Bounds on partial derivatives of NURBS surfaces

In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.

Symmetric alteration of four knots of B-spline and NURBS surfaces

Modifying the knots of a B-spline curve, the shape of the curve will be changed. In this paper, we present the effect of the symmetric alteration of four knots of the B-spline and the NURBS surfaces, i.e., symmetrical alteration of the knots of surface, the extended paths of points of the surface will converge to a point which should be expressed with several control points. This theory can be used in the constrained shape modification of B-spline and NURBS surfaces.

Representing Quadric Surfaces Using NURBS Surfaces

A method for representing quadric surfaces using NURBS is presented. By means of the necessary and sufficient conditions for NURBS cu-rves to precisely represent circular arcs and other conics, quadric surfaces can be represented by NURBS surfaces with fewer control vertices. The method can be used not only for NURBS surface representation of quadric surfaces, but also for rounding polyhedrons. Many examples are given in the paper.

CONSTRUCTING RECTANGULAR NURBS SURFACES IN TERMS OF THREE OF ITS FOUR BOUNDARY CURVES

in the paper, first, using the transinfinite interpolation strategy rectangular free formsurfaces with C0 and C1 cross boundary continuity are respectively generated by three of its fourboundary curves and corresponding cross derivatives. Then, their singularity is removed by anapproximation approach. Finally the surfaces are converted to the NURBS surfaces of degrees 5 × 6and 7×10 respectively. At the end of the paper, conclusions and examples are given.

Reconstructing DEM using TLS point cloud data and NURBS surface

Underground coal mining inevitably results in land surface subsidence.Acquiring information on land surface subsidence is important in the detection of surface change.However,conventional data acquisition techniques cannot always retrieve information on whole subsidence area.This study focuses on the reconstruction of a digital elevation model（DEM） with terrestrial laser scanning（TLS） point cloud data.Firstly,the methodology of the DEM with terrestrial 3-dimensional laser scanning is introduced.Then,a DEM modeling approach that involves the application of curved non-uniform rational B-splines（NURBS） surface is put forward.Finally,the performance of the DEM modeling approach with different surface inverse methods is demonstrated.The results indicate that the DEM based on the point cloud data and curved NURBS surface can achieve satisfactory accuracy.In addition,the performance of the hyperbolic paraboloid appears to be better than that of the elliptic paraboloid.The reconstructed DEM is continuous and can easily be integrated into other programs.Such features are of great importance in monitoring dynamic ground surface subsidence.

DIAGONAL POINT BY POINT SURFACE DEVELOPMENT METHOD BASED ON NURBS SURFACE FOR BLANK SHAPE ESTIMATION OFCOVER PANEL

According to the characteristics of a complex cover panel, its geometry shape is described by the NURBS surface with great description capability. With the reference to the surface classification determined by Gauss curvature, the proportion of the mid-surface area between before and after being developed is derived from the displacement variation of the mid-surface in the normal vector direction of the sheet metal during the sheet metal forming process. Hereby, based on the curve development theory in differential geometry, a novel diagonal point by point surface development method is put forward to estimate a complex cover panel＇s blank contour efficiently. By comparing the sample＇s development result of diagonal point by point surface development method with that of available one-step method, the validity of the proposed surface development method is verified.

IMPROVED DELAUNAY TRIANGULATION FOR TRIMMED NURBS SURFACE

An improved algorithm of Delaunay triangulation is proposed by expanding the scope from a convex polygon to an arbitrary polygon area in which holes can be contained in the subdivision procedure. The data structure of generated triangles and the exuviationslike method play a key role, and a single connectivity domain (SCD) without holes is constructed as the initial part of the algorithm. Meanwhile, some examples show that the method can be applied to the triangulation of the trimmed NURBS surface. The result of surface tessellation can be used in many applications such as NC machining, finite element analysis, rendering and mechanism interference detection.

Constructing M-sided Patches with NURBS

The paper first summarizes the development of generating N-sided surface patches , andthen proposes a method of constructing general N-sided patches whose boundary curves and crossderivatives are Bspline curves of degree 3. The patch constructed is made up of n rectangularNURBS subpatches of degrees 7 ×7. The method is implemented in two steps : first , an N-sidedGregory patch is constructed, then, using our recent research on blended B-spline surface, nrectangular NURBS subpatches of degrees 7×7 are constructed to form an Nsided patch in termsof the data calculated in the first step. At the end of the paper , conclusions and some examples aregiven.

Approximation of NURBS Curves and Surfaces Using Adaptive Equidistant Parameterizations

Non-uniform rational B-spline (NURBS) curves and surfaces are very important tools for model- ling curves and surfaces. Several important details, such as the choice of the sample points, of the parame- terization, and of the termination condition, are however not well described. These details have a great in- fluence on the performance of the approximation algorithm, both in terms of quality as well as time and space usage. This paper described how to sample points, examining two standard parameterizations: equi- distant and chordal. A new and local parameterization, namely an adaptive equidistant model, was pro- posed, which enhances the equidistant model. Localization can also be used to enhance the chordal parameterization. For NURBS surfaces, one must choose which direction will be approximated first and must pay special attention to surfaces of degree 1 which have to be handled as a special case.

Extracting Silhouette Curves of NURBS Surfaces by Tracing Silhouette Points

Extracting silhouette curves is a key part of hidden curve removal algorithms. It relates directly to the quality of graphics generated by the algorithm, and to the speed and space efficiency of the algorithm. This paper proposes an algorithm for extracting silhouette curves that numerically determines the starting and ending silhouette points, then traces each starting point to obtain the whole silhouette curve. Silhouette loops are obtained by tracing silhouette points on isoparametric lines. This algorithm is suitable for extracting silhouette curves at any given tolerance without surface subdivision.The algorithm is fast and requires minimal storage space.

Accuracy assessment of mine walls＇ surface models derived from terrestrial laser scanning

The monitoring of highwall slopes at open-pit mines is an important task to ensure safe mining. For this reason, several techniques such as total station, radar, terrestrial Light Detection and Ranging （LIDAR） can be employed for surface measurement. The objective of this study is to investigate mesh algorithms, which can be used to interpolate 3D models of pit walls. Experiments were carried out at Coc Sau open-pit mine at Quang Ninh province of Vietnam, and at experimental mine of Akademia Gtrniczo-Hutnicza University of Science and Technology in Cracow, Poland. First, 3D point cloud data for the study area was acquired by using terrestrial LIDAR, then was used to generate mesh surfaces using three algorithms--Delaunay 2.5D XY Plane, Delaunay 2.5D Best Fitting Plane, and Mesh from Points. After that, the results were rectified and optimized. Subsequently, the optimized meshes were used for generation of non-uniform rational basis spline （NURBS） surfaces. Then, the NURBS surface accuracy was assessed. The results showed that the average distance between surface and point cloud was within range of 5.6-5.8 mm with deviation of 6.2-6.8 mm, depending on the used mesh. Additionally, the quality of surfaces depends on the quality of input data set and the algorithm used to generate mesh network, and the accuracy of computed NURBS surfaces fitting into pointset was 4-5 times lower than that of optimized mesh fitting. However, the accuracy of the final product allows determining displacements on the level of centimeters.

NEW DESIGN METHOD FOR HYPOID GEARS

A digital model is presented for the purpose of design, manufacture and measurement of hypoid gear, based on the non-uniform rational B-spline surface （NURBS） method. The digital model and the function-oriented active design technique are combined to form a new design method for hypoid gears. The method is well adaptable to CNC bevel gear cutting machines and CNC-controlled gear inspection machines, and can be used to create the initial machine tool cutting location data or program measurement path. The presented example verifies the method is correct.