Functionsmentioning

confidence: 99%

Support Function Representation for Curvature Dependent Surface Sampling

Sampoli

Jüttler

2009

Applied and Industrial Mathematics in Italy III

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In many applications it is required to have a curvature-dependent surface sampling, based on a local shape analysis. In this work we show how this can be achieved by using the support function (SF) representation of a surface. This representation, a classical tool in Convex Geometry, has been recently considered in CAD problems for computing surface offsets and for analyzing curvatures. Starting from the observation that triangular Bézier spline surfaces have quite simple support functions, we approximate any given free-form surface by a quadratic triangular Bézier spline surface. Then the corresponding approximate SF representation can be efficiently exploited to produce a curvature dependent sampling of the approximated surface.

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“…Later, it has been proved in [34] that surfaces with Linear field of Normal vectors (LN surfaces), introduced in [35], provide rational convolution surfaces with an arbitrary rational surface. Since spheres admit rational descriptions, LN surfaces possess exact rational offsets.…”

Section: Introductionmentioning

confidence: 99%

Volumes with piecewise quadratic medial surface transforms: Computation of boundaries and trimmed offsets

Bastl

Jüttler

Kosinka

et al. 2010

Computer-Aided Design

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MOS surfaces are rational surfaces in R 3,1 which provide rational envelopes of the associated two-parameter family of spheres. Moreover, all the offsets admit rational parameterizations as well. Recently, it has been proved that quadratic triangular Bézier patches in R 3,1 are MOS surfaces. Following this result, we describe an algorithm for computing an exact rational envelope of a two-parameter family of spheres given by a quadratic patch in R 3,1 . The main focus of this paper is given to geometric aspects of the algorithm. Since these patches are capable of producing C 1 smooth approximations of medial surface transforms of spatial domains, we use this algorithm to generate rational approximations of envelopes of general medial surface transforms. One of the main advantages of this approach to offsetting is the fact that the trimming procedure becomes considerably simpler.

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Rational surfaces with linear normals and their convolutions with rational surfaces

Cited by 41 publications

References 14 publications

Computing Minkowski Sum of Periodic Surface Models

Computing Minkowski Sum of Periodic Surface Models

Support Function Representation for Curvature Dependent Surface Sampling

Volumes with piecewise quadratic medial surface transforms: Computation of boundaries and trimmed offsets

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