An efficient method for tracing planar implicit curves
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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 wordsPlanar implicit curve Curve tracing Continuation method Geometric modeling
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