JOURNAL BROWSE
Search
Advanced SearchSearch Tips
Comparison of Offset Approximation Methods of Conics with Explicit Error Bounds
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
  • Journal title : Journal of the Chosun Natural Science
  • Volume 9, Issue 1,  2016, pp.10-15
  • Publisher : The Research Institute of Chosun Natural Science
  • DOI : 10.13160/ricns.2016.9.1.10
 Title & Authors
Comparison of Offset Approximation Methods of Conics with Explicit Error Bounds
Bae, Sung Chul; Ahn, Young Joon;
  PDF(new window)
 Abstract
In this paper the approximation methods of offset curve of conic with explicit error bound are considered. The quadratic approximation of conic(QAC) method, the method based on quadratic circle approximation(BQC) and the Pythagorean hodograph cubic(PHC) approximation have the explicit error bound for approximation of offset curve of conic. We present the explicit upper bound of the Hausdorff distance between the offset curve of conic and its PHC approximation. Also we show that the PHC approximation of any symmetric conic is closer to the line passing through both endpoints of the conic than the QAC.
 Keywords
Conic;Offset Approximation;Explicit Error Bound;Convolution Curve;Pythagorean Hodograph;
 Language
English
 Cited by
 References
1.
W. Bohm, G. Farin, and J. Kahmann, "A survey of curve and surface methods in CAGD", Comput. Aided Geom. D., Vol. 1, pp 1-60, 1984. crossref(new window)

2.
E. T. Lee, "The rational Bezier representation for conics", In G. E. Farin, eds., Geometric Modeling: Algorithms and New Trends. SIAM, pp. 3-19, 1987.

3.
W. H. Frey and D. A. Field, "Designing Bezier conic segments with monotone curvature", Comput. Aided Geom. D., Vol. 17, pp 457-483, 2000. crossref(new window)

4.
Y. J. Ahn and H. O. Kim, "Curvatures of the quadratic rational Bezier curves", Comput. Math. Appl., Vol., 36, pp 71-83, 1998.

5.
G.-J. Wang and G.-Z. Wang, "The rational cubic Bezier representation of conics", Comput. Aided Geom. D., Vol. 9, pp 447-455, 1992. crossref(new window)

6.
L. Fang, "A rational quartic Bezier representation for conics", Comput. Aided Geom. D., Vol 19, pp 297-312, 2002. crossref(new window)

7.
J. Sanchez-Reyes, "Geometric recipes for constructing Bezier conics of given centre or focus", Comp. Aided Geom. Desi., Vol. 21, pp 111-116, 2004. crossref(new window)

8.
C. Xu, T.-W. Kim and G. Farin, "The eccentricity of conic sections formulated as rational Bezier quadratics", Comput. Aided Geom. D., Vol. 27, 458-460, 2010. crossref(new window)

9.
A. Cantona, L. Fernandez-Jambrina and E. R. Maria, "Geometric characteristics of conics in Bezier form", Comput. Aided Design, Vol. 43, 1413-1421, 2011. crossref(new window)

10.
J. Sanchez-Reyes, "Simple determination via complex arithmetic of geometric characteristics of Bezier conics", Comput. Aided Geom. D., Vol. 28, pp 345-348, 2011. crossref(new window)

11.
M. Floater, "High-order approximation of conic sections by quadratic splines", Comput. Aided Geom. D., Vol. 12, pp 617-637, 1995. crossref(new window)

12.
M. S. Floater, "An O($h^{2n}$) Hermite approximation for conic sections", Comput. Aided Geom. D., Vol. 14, pp 135-151, 1997. crossref(new window)

13.
L. Fang, "$G^3$ approximation of conic sections by quintic polynomial curves", Comput. Aided Geom. D., Vol. 16, pp 755-766, 1999. crossref(new window)

14.
Y. J. Ahn, "Approximation of conic sections by curvature continuous quartic Bezier curves", Comput. Math. Appl., Vol. 60, pp 1986-1993, 2010. crossref(new window)

15.
R. T. Farouki, "Conic approximation of conic offsets", J. Symb. Comput., Vol. 23, pp 301-313. 1997. crossref(new window)

16.
G. Salmon, "A treatise on conic sections", New York: Chelsea, 1954.

17.
W. Lu, "Offset-rational parametric plane curves", Comput. Aided Geom. D., Vol. 12, pp 601-616, 1995. crossref(new window)

18.
G. Farin, "Curvature continuity and offsets for piecewise conics", ACM T. Graphic., Vol. 8, pp. 89-99, 1989. crossref(new window)

19.
R. Lee and Y. J. Ahn, "Construction of logarithmic spiral-like curve using $G^2$ quadratics spline with self similarity", J. Chosun Natural Sci., Vol. 7, pp 124-129, 2014. crossref(new window)

20.
Y. J. Ahn, C. M. Hoffmann, and Y. S. Kim, "Curvature-continuous offset approximation based on circle approximation using quadratic Bezier biarcs", Comput. Aided Design, Vol. 43, pp. 1011-1017, 2011. crossref(new window)

21.
I.-K. Lee, M.-S. Kim, and G. Elber, "Planar curve offset based on circle approximation", Comput. Aided Design, Vol. 28, pp 617-630, 1996. crossref(new window)

22.
S. W. Kim, S. C. Bae, and Y. J. Ahn, "An algorithm for $G^2$ offset approximation based on circle approximation by $G^2$ quadratic spline", Comput. Aided Design, Vol. 73, pp 36-40, 2016. crossref(new window)

23.
R. T. Farouki and T. Sakkalis, "Pythagorean hodographs", IBM J. Res. Dev., Vol. 34, pp 736-752, 1990. crossref(new window)

24.
R. T. Farouki, "Pythagorean-hodograph Curves", Berlin: Springer, pp. 381-391, 2008.

25.
D. S. Meek and D. J. Walton, "Geometric Hermite interpolation with Tschirnhausen cubics", J. Comput. Appl. Math., Vol. 81, pp. 299-309, 1997. crossref(new window)