Rational curves and surfaces with rational offsets

Helmut Pottmann

doi:10.1016/0167-8396(94)00008-G

Rational curves and surfaces with rational offsets

Helmut Pottmann^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

179 Scopus citations

Abstract

Given a rational algebraic surface in the rational parametric representation s(u,v) with unit normal vectors n(u,v) = (s_u × s_v) t ́|s_u × s_v t ́| the offset surface at distance d is s_d(u,v) = s(u,v) + dn(u,v). This is in general not a rational representation, since t ́|s_u × s_v is in general not rational. In this paper, we present an explicit representation of all rational surfaces with a continuous set of rational offsets s_d(u,v). The analogous question is solved for curves, which is an extension of Farouki's Pythagorean hodograph curves to the rationals. Additionally, we describe all rational curves c(t) whose arc length parameter s(t) is a rational function of t. Offsets arise in the mathematical description of milling processes and in the representation of thick plates, such that the presented curves and surfaces possess a very attractive property for practical use.

Original language	English (US)
Pages (from-to)	175-192
Number of pages	18
Journal	Computer Aided Geometric Design
Volume	12
Issue number	2
DOIs	https://doi.org/10.1016/0167-8396(94)00008-G
State	Published - Mar 1995
Externally published	Yes

Keywords

Dual Bézier curves and surfaces
Isophote
Offset curve
Offset surface
Rational Bézier representation
Rational curve
Rational surface
Spherical Bézier patch

ASJC Scopus subject areas

Modeling and Simulation
Automotive Engineering
Aerospace Engineering
Computer Graphics and Computer-Aided Design

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10.1016/0167-8396(94)00008-G

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title = "Rational curves and surfaces with rational offsets",

abstract = "Given a rational algebraic surface in the rational parametric representation s(u,v) with unit normal vectors n(u,v) = (su × sv) t{\' }|su × sv t{\' }| the offset surface at distance d is sd(u,v) = s(u,v) + dn(u,v). This is in general not a rational representation, since t{\' }|su × sv is in general not rational. In this paper, we present an explicit representation of all rational surfaces with a continuous set of rational offsets sd(u,v). The analogous question is solved for curves, which is an extension of Farouki's Pythagorean hodograph curves to the rationals. Additionally, we describe all rational curves c(t) whose arc length parameter s(t) is a rational function of t. Offsets arise in the mathematical description of milling processes and in the representation of thick plates, such that the presented curves and surfaces possess a very attractive property for practical use.",

keywords = "Dual B{\'e}zier curves and surfaces, Isophote, Offset curve, Offset surface, Rational B{\'e}zier representation, Rational curve, Rational surface, Spherical B{\'e}zier patch",

author = "Helmut Pottmann",

year = "1995",

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doi = "10.1016/0167-8396(94)00008-G",

language = "English (US)",

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pages = "175--192",

journal = "Computer Aided Geometric Design",

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T1 - Rational curves and surfaces with rational offsets

AU - Pottmann, Helmut

PY - 1995/3

Y1 - 1995/3

N2 - Given a rational algebraic surface in the rational parametric representation s(u,v) with unit normal vectors n(u,v) = (su × sv) t ́|su × sv t ́| the offset surface at distance d is sd(u,v) = s(u,v) + dn(u,v). This is in general not a rational representation, since t ́|su × sv is in general not rational. In this paper, we present an explicit representation of all rational surfaces with a continuous set of rational offsets sd(u,v). The analogous question is solved for curves, which is an extension of Farouki's Pythagorean hodograph curves to the rationals. Additionally, we describe all rational curves c(t) whose arc length parameter s(t) is a rational function of t. Offsets arise in the mathematical description of milling processes and in the representation of thick plates, such that the presented curves and surfaces possess a very attractive property for practical use.

AB - Given a rational algebraic surface in the rational parametric representation s(u,v) with unit normal vectors n(u,v) = (su × sv) t ́|su × sv t ́| the offset surface at distance d is sd(u,v) = s(u,v) + dn(u,v). This is in general not a rational representation, since t ́|su × sv is in general not rational. In this paper, we present an explicit representation of all rational surfaces with a continuous set of rational offsets sd(u,v). The analogous question is solved for curves, which is an extension of Farouki's Pythagorean hodograph curves to the rationals. Additionally, we describe all rational curves c(t) whose arc length parameter s(t) is a rational function of t. Offsets arise in the mathematical description of milling processes and in the representation of thick plates, such that the presented curves and surfaces possess a very attractive property for practical use.

KW - Dual Bézier curves and surfaces

KW - Isophote

KW - Offset curve

KW - Offset surface

KW - Rational Bézier representation

KW - Rational curve

KW - Rational surface

KW - Spherical Bézier patch

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U2 - 10.1016/0167-8396(94)00008-G

DO - 10.1016/0167-8396(94)00008-G

M3 - Article

AN - SCOPUS:0029271014

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VL - 12

SP - 175

EP - 192

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

IS - 2

ER -

Rational curves and surfaces with rational offsets

Abstract

Keywords

ASJC Scopus subject areas

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