B-spline curve fitting using dominant points [CAD 2007, LNCS 2006, CGACC 2005]
This work presents a new approach for B-spline curve fitting to a sequence of points, which is motivated by an insight that properly selected points, called dominant points, can play an important role in producing better curve approximation. The approach is substantially different from the conventional ones in knot placement and dominant point selection. It can generate a B-spline curve in good quality with less deviation. Adopted in the error-bounded curve approximation, it can play an important role in generating B-spline curves with much less control points. |
Morphological development and transformation of Bezier curves based on ribs and fans [SPM 2007]| This work presents novel methods to generate a sequence of shapes that represents the pattern of morphological development or transformation of a Bezier curve. The methods utilize the intrinsic geometric structures of a Bezier curve that are derived from rib and fan decomposition (RFD). Morphological development based on RFD shows a characteristic pattern of structural growth of a Bezier curve. Morphological transformation based RFD utilizes development patterns of given curves. |  |
Look at an example of morphological transformation between two different set of Bezier curves. |
Ribs and fans of Bezier curves and surfaces [CADA 2005, JCST 2006]
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Each Bezier curve or surface can be decomposed into new geometric entities called ribs and fans, which is referred to as rib and fan decomposition (RFD). |
(a) quartic Bezier curve b(t) = r4(t) and its cubic rib r3(t) with their control polygons
(b) the control vectors of the fan f2(t)
(c) fan lines of the scaled fan 2t(1-t) on the rib r3(t)
(d) fan lines of all the scaled fans on the corresponding ribs
Error-bounded biarc approximation of planar curves [CAD 2004]
Given a planar parametric curve and a desired tolerance t, this work presents a method for generating a G1 arc spline made of biarcs such that the Hausdorff distance between the curve and the arc spline is smaller than the tolerance t. We can get such a G1 arc spline as follows:
(a) cubic B-spline curve (b) approximated polygon (c) biarcs obtained via biarc fitting with r=1 (d) biarcs obtained via optimal single biarc fitting |
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Optimal single biarc fitting to a polygon [CADA 2004]
Given a polygon and its two end tangents, this work presents a method for generating a biarc that minimizes the distance between the biarc and the polygon. (a) Biarc obtained with Piegl's choice (r=1, e=0.07228) (b) Biarc obtained with Sabin's choice (r=1.726, e=0.2084) (c) Biarc obtained with the optimal choice (r=0.784, e=0.02179) |
Error-bounded B-spline curve approximation to planar curves [CAGD 2004]|
Given a planar curve and a desired tolerance t, this work presents a method for generating a B-spline curve with fewer control points such that the Hausdorff distance between the given curve and the B-spline curve is smaller than the tolerance t. |
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We can get such a B-spline curve as follows:
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 Colored images are created with ray tracing techniques. We can use this method in designing an aspheric optical lens of a beam projector as follows:
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Closed B-spline curve interpolation to point data [CAD 2001]|
Given a closed polygon and a degree of B-splines, this work presents a method for generating a closed B-spline curve that pass through all the polygonal points. [Note] Node and knot placement for closed B-spline curve interpolation to point data heavily depends on whether the degree of B-splines is odd or not. For odd degree B-splines, the natural method of setting knots to coincide with nodes (i.e. parameters) works very well and provides the good quality. However, when the degree is even, the usual methods including the natural method can have problems and result in the bad quality. This paper presents a method, called the shifting method, which works well for even degree B-spline interpolation. It has nearly the same properties as the natural method does for odd degree B-splines. It is simple and provides the good quality of a resultant curve.  (a) Cubic B-spline curve obtained with the natural method using uniform parameters (b) Cubic B-spline curve obtained with the natural method using centripetal parameters (c) Cubic B-spline curve obtained with the natural method using chord length parameters (d) Quadratic B-spline curve obtained with the shifting method using uniform parameters (e) Quadratic B-spline curve obtained with the shifting method using centripetal parameters (f) Quadratic B-spline curve obtained with the shifting method using chord length parameters |