B—SPLINE—SURFACE

The attribute upper index on u control points has the value K1 The attribute upper indexL.oa v control points has the value K2 

If only u rational =. T. (and v rational =.F.) the weights will depend on i only and the dimension of weights becomes (0..L1). 

The restrictions on knot multiplicities and the interpretation of multiple knots are defined as for B SPLINE CURVE. 

N (u) and (v) are the normalised B-spline basis functions defined on the knot sets  

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Greiner, G., and Hermann, K. Interpolating and Approximating Scattered 3D Data with Hierarchical Tensor Product B-splines, Surface Fitting and Multiresolution Methods, A. Le Mehaute, C. Rabut, and L. L. Schumaker (eds),Vanderbilt University Press, Nashville, TN, (1997) pp. 163-172. 

A sculptured surface is obtained by interpolation with two-dimensional B-spline-functions. A B-spline surface is considered as a collection of surface patches and the whole surface is a mosaic of these patches linked together with proper continuity (Figure 9). Due to its computational efficiency a uniform bicubic B-spline surface has been implemented. The bicubic B-spline patch can be written in matrix form as... 

Eck, M. Hoppe, H. (1996). Automatic reconstruction of b-spline surfaces of arbitrary topological type, ACM Siggraph, pp. 325-334. 

CYLINDRICAL SURFACE, CONICAL SURFACE, TOROIDAL SURFACE, PLANAR SURFACE, SURFACE OF REVOLUTION(DIM), SURFACE OF TRANSLATION(DIM), B SPLINE SURFACE ) ... 

The form number is optional, When present, it it used to identify special cases of quadric surfaces of degree 2, ruled surfaces, or surfaces of revolution. The receiving system will have to check whether the received B SPLINE SURFACE geometry actually conforms with this shape within an accuracy to be specified as a directive to the post-processor. If the... 

This is a composite surface having a simple rectangular topology with n u patches in u direction and n v patches in v direction. Each composite surface must be bounded and topological rectangular and therefore be either a B SPLINE SURFACE or a RECTANGULAR TRIMMED SURFACE. It is required that there is at least positional continuity between the adjacent patches. 

Weir, D., Milroy, M., Bradley, C., and Vickers, G. 1996. Reverse engineering physical models employing wrap-around B-spline surfaces and quadrics. Proc. Inst. Mech. Eng. B. 210 147-57. 

The spline surface S(x,y) consists of a set of bicubic polynomials, one in each panel, joined together with continuity up to the second derivative across the panel boundaries. Because each B-spline only extends over four adjacent knot intervals, the functions B.(x)C.(y) are each non-zero only over a rectangle composed of 16 adjacent panels in a 4 x 4 arrangement. The amount of calculation required to evaluate the coefficients y may be reduced by making use of this property. As before, least-squares methods may be used if the number of data exceeds (h+4)(jJ+4), which is usually the case. 

T. J.Cashman, N.A.Dodgson and M.A.Sabin Non-uniform B-spline subdivision using rehne and smooth. ppl21-137 in Mathematics of Surfaces XII, (eds Martin, Sabin and Winkler), Springer LNCS4647, 2007 ISBN 3-540-73842-8... 

Another advantage of the B-spline filter is that there are very fast and accurate recurrence schemes for evaluating and differentiating the resultant surfaces (de Boor, (1972) Cox (1972)). This means that once a good surface of experimental data has been determined from a TS-PFR experiment, there is a fast process for calculating rates at any desired point on that surface within the range of the experimental data. This capability will produce the rate surface and is the source of the unlimited number of rates available from TS-PFR experiments. 

In the presence of rigid bodies in contact with the free surface, such as the case of the railroad ties, the same method can be used along with the Rigid Surface Boundary Element developed by Rizos (2000) to compute the BIRE functions of the soil-tie interface. To this end, the surface of the soil region is discretized with 8-node surface elements and the 4-node rigid surface boundary elements reported in Rizos (2000) are placed at soil-tie interface to represent the ties. The response of each tie is described by three translations and three rotations of the center of the tie. In order to compute the BIRE of the soil-tie system, each of the six degrees of freedom (dof) of the loaded tie is excited by a B-Spline impulse force and the response of all ties in the system is computed following the procedures introduced in O Brien and Rizos (2005). The computed BIRE are characteristic responses of the system and are expressed symbolically in matrix form as... 

Figure 9. B-spline patch as a mosaic element of a surface.

If a substantial number of 2-D curves are available at varying Z depths, a 3-D model can be created by curve fitting the points along the Z direction (axis) simultaneously with the X and Y directions. The most common way to create a digitized model for 3-D objects is from a 3-D model. In this method a measurement device such as a coordinate measuring machine is programmed to traverse to known X and Y locations, and the Z height of the model is measured and recorded at those locations. This series of points is then fitted with smooth curves such as b-splines, which are then used to develop surfaces like b-surfaces. 

In Fig. 2, we summarize the detailed steps involved in developing the atomistic pore model from the lattice model. From the lattice configuration (Fig. 2a), we isolate a pore and model its surfece as a b-sphne [24], The b-spline approach serves as a reasonable interpolation to obtain a pore surfece with sub-atomistic resolution <0.1 nm. The control points in this b-spline model are the lattice surface coordinates, which allows us to retain the sur ce undulations and irregularities fiom the mimetic simulation. The atomistic pore is now... 

Fig. 2. Steps in developing the atomistic pore from the mesoscaJe model, (a) snapshot of lattice simulation box (b) isolated pore (c) pore surface modeled as b-spline (d) silica block (e) mesopores carved out (f) model material after carving out the micropores and pore surface relaxation. White particles are the hydrogen atoms on the pore surface.

The result of polygon mesh optimization stage is a fully closed model (fig. 6) ready to generate NURBS (Non-uniform Rational B-Spline) curves or surfaces network. 

This quantity is computed on a smoothed /-field /, which is obtained by convo-lutmg/with a B-spline of degree 2. The normal computed with an 18-points stencil at the center of the cell face. These face-centered gradients of / are then used to calculate the capillary stress tensor at the cell faces. Central differences yield the actual surface tension force in the cell centers. This surface tension force is then evenly distributed to the cell faces, where the velocities are stored. A detailed description of the above methods and the finite-volume method for fluid dynamics used in FS3D can be found in [25]. 

Fig. 6. Cubic B-spline interpolation versus surface spline interpolation.

An explicit convergence proof for Sobolev norms can be found in [15). Let us fix the sampling nodes and increase the domain fi. What we observe is that, similar to the surface spline, the bicubic B-spline interpolant tends to a linear polynomial. 

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