Offset curves and surfaces: a brief survey

Pham, Binh

doi:10.1016/0010-4485(92)90059-j

Cited by 152 publications

(48 citation statements)

References 19 publications

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“…Several methods for the computation of the offsets for curves are compared in [15]. As pointed out in [59], offsetting general surfaces is more complicated, and an offset surface is often approximated [21], although this approximation becomes inaccurate near its selfintersecting area [2,59]. Another approach for computing offsets of NURBS curves and surfaces is given in [62].…”

Section: Offset Curves and Surfacesmentioning

confidence: 99%

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Computational Methods for Geometric Processing. Applications to Industry

Iglesias

Gálvez

Puig-Pey

2001

Computational Science — ICCS 2001

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Section: Offset Curves and Surfacesmentioning

confidence: 99%

“…Finally, the set of rational curves and surfaces is not closed under offsetting [18]. Therefore, considerable attention has been paid to identify the curves and surfaces which admit rational offsets [22,59,64]. The case of polynomial and rational curves with rational offsets is analyzed in [48].…”

Section: Offset Curves and Surfacesmentioning

confidence: 99%

Computational Methods for Geometric Processing. Applications to Industry

Iglesias

Gálvez

Puig-Pey

2001

Computational Science — ICCS 2001

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“…(Offset curves arise in a variety of CAD/CAM applications; the fact that general polynomial curves do not have rational offsets [7] has prompted a profusion of more or less heuristic approximation schemes; see [18].) It may be feared that condition (2) will transpire to be unduly restrictive in terms of the shape flexibility of curves that enjoy the above attributes; such curves would then be of little practical value.…”

Section: ) Rd(t) = T(t) + Dn(t)mentioning

confidence: 99%

Hermite Interpolation by Pythagorean Hodograph Quintics

Farouki¹,

Neff²

1995

Mathematics of Computation

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Abstract. The Pythagorean hodograph (PH) curves are polynomial parametric curves {x(t), y(t)} whose hodograph (derivative) components satisfy the Pythagorean condition x'2(t)+y'2(t) = a2(t) for some polynomial a(t). Thus, unlike polynomial curves in general, PH curves have arc lengths and offset curves that admit exact rational representations. The lowest-order PH curves that are sufficiently flexible for general interpolation/approximation problems are the quintics. While the PH quintics are capable of matching arbitrary first-order Hermite data, the solution procedure is not straightforward and furthermore does not yield a unique result-there are always four distinct interpolants (of which only one, in general, has acceptable "shape" characteristics). We show that formulating PH quintics as complex-valued functions of a real parameter leads to a compact Hermite interpolation algorithm and facilitates an identification of the "good" interpolant (in terms of minimizing the absolute rotation number). This algorithm establishes the PH quintics as a viable medium for the design or approximation of free-form curves, and allows a one-for-one substitution of PH quintics in lieu of the widely-used "ordinary" cubics.

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“…Hence, shape (curve, surface or volume) offsets and corresponding algorithms have been widely studied in Computer-Aided Design and Manufacturing, see [1,2,3,4,5,6,7,8] and references therein for more details. However, free-form NURBS shapes do not possess rational offsets in general and thus suitable approximation techniques, often based on rational curves or surfaces of relatively low degree, are used -see e.g.…”

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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Offset curves and surfaces: a brief survey

Cited by 152 publications

References 19 publications

Computational Methods for Geometric Processing. Applications to Industry

Computational Methods for Geometric Processing. Applications to Industry

Hermite Interpolation by Pythagorean Hodograph Quintics

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

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