1,467 research outputs found
On multi-degree splines
Multi-degree splines are piecewise polynomial functions having sections of
different degrees. For these splines, we discuss the construction of a B-spline
basis by means of integral recurrence relations, extending the class of
multi-degree splines that can be derived by existing approaches. We then
propose a new alternative method for constructing and evaluating the B-spline
basis, based on the use of so-called transition functions. Using the transition
functions we develop general algorithms for knot-insertion, degree elevation
and conversion to B\'ezier form, essential tools for applications in geometric
modeling. We present numerical examples and briefly discuss how the same idea
can be used in order to construct geometrically continuous multi-degree
splines
A practical method for computing with piecewise Chebyshevian splines
A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial splines, the related parametric curves exhibit the desired properties of convex hull inclusion, variation diminution and intuitive relation between the curve shape and the location of the control points. For a good-for-design space, in this paper we construct a set of functions, called transition functions, which allow for efficient computation of the B-spline basis, even in the case of nonuniform and multiple knots. Moreover, we show how the spline coefficients of the representations associated with a refined knot partition and with a raised order can conveniently be expressed by means of transition functions.
This result allows us to provide effective procedures that generalize the classical knot insertion and degree raising algorithms for polynomial splines. We further discuss how the approach can straightforwardly be generalized to deal with geometrically continuous piecewise Chebyshevian splines as well as with splines having section spaces of different dimensions. From a numerical point of view, we show that the proposed evaluation method is easier to implement and has higher accuracy than other existing algorithms
Blending techniques in Curve and Surface constructions
Source at https://www.geofo.no/geofoN.html. <p
q-Bernstein polynomials and Bézier curves
AbstractWe define q-Bernstein polynomials, which generalize the classical Bernstein polynomials, and show that the difference of two consecutive q-Bernstein polynomials of a function f can be expressed in terms of second-order divided differences of f. It is also shown that the approximation to a convex function by its q-Bernstein polynomials is one sided.A parametric curve is represented using a generalized Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We study the nature of degree elevation and degree reduction for this basis and show that degree elevation is variation diminishing, as for the classical Bernstein basis
Gn blending multiple surfaces in polar coordinates
International audienceThis paper proposes a method of Gn blending multiple parametric surfaces in polar coordinates. It models the geometric continuity conditions of parametric surfaces in polar coordinates and presents a mechanism of converting a Cartesian parametric surface into its polar coordinate form. The basic idea is first to reparameterize the parametric blendees into the form of polar coordinates. Then they are blended simultaneously by a basis function in the complex domain. To extend its compatibility, we also propose a method of converting polar coordinate blending surface into N NURBS patches. One application of this technique is to fill N-sided holes. Examples are presented to show its feasibility and practicability
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