Cubic helical splines with Frenet-frame continuity

Chang Yong Han; Song Hwa Kwon

doi:10.1016/j.cagd.2011.08.003

Cubic helical splines with Frenet-frame continuity

Chang Yong Han
, Song Hwa Kwon

Mathematics

Kyung Hee University

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

8 Scopus citations

Abstract

We present a scheme of interpolating a sequence of points ^p0,⋯,^pn in ^R3 by a piecewise-cubic- helical spline with continuous Frenet frame. Each cubic helical segment ^rk(ξ) connecting pk-₁ and ^pk is constructed such that its end tangents are symmetric with respect to the displacement vector Δ^pk=^pk-pk-₁. The existence condition of such a segment ^rk(ξ) is formulated in terms of the configuration involving Δ^pk and the Frenet frame of ^rk(ξ) at pk-₁. A Frenet-frame-continuous spline is obtained by matching the Frenet frames of sequentially constructed segments at their juncture points.

Original language	English
Pages (from-to)	395-406
Number of pages	12
Journal	Computer Aided Geometric Design
Volume	28
Issue number	7
DOIs	https://doi.org/10.1016/j.cagd.2011.08.003
State	Published - Oct 2011

Keywords

Cubic helix
Frenet frame
Geometric continuity
Hermite interpolation
Pythagorean-hodograph
Quaternion
Spline
Tangent indicatrix

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10.1016/j.cagd.2011.08.003

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title = "Cubic helical splines with Frenet-frame continuity",

abstract = "We present a scheme of interpolating a sequence of points p0,⋯,pn in R3 by a piecewise-cubic- helical spline with continuous Frenet frame. Each cubic helical segment rk(ξ) connecting pk-1 and pk is constructed such that its end tangents are symmetric with respect to the displacement vector Δpk=pk-pk-1. The existence condition of such a segment rk(ξ) is formulated in terms of the configuration involving Δpk and the Frenet frame of rk(ξ) at pk-1. A Frenet-frame-continuous spline is obtained by matching the Frenet frames of sequentially constructed segments at their juncture points.",

keywords = "Cubic helix, Frenet frame, Geometric continuity, Hermite interpolation, Pythagorean-hodograph, Quaternion, Spline, Tangent indicatrix",

author = "Han, \{Chang Yong\} and Kwon, \{Song Hwa\}",

year = "2011",

month = oct,

doi = "10.1016/j.cagd.2011.08.003",

language = "English",

volume = "28",

pages = "395--406",

journal = "Computer Aided Geometric Design",

issn = "0167-8396",

publisher = "Elsevier B.V.",

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TY - JOUR

T1 - Cubic helical splines with Frenet-frame continuity

AU - Han, Chang Yong

AU - Kwon, Song Hwa

PY - 2011/10

Y1 - 2011/10

N2 - We present a scheme of interpolating a sequence of points p0,⋯,pn in R3 by a piecewise-cubic- helical spline with continuous Frenet frame. Each cubic helical segment rk(ξ) connecting pk-1 and pk is constructed such that its end tangents are symmetric with respect to the displacement vector Δpk=pk-pk-1. The existence condition of such a segment rk(ξ) is formulated in terms of the configuration involving Δpk and the Frenet frame of rk(ξ) at pk-1. A Frenet-frame-continuous spline is obtained by matching the Frenet frames of sequentially constructed segments at their juncture points.

AB - We present a scheme of interpolating a sequence of points p0,⋯,pn in R3 by a piecewise-cubic- helical spline with continuous Frenet frame. Each cubic helical segment rk(ξ) connecting pk-1 and pk is constructed such that its end tangents are symmetric with respect to the displacement vector Δpk=pk-pk-1. The existence condition of such a segment rk(ξ) is formulated in terms of the configuration involving Δpk and the Frenet frame of rk(ξ) at pk-1. A Frenet-frame-continuous spline is obtained by matching the Frenet frames of sequentially constructed segments at their juncture points.

KW - Cubic helix

KW - Frenet frame

KW - Geometric continuity

KW - Hermite interpolation

KW - Pythagorean-hodograph

KW - Quaternion

KW - Spline

KW - Tangent indicatrix

UR - https://www.scopus.com/pages/publications/80053207326

U2 - 10.1016/j.cagd.2011.08.003

DO - 10.1016/j.cagd.2011.08.003

M3 - Article

AN - SCOPUS:80053207326

SN - 0167-8396

VL - 28

SP - 395

EP - 406

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

IS - 7

ER -

Cubic helical splines with Frenet-frame continuity

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Keywords

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