Approximation of involute curves for CAD-system processing

Fumitaka Higuchi, Shuuichi Gofuku, Takashi Maekawa, Harish Mukundan, Nicholas M. Patrikalakis

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

In numerous instances, accurate algorithms for approximating the original geometry is required. One typical example is a circle involute curve which represents the underlying geometry behind a gear tooth. The circle involute curves are by definition transcendental and cannot be expressed by algebraic equations, and hence it cannot be directly incorporated into commercial CAD systems. In this paper, an approximation algorithm for circle involute curves in terms of polynomial functions is developed. The circle involute curve is approximated using a Chebyshev approximation formula (Press et al. in Numerical recipes, Cambridge University Press, Cambridge, 1988), which enables us to represent the involute in terms of polynomials, and hence as a Bézier curve. In comparison with the current B-spline approximation algorithms for circle involute curves, the proposed method is found to be more accurate and compact, and induces fewer oscillations.

Original languageEnglish
Pages (from-to)207-214
Number of pages8
JournalEngineering with Computers
Volume23
Issue number3
DOIs
Publication statusPublished - 2007 Sep 1
Externally publishedYes

Fingerprint

Approximation algorithms
Computer aided design
Chebyshev approximation
Polynomials
Circle
Curve
Geometry
Gear teeth
Approximation
Processing
Splines
Approximation Algorithms
Chebyshev Approximation
Spline Approximation
Transcendental
Polynomial function
B-spline
Algebraic Equation
Oscillation
Polynomial

Keywords

  • Bézier curves
  • Chebyshev approximation formula
  • Circle involute curves
  • Involute gears

ASJC Scopus subject areas

  • Software
  • Modelling and Simulation
  • Engineering(all)
  • Computer Science Applications

Cite this

Higuchi, F., Gofuku, S., Maekawa, T., Mukundan, H., & Patrikalakis, N. M. (2007). Approximation of involute curves for CAD-system processing. Engineering with Computers, 23(3), 207-214. https://doi.org/10.1007/s00366-007-0060-3

Approximation of involute curves for CAD-system processing. / Higuchi, Fumitaka; Gofuku, Shuuichi; Maekawa, Takashi; Mukundan, Harish; Patrikalakis, Nicholas M.

In: Engineering with Computers, Vol. 23, No. 3, 01.09.2007, p. 207-214.

Research output: Contribution to journalArticle

Higuchi, F, Gofuku, S, Maekawa, T, Mukundan, H & Patrikalakis, NM 2007, 'Approximation of involute curves for CAD-system processing', Engineering with Computers, vol. 23, no. 3, pp. 207-214. https://doi.org/10.1007/s00366-007-0060-3
Higuchi F, Gofuku S, Maekawa T, Mukundan H, Patrikalakis NM. Approximation of involute curves for CAD-system processing. Engineering with Computers. 2007 Sep 1;23(3):207-214. https://doi.org/10.1007/s00366-007-0060-3
Higuchi, Fumitaka ; Gofuku, Shuuichi ; Maekawa, Takashi ; Mukundan, Harish ; Patrikalakis, Nicholas M. / Approximation of involute curves for CAD-system processing. In: Engineering with Computers. 2007 ; Vol. 23, No. 3. pp. 207-214.
@article{156c291945b447ec8375ca791a1bfb88,
title = "Approximation of involute curves for CAD-system processing",
abstract = "In numerous instances, accurate algorithms for approximating the original geometry is required. One typical example is a circle involute curve which represents the underlying geometry behind a gear tooth. The circle involute curves are by definition transcendental and cannot be expressed by algebraic equations, and hence it cannot be directly incorporated into commercial CAD systems. In this paper, an approximation algorithm for circle involute curves in terms of polynomial functions is developed. The circle involute curve is approximated using a Chebyshev approximation formula (Press et al. in Numerical recipes, Cambridge University Press, Cambridge, 1988), which enables us to represent the involute in terms of polynomials, and hence as a B{\'e}zier curve. In comparison with the current B-spline approximation algorithms for circle involute curves, the proposed method is found to be more accurate and compact, and induces fewer oscillations.",
keywords = "B{\'e}zier curves, Chebyshev approximation formula, Circle involute curves, Involute gears",
author = "Fumitaka Higuchi and Shuuichi Gofuku and Takashi Maekawa and Harish Mukundan and Patrikalakis, {Nicholas M.}",
year = "2007",
month = "9",
day = "1",
doi = "10.1007/s00366-007-0060-3",
language = "English",
volume = "23",
pages = "207--214",
journal = "Engineering with Computers",
issn = "0177-0667",
publisher = "Springer London",
number = "3",

}

TY - JOUR

T1 - Approximation of involute curves for CAD-system processing

AU - Higuchi, Fumitaka

AU - Gofuku, Shuuichi

AU - Maekawa, Takashi

AU - Mukundan, Harish

AU - Patrikalakis, Nicholas M.

PY - 2007/9/1

Y1 - 2007/9/1

N2 - In numerous instances, accurate algorithms for approximating the original geometry is required. One typical example is a circle involute curve which represents the underlying geometry behind a gear tooth. The circle involute curves are by definition transcendental and cannot be expressed by algebraic equations, and hence it cannot be directly incorporated into commercial CAD systems. In this paper, an approximation algorithm for circle involute curves in terms of polynomial functions is developed. The circle involute curve is approximated using a Chebyshev approximation formula (Press et al. in Numerical recipes, Cambridge University Press, Cambridge, 1988), which enables us to represent the involute in terms of polynomials, and hence as a Bézier curve. In comparison with the current B-spline approximation algorithms for circle involute curves, the proposed method is found to be more accurate and compact, and induces fewer oscillations.

AB - In numerous instances, accurate algorithms for approximating the original geometry is required. One typical example is a circle involute curve which represents the underlying geometry behind a gear tooth. The circle involute curves are by definition transcendental and cannot be expressed by algebraic equations, and hence it cannot be directly incorporated into commercial CAD systems. In this paper, an approximation algorithm for circle involute curves in terms of polynomial functions is developed. The circle involute curve is approximated using a Chebyshev approximation formula (Press et al. in Numerical recipes, Cambridge University Press, Cambridge, 1988), which enables us to represent the involute in terms of polynomials, and hence as a Bézier curve. In comparison with the current B-spline approximation algorithms for circle involute curves, the proposed method is found to be more accurate and compact, and induces fewer oscillations.

KW - Bézier curves

KW - Chebyshev approximation formula

KW - Circle involute curves

KW - Involute gears

UR - http://www.scopus.com/inward/record.url?scp=34547432456&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547432456&partnerID=8YFLogxK

U2 - 10.1007/s00366-007-0060-3

DO - 10.1007/s00366-007-0060-3

M3 - Article

AN - SCOPUS:34547432456

VL - 23

SP - 207

EP - 214

JO - Engineering with Computers

JF - Engineering with Computers

SN - 0177-0667

IS - 3

ER -