article

Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data

Author:

A. KurnosenkoAuthors Info & Claims

Computer Aided Geometric Design, Volume 27, Issue 3

Pages 262 - 280

https://doi.org/10.1016/j.cagd.2009.12.004

Published: 01 March 2010 Publication History

Abstract

A method of two-point G^2 Hermite interpolation with spirals is proposed. To construct a sought for curve, the inversion is applied to an arc of some other spiral. To illustrate the method, inversions of parabola are considered in detail. The resulting curve is 4th degree rational. The method allows the matching of a wide range of boundary conditions, including those which require an inflection. Although not all G^2 Hermite data can be matched with a spiral generated from a parabolic arc, introducing one intermediate G^2 data solves the problem. Expanding the method by involving other spirals arcs is also discussed.

References

[1]

The generalised Cornu spiral and its application to span generation. J. Comp. Appl. Math. v102. 37-47.

Crossref

Google Scholar

[2]

Rational cubic spirals. Comp.-Aided Design. v40. 3-12.

Crossref

Google Scholar

[3]

Curves and Surfaces for Computer Geometric Aided Design: A Practical Guide. Academic Press.

Crossref

Google Scholar

[4]

Designing Bezièr conic segments with monotone curvature. Comp. Aided Geom. Design. v17. 457-483.

Crossref

Google Scholar

[5]

An involute spiral that matches G2 Hermite data in the plane. Comp. Aided Geom. Design. v26 i7. 733-756.

Crossref

Google Scholar

[6]

Differential Geometry. Dover Publications, New York.

Google Scholar

[7]

An inversion invariant of a pair of circles. J. Math. Sci. v110 i4. 2848-2860.

Google Scholar

[8]

General properties of spiral plane curves. J. Math. Sci. v161 i3. 405-418.

Google Scholar

[9]

Short spirals. Zapiski Nauch. Sem. POMI. v372. 34-43.

Google Scholar

[10]

Theory of Functions of a Complex Variable, vol. 1. Prentice-Hall.

Google Scholar

[11]

Planar spirals that match G2 Hermite data. Comp. Aided Geom. Design. v15. 103-126.

Crossref

Google Scholar

[12]

Weisstein, Eric W. Parabola inverse curve. From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/ParabolaInverseCurve.html

Google Scholar

Cited By

View all

Wu WYang X(2016)Geometric Hermite interpolation by a family of intrinsically defined planar curvesComputer-Aided Design10.1016/j.cad.2016.04.00177:C(86-97)Online publication date: 1-Aug-2016
https://dl.acm.org/doi/10.1016/j.cad.2016.04.001
Li ZAit-Haddou RBiard L(2015)Pythagorean hodograph spline spirals that match G 3 Hermite data from circlesJournal of Computational and Applied Mathematics10.1016/j.cam.2014.10.005278:C(162-180)Online publication date: 15-Apr-2015
https://dl.acm.org/doi/10.1016/j.cam.2014.10.005
Georgiev G(2010)Shape curvatures of planar rational spiralsProceedings of the 7th international conference on Curves and Surfaces10.1007/978-3-642-27413-8_17(270-279)Online publication date: 24-Jun-2010
https://dl.acm.org/doi/10.1007/978-3-642-27413-8_17

Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation

Recommendations

Two-point G2 Hermite interpolation with spirals by inversion of hyperbola

Planar, 4th degree rational spirals, matching given G^2 Hermite data, are constructed by inversion of hyperbola. The proposed method handles any data for convex spiral arcs, and a great deal of non-convex data. Non-convex data are either with an ...
Fitting G2 multispiral transition curve joining two straight lines

In this paper, we describe an algorithm for generating a C-shaped G^2 multispiral transition curve between two non-parallel straight lines. The G^2 multispiral is a curve that consists of two or more log-aesthetic curve segments connected with curvature ...
Admissible regions for rational cubic spirals matching G2 Hermite data

This paper finds reachable regions for a single segment of parametric rational cubic Bezier spiral matching G^2 Hermite data. First we derive spiral conditions for rational cubics and then we use a free parameter to find the admissible region for a ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Computer Aided Geometric Design

Computer Aided Geometric Design Volume 27, Issue 3

March, 2010

62 pages

ISSN:0167-8396

Issue’s Table of Contents

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 March 2010

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

3
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 30 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Wu WYang X(2016)Geometric Hermite interpolation by a family of intrinsically defined planar curvesComputer-Aided Design10.1016/j.cad.2016.04.00177:C(86-97)Online publication date: 1-Aug-2016
https://dl.acm.org/doi/10.1016/j.cad.2016.04.001
Li ZAit-Haddou RBiard L(2015)Pythagorean hodograph spline spirals that match G 3 Hermite data from circlesJournal of Computational and Applied Mathematics10.1016/j.cam.2014.10.005278:C(162-180)Online publication date: 15-Apr-2015
https://dl.acm.org/doi/10.1016/j.cam.2014.10.005
Georgiev G(2010)Shape curvatures of planar rational spiralsProceedings of the 7th international conference on Curves and Surfaces10.1007/978-3-642-27413-8_17(270-279)Online publication date: 24-Jun-2010
https://dl.acm.org/doi/10.1007/978-3-642-27413-8_17

Abstract

References

Cited By

Recommendations

Two-point G2 Hermite interpolation with spirals by inversion of hyperbola

Fitting G2 multispiral transition curve joining two straight lines

Admissible regions for rational cubic spirals matching G2 Hermite data

Comments

Information

Published In

Publisher

Publication History

Author Tags

Qualifiers

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

View options

Figures

Other

Share

Share this Publication link

Share on social media

Affiliations