Journal of Zhejiang University-SCIENCE A

, Volume 7, Issue 7, pp 1115–1123

# An efficient method for tracing planar implicit curves

• Yu Zheng-sheng
• Cai Yao-zhi
• Oh Min-jae
• Kim Tae-wan
• Peng Qun-sheng
Article

## Abstract

This paper presents a method for tracing a planar implicit curve f(x, y)=0 on a rectangular region based on continuation scheme. First, according to the starting track-point and the starting track-direction of the curve, make a new function F(x, y)=0 where the same curve with f(x, y)=0 is defined. Then we trace the curve between the two domains where F(x, y)>0 and F(x, y)<0 alternately, according to the two rules presented in this paper. Equal step size or adaptive step size can be used, when we trace the curve. An irregular planar implicit curve (such as the curve with large curvatures at some points on the curve), can be plotted if an adaptive step size is used. Moreover, this paper presents a scheme to search for the multiple points on the curve. Our method has the following advantages: (1) it can plot C0 planar implicit curves; (2) it can plot the planar implicit curves with multiple points; (3) by the help of using the two rules, our method does not need to compute the tangent vector at the points on the curve, and directly searches for the direction of the tracing curve; (4) the tracing procedure costs only one of two evaluations of function f(x, y)=0 per moving step, while most existing similar methods cost more evaluations of the function.

### Key words

Planar implicit curve Curve tracing Continuation method Geometric modeling

TP39

## Preview

### References

1. Barnhill, R.E., Farin, G., Jordan, M., Piper, B.R., 1987. Surface/surface intersection. Computer Aided Geometric Design, 4(1–2):3–16. [doi:10.1016/0167-8396(87)90020-3]
2. Bresenham, J.E., 1965. Algorithm for computer control of a digital plotter. IBM Systems Journal, 4(1):25–30.
3. Bresenham, J.E., 1977. A linear algorithm for incremental digital display of circular arcs. Comm. ACM, 20(2):100–106. [doi:10.1145/359423.359432]
4. Cai, Y.Z., 1990. Numerical Control Rendering Using Positive-negative Method. Zhejiang University Press, Hangzhou.Google Scholar
5. Chandler, R.E., 1988. A tracking algorithm for implicitly defined curves. IEEE Computer Graphics and Applications, 8(2):83–89. [doi:10.1109/38.506]
6. Cohen, E., 1976. A method for plotting curves defined by implicit equation. Computer Graphics (SIGGRAPH’76), 10(2):263–265. [doi:10.1145/965143.563321]
7. Lennon, W.J., Jordan, B.W., Holm, B.C., 1973. An improved algorithm for the generation of nonparametric curves. IEEE Transactions on Computers, C-22:1052–1060.
8. Lopes, H., Oliverira, J.B., de Figueiredo, L.H., 2002. Robust adaptive polygonal approximation of implicit curves. Computer & Graphics, 26(6):841–852. [doi:10.1016/S0097-8493(02)00173-5]
9. Martin, R., Shou, H., Voiculescu, I., Bowyer, A., Wang, G., 2002. Comparison of interval methods for plotting algebraic curves. Computer Aided Geometric Design, 19(7):553–587. [doi:10.1016/S0167-8396(02)00146-2]
10. Shou, H.H., Martin, R.R., Wang, G.J., Bowyer, A., Voiculescu, I., 2005. A Recursive Taylor Method for Algebraic Curves and Surfaces. In: Dokken, T., Jüttler, B. (Eds.), Computational Methods for Algebraic Spline Surface (COMPASS). Springer, p. 135–155.Google Scholar
11. van Aken, J., 1984. An efficient ellipse-drawing algorithm. IEEE Computer Graphics and Applications, 3:24–35.
12. van Aken, J., Novak, M., 1985. Curve-drawing algorithms for raster displays. ACM Transactions on Graphics, 4(2):147–169. [doi:10.1145/282918.282943]

## Authors and Affiliations

• Yu Zheng-sheng
• 1
• 2
• Cai Yao-zhi
• 3
• Oh Min-jae
• 2
• Kim Tae-wan
• 2
• Peng Qun-sheng
• 4
1. 1.Computer Science SchoolHangzhou Dianzi UniversityHangzhouChina
2. 2.Department of Naval Architecture and Ocean EngineeringSeoul National UniversitySeoulKorea
3. 3.Applied Mathematics DepartmentZhejiang UniversityHangzhouChina
4. 4.State Key Laboratory of CAD & CGZhejiang UniversityHangzhouChina