Paths Of C-curves And H-curves

C-Bézier curve and C-B-spline curve are called C-curves. H-Béziercurve and H-B-spline curve are called H-curves Because they can dealwith free form curves, conic sections and transcendental curves inuniform. They are widely used in geometric sculpt, and there is a shapeparameter in both of them. The geometric effect of this parameter isdiscussed in this paper.Take the C-curves as a example, after making a proper substitutionof variable, t can be defined from 0 to 1, andαis from 0 toπ. When t isrunning from 0 to 1, we get a C-curve, but now we fix up t, andαisrunning from 0 toπ, the points of the C-curve will move on certaincurves called Path. When t=1/2, Path of the C-curves and H-curves ofdegree two and three are lines, but not for the Path of degree four, weneed more about the control polygon. Paths are commonly curves. Butthe reference [38] pointed out that they can closely be approximated bylines. By this method we can get a C-curve passing a given point easily.We study the technique further in this paper and we study thegeometric property of the Path of the C-curves and H-curves of degreetwo and four and the application in sculpt in this paper.