Interior Knots Research Articles

G1 continuous conditions of biquartic B-spline surfaces

The necessary and sufficient conditions of G 1 continuity between two biquartic B-spline patches with single interior knots are obtained, and the intrinsic conditions of the common boundary control points are also presented. Further, some remarkable difference on the conditions for G 1 continuity of two adjacent Bézier patches and those of two adjacent B-spline patches are studied. As a result, the construction of local scheme by using biquartic B-spline surfaces with the single interior knots does not exist.

Reparameterization of piecewise rational Bezier curves and its applications

. Although the curve segments are C1 continuous in three dimensions, they may be C0 continuous in four dimensions. In this case, the multiplicity of each interior knot cannot be reduced and the B-spline basis function becomes C0 continuous. Using a surface generation method, such as skinning these kinds of rational B-spline curves to construct an interpolatory surface, may generate surfaces with C0 continuity. This paper presents a reparameterization method for reducing the multiplicity of each interior knot to make the curve segments C1 continuous in four dimensions. The reparameterized rational B-spline curve has the same shape and degree as before and also has a standard form. Some applications in skinned surface and ruled surface generation based on the reparameterized curves are shown.

A method for approximate NURBS curve compatibility based on multiple curve refitting

This paper presents an approximate method for making a set of NURBS curves of various types compatible. The method is based on refitting input curves simultaneously on a common knot vector while making the refitted curves have parametrically regular transition across curves. It adopts a concept of reducing the number of interior knots of the common knot vector to acquire more compact representation for the resultant curves. It also incorporates energy minimization in the curve refitting process to stabilize the solution of the curve fitting problem and to get smooth curves. The proposed method can be utilized in several construction tasks including surface skinning and surface creation from a mesh of curves. It plays an important role in improving the quality of surfaces and realizing efficient reduction of their control points especially when the parameterization and the compatibility of input curves are not good enough. Some experimental results demonstrate its usefulness and quality.

Algorithms for G 1 connection of multiple parametric bicubic NURBS surfaces

The objective of this paper is to introduce an innovative approach for constructing effective algorithms for removing gaps between parametric NURBS surfaces in three-space, while maintaining geometrical smoothness for the combined (or compound) surface. Similar to the degenerate case of tensor-product B-spline surfaces, if the underlying knot sequences along the connecting boundaries of two NURBS surfaces are proportional, then the parametric surfaces can be connected in a G1 fashion. This approach can be easily extended to connecting three or four parametric NURBS surfaces. We will demonstrate the feasibility of our approach by focusing on the C1 bicubic setting with knot sequences being equally-spaced and having double interior knots.

The interior knot structure of Pinus sylvestris stems

Knots negatively influence the use of pine wood for many products. Knowledge on the interior knot structure of the stems is therefore essential for many decisions regarding wood utilization. This knot structure was studied using data from 185 stems of Scots pine (Pinus sylvestris L.) scanned by computerized tomography. First with respect to distance between whorls and number of knots per whorl, and second, by dividing the stems into knot structure sections, to determine if knot diameters follow some kind of predictable and uniform pattern. The results showed that distance between whorls and number of knots per whorl both increased with increasing site index and reached their maximum values at 2–4 m height in the stems. The study also showed that a division of the stems into four knot‐structure sections can be a useful basis for description of the interior knot structure. These sections were named establishment section, constant growth section, retardation section, and crown section.

A local algorithm for constructing non-negative cubic splines

An interpolating spline which interpolates positive function values is not necessarily positive for all arguments; a typical example of this occurs in the spline interpolation of the normal density function of probability theory. We describe an algorithm which produces non-negative interpolating splines provided the function to be interpolated is positive. We start with the interpolating natural spline and redefine it in those intervals for which the spline has negative values; we do this by adding few extra knots to those intervals. First we show that for any choice of extra knots it is possible to construct a spline which is non-negative. We call such a spline feasible. Second we show that within the set of feasible splines there is a best spline, in the sense that the functional which reflects the strain energy is minimized. Finally, by using a necessary optimality criterion we obtain an optimal spline which minimizes the strain energy also with respect to the free knots. The resulting formulae give explicit results and are very simple. Our numerical results show that the final step of selecting the interior knots appropriately gives very satisfactory results in a very efficient manner. The algorithm can be applied also to produce splines which stay below upper or stay above lower constant bounds. It is also very useful as a starting procedure for finding the globally best spline.

Cubic spline techniques for fitting X-ray cross sections

Cubic spline least squares fitting algorithms are developed for continous functions and functions with jump discontinuities. The cubic spline coefficients for incoherent and coherent scattering cross sections for elements from sodium through cobalt are reported for energies from 1 to 150 keV, and when two interior knots (three regions) are used they yield accuracies better than those obtined by a single cubic polynomial. Specialized discontinuous cubic spline functions are used to fit X-ray photoelectric cross sections to allow accurate fitting across the absorption edge discontinuities for a wide range of energies with a very simple function. Cubic spline coefficients for photoelectric cross sections of all elements in the atomic number range from 11 to 27 and in the energy range from 1 to 150 keV are reported. The cubic spline representation provides a computationally faster method of using X-ray cross sections than polynomial interpolation from discrete data sets and requires much less computer storage.

Unicity of Best Mean Approximation by Second Order Splines with Variable Knots

Let $S_N^2$ denote the nonlinear manifold of second order splines defined on [0, 1] having at most $N$ interior knots, counting multiplicities. We consider the ques tion of unicity of best approximations to a function $f$ by elements of $S_N^2$. Approximation relative to the ${L_2}[0,1]$ norm is treated first, with the results then extended to the best ${L_1}$ and best one-sided ${L_1}$ approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: a sufficiently smooth function $f$ satisfying $f” > 0$ has a unique best approximant from $S_N^2$ provided either $\log f”$ is concave, or $N$ is sufficiently large, $N \geqslant {N_0}(f)$; for any $N$, there is a smooth function $f$, with $f” > 0$, having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping.