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1.
Procedurally representing lofted surfaces   总被引:1,自引:0,他引:1  
The use of tensor-product B-spline surfaces to construct lofted surfaces is discussed. It is shown that several problems must be overcome if tensor-product B-splines are to represent complex lofted surfaces. A method that seems to satisfy all the necessary requirements for parametrically describing such a surface is given. It uses a novel type of surface implemented in a production design system  相似文献   

2.
The formulations for parametric curves and surfaces that are based on control points are revised to use control lines and control planes instead. Curves defined by control lines are called control-line curves or plus curves, and surfaces defined by control planes are called control-plane surfaces or plus surfaces; the plus implies that in addition to the control points, gradient vectors at the control points are used to design curves and surfaces. The new curve and surface formulations provide more flexibility than traditional formulations in geometric design. Properties of plus curves and surfaces are investigated and an application of plus surfaces in smooth parametric representation of polygon meshes is introduced.  相似文献   

3.
Spanning tree contours, a special class of Truchet contour based upon a random spanning tree of a Truchet tiling's underlying graph, are presented. This spanning tree method is extended to three dimensions to define a Truchet surface with properties similar to its two-dimensional counterpart. Both contour and surface are smooth, have known minimum curvature and known maximum distance to interior points, and high ratios of perimeter to area and area to volume, respectively. Expressions for calculating contour length, contour area, surface area and surface volume directly from the spanning tree are given.  相似文献   

4.
在分析了Tiller给出的B样条曲线节点消去算法的基础上,提出了改进算法。改进算法充分地利用了B样条曲线的局部性质,无需考虑节点消去的顺序,一次消去多个节点。实验表明,与Tiller的算法相比较,改进后的算法效率有较大提高。  相似文献   

5.
A method for constructing rational Pythagorean-hodograph (PH) curves in R3 is proposed, based on prescribing a field of rational unit tangent vectors. This tangent field, together with its first derivative, defines the orientation of the curve osculating planes. Augmenting this orientation information with a rational support function, that specifies the distance of each osculating plane from the origin, then completely defines a one-parameter family of osculating planes, whose envelope is a developable ruled surface. The rational PH space curve is identified as the edge of regression (or cuspidal edge) of this developable surface. Such curves have rational parametric speed, and also rational adapted frames that satisfy the same conditions as polynomial PH curves in order to be rotation-minimizing with respect to the tangent. The key properties of such rational PH space curves are derived and illustrated by examples, and simple algorithms for their practical construction by geometric Hermite interpolation are also proposed.  相似文献   

6.
7.
Curvature continuous curves and surfaces   总被引:5,自引:0,他引:5  
A simple methods is given for constructing the Bézier points of curvature continuous cubic spline curves and surfaces from their B-spline control points. The method is similar to the well-known construction of Bézier points of C2 splines from their B-spline control points. The new construction allows the use of all results of the powerful Bernstein-Bézier technique in the realm of geometric splines.

A simple introduction to nu- and beta-splines is also derived, as well as some simple geometric properties of beta-splines.  相似文献   


8.
Rasterizing algebraic curves and surfaces   总被引:2,自引:0,他引:2  
A new, recursive, space-subdivision algorithm for rasterizing algebraic curves and surfaces gets its accuracy from a newly devised, computationally efficient, and asymptotically correct test. The approach followed is essentially the interval arithmetic method for rendering implicit curves. The author's contribution is a particularly efficient way to construct inclusion functions for polynomials. An ideal algorithm is given for rendering an algebraic curve Z(f)={(x,y):f(x,y)=0} in a square box of side n. The algorithm scans the square and paints only those pixels cut by the curve. This algorithm is ideal, because every correct algorithm should paint exactly the same pixels, but it is impractical. It requires n2 test evaluations, one for each pixel in the square. However, since in general it will be rendering a curve on a planar region, the number of pixels it is expected to paint is only O(n). We need a more efficient algorithm. There are two issues to examine. The first is how to reduce the computational complexity by recursive subdivision. The second is how to test whether the curve Z(f) cuts a square  相似文献   

9.
A representation for parametric cubic curves and surfaces is presented which incorporates the polygonal approach popularized by the Bézier and B-spline schemes. Since the curves more closely mimic the polygon than their counterparts employing the Bézier or B-spline schemes, the method is potentially useful for creating and manipulating geometric models. In connection with the logical extension of the approach from curves to surfaces, a discussion of the relationship between continuity and redundant data storage is included.  相似文献   

10.
Among curves and surfaces defined by parametric polynomials, the cases dealt with here are those which only have to comply with the requirement to run through a certain number of points previously located in space. The process can be totally automatic, but the results are liable to be altered by arbitrary decisions.  相似文献   

11.
The de Casteljau evaluation algorithm applied to a finite sequence of control points defines a Bézier curve. This evaluation procedure also generates a subdivision algorithm and the limit of the subdivision process is this same Bézier curve. Extending the de Casteljau subdivision algorithm to an infinite sequence of control points defines a new family of curves. Here, limits of this stationary non-uniform subdivision process are shown to be equivalent to curves whose control points are the original data points and whose blending functions are given by the Poisson distribution. Thus this approach generalizes standard subdivision techniques from polynomials to arbitrary analytic functions. Extensions of this new subdivision scheme from curves to tensor product surfaces are also discussed.  相似文献   

12.
Free-Form Deformation Techniques (FFD) are commonly used to generate animations, where a polygonal approximation of the final object suffices for visualization purposes. However, for some CAD/CAM applications, we need an explicit expression of the object, rather than a collection of sampled points. If both object and deformation are polynomial, their composition yields a result that is also polynomial, albeit very high degree, something undesirable in real applications. To solve this problem, we transform each curve or surface composing the object, usually expressed in the Bernstein basis, to a modified Newton form. In this representation, the two-point analogue of Taylor expansions, the composition admits a simple expression in terms of discrete convolutions, and degree reduction corresponding to Hermite approximation is trivial by dropping high-degree coefficients. Furthermore, degree-reduction can be incorporated into the composition. Finally, the deformed curve or surface is converted back to the Bernstein form. This method extends to general non-polynomial deformation, such as bending and twisting, by computing a polynomial approximant of the deformation.  相似文献   

13.
Implicit curves and surfaces in CAGD   总被引:5,自引:0,他引:5  
The role of implicit curves and surfaces in computer-aided geometric design (CAGD) are described. The ways in which the study of implicit algebraic curves and surfaces draws on algebraic geometry are reviewed. The implicitization of parametric curves and surfaces, parameterization of implicits, and techniques used to circumvent conversions between implicit and parametric representations are discussed  相似文献   

14.
Parabolic curves of evolving surfaces   总被引:1,自引:1,他引:1  
In this article we show how certain geometric structures which are also associated with a smooth surface evolve as the shape of the surface changes in a 1-parameter family. We concentrate on the parabolic set and its image under the Gauss map, but the same techniques also classify the changes in the dual of the surface. All these have significance for computer vision, for example through their connection with specularities and apparent contours. With the aid of our complete classification, which includes all the phenomena associated with multi-contact tangent planes as well as those associated with parabolic sets, we re-examine examples given by J. Koenderink in his book (1990) under the title of Morphological Scripts.We also explain some of the connections between parabolic sets and ridges of a surface, where principal curvatures achieve turning values along lines of curvature.The point of view taken is the analysis of the contact between surfaces and their tangent planes. A systematic investigation of this yields the results using singularity theory. The mathematical details are suppressed here and appear in Bruce et al. (1993).The third author was supported by the Esprit grant VIVA while this paper was in preparation.  相似文献   

15.
16.
运用积分定义的方式,构造了带多形状参数的均匀CB样条曲线曲面,随着基函数次数的升高,形状参数的范围可以扩展,具体讨论了3~9次时形状参数的取值范围.它们包含均匀CB样条曲线曲面为其特例且具有均匀CB样条曲线曲面的主要性质.改变形状参数的值,能整体或局部调控曲线曲面的形状,比均匀CB样条具有更强的造型能力,在CAD/CAM中具有很好的应用前景.  相似文献   

17.
We present iterative algorithms for B-spline scale-space smoothing of geometric data and recovery of high frequency information in the smoothing process. The scale-space representation is based on a directional smoothing process using B-splines. If the geometric data are approximated or modelled by uniform B-splines or box-splines then the scale-space smoothing produces B-spline curves or box-spline surfaces. The method is applicable to geometric data processing and geometric modelling of free-form curves and surfaces from quadrilateral polyhedra with extraordinary vertices.  相似文献   

18.
This paper deals with subdivision depth computation technique for n-ary subdivision curves/surfaces. This technique also includes error bound evaluation technique for n-ary subdivision curves/surfaces with their control polygon. Both techniques provide error control tools in subdivision schemes.  相似文献   

19.
Curvature and the fairness of curves and surfaces   总被引:11,自引:0,他引:11  
The use of curvature plots for the design of curves that have to meet aesthetic requirements is discussed. The aim is to emphasize the usefulness of curvature as a measure for curve fairness. A local method to optimize the curvature plot of a given cubic spline curve is presented. It automatically determines where the curve is to be faired and can be applied repeatedly. A straightforward generalization to surfaces is easy to formulate  相似文献   

20.
Offset of curves on tessellated surfaces   总被引:2,自引:0,他引:2  
Geodesic offset of curves on surfaces is an important and useful tool of computer aided design for applications such as generation of tool paths for NC machining and simulation of fibre path on tool surfaces in composites manufacturing. For many industrial and graphic applications, tessellation representation is used for curves and surfaces because of its simplicity in representation and for simpler and faster geometric operations. The paper presents an algorithm for computing offset of curves on tessellated surfaces. A curve on tessellation (COT) is represented as a sequence of 3D points, with each line segment of every two consecutive points lying exactly on the tessellation. With an incremental approach of the algorithm to compute offset COT, the final offset curve position is obtained through several intermediate offset curve positions. Each offset curve position is obtained by offsetting all the points of COT along the tessellation in such a way that all the line segments gets offset exactly along the faces of tessellation in which the line segments are contained. The algorithm, based entirely on tessellation representation, completely eliminates the formation of local self-intersections. Global self-intersections if any, are detected and corrected explicitly. Offset of both open and closed tessellated curves, either in a plane or on a tessellated surface, can be generated using the proposed approach. The computation of offset COT is very accurate within the tessellation tolerance.  相似文献   

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