Elaine Cohen
Abstract
Recently there has been a great deal of interest in the use of “tension” parameters to augment control mesh vertices as design handles for piecewise polynomials. A particular local cubic basis called B-splines, which has been termed a “generalization of B-splines, v has been proposed as an appropriate basis. These functions are defined only for floating knot sequences. This paper uses the known property of B-splines that with appropriate knot vectors span what are called here spaces of tensioned splines, and that particular combinations of them, called LT-splines, form bases for the spaces of tensioned splines. In addition, this paper shows that these new proposed bases have the variation diminishing property, the convex hull property, and straightforward knot insertion algorithms, and that both curves and individual basis functions can be easily computed. Sometimes it is desirable to interpolate points and also use these tension parameters, so interpolation methods using the LT-spline bases are presented. Finally, the above properties are established for uniform and nonuniform knot vectors, open and floating end conditions, and homogeneous and nonhomogeneous tension parameter pairs.
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Index Terms
- A new local basis for designing with tensioned splines
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Reviews
Gabriel Constantin Barzescu
The theoretical and practical importance of tensioned spline functions for interactive design of smooth technical curves and surfaces has been increasing since their first formulation in 1986 [1]. Addressing readers with a solid background in the field, the paper presents a new approach to the topic of tensioned splines, including the definition of local tensioned (LT) splines, which form bases for the spaces of tensioned splines. This approach borrows the computational and theoretical properties of the well-known B-splines. In the introduction, the author briefly describes the state of the art in the research on the use of tension parameters to augment control of cubic parametric splines. Then the variation-diminishing property and the convex hull property are given, and algorithms for knot insertion and computing of both curves and individual basis functions are presented. The paper also analyses interpolation methods using the LT-spline bases. Finally, the cases of uniform and nonuniform knot vectors, open and floating end conditions, homogeneous and nonhomogeneous tension parameter pairs, and tensioned surfaces are considered. Original contributions together with a critical analysis of the state of the art in the theory of tensioned splines make this paper one of the strongest in the field. A special feature of this valuable research contribution is a unifying conceptual and computational framework for the interactive design of curves and surfaces with tensioned splines. A historical reference to the first pioneer of tensioned splines [1] would have been helpful.
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