Interference-free tool-path generation in the NC machining of parametric compound surfaces

A method is presented for generating interference-free tool paths from parametric compound surfaces. A parametric compound surface is a surface that consists of parametric surface elements. The method is largely composed of two steps: points are obtained from a compound surface to be converted into a triangular polyhedron; tool paths are then generated from the polyhedron. An efficient algorithm is used in the calculation of cutter-location data, and planar tool paths, which are suitable for metal cutting, are produced. The time taken to obtain all the tool paths from a surface model that consists of a large number of parametric surfaces is short. Some real applications are presented.     

An algorithm for the interpolation of hybrid curves

Real time tool path generation consists of off-line design and real time interpolation of tool paths. An hybrid curve is the intersection of a parametric surface and an implicit surface. Previous work in tool path interpolation focused mainly in the interpolation of parametric curves. Tool paths designed by drive surface methods are hybrid curves which, in general, cannot be represented as parametric curves. An algorithm for the interpolation of hybrid curves is proposed in this paper. The algorithm is based on interpolation of the projection of the hybrid curve into the parametric domain. Each increment involves a second-order interpolation step augmented by iterative error reduction.Simulations of hybrid curve interpolation have been carried out. They are based on practical surfaces represented as NURB surfaces and implicit surfaces including a plane, a cylinder and a high order algebraic surface. They demonstrate that under typical machining conditions, interpolation error is well within the accuracy requirements of typical machining and that the use of one iteration error reduction can significantly reduce the path deviation. These show that the proposed algorithm is potentially useful for tool path interpolation for the machining of parametric surfaces.     

Transition to canonical principal parameters on minimal surfaces

Ganchev has recently proposed a new approach to minimal surfaces. Introducing canonical principal parameters for these surfaces, he has proved that the normal curvature determines the surface up to its position in the space. Here we prove a theorem that permits to obtain equations of a minimal surface in canonical principal parameters and we make some applications on parametric polynomial minimal surfaces. Thus we show that Ganchev's approach implies an effective method to prove the coincidence of two minimal surfaces given in isothermal coordinates by different parametric equations.     

Bézier representation for cubic surface patches

The paper describes a new method for creating rectangular Bézier surface patches on an implicit cubic surface. Traditional techniques for representing surfaces have relied on parametric representations of surfaces, which, in general, generate surfaces of implicit degree 8 in the case of rectangular Bézier surfaces with rational biquadratic parameterization. The method constructs low-degree algebraic surface patches by reducing the implicit degree from 8 to 3. The construction uses a rectangular biquadratic Bézier control polyhedron that is embedded within a tetrahedron and satisfies a projective constraint. The control polyhedron and the resulting cubic surface patch satisfy all of the standard properties of parametric Bézier surfaces, including interpolation of the corners of the control polyhedron and the convex-hull property.     

Feature-based decomposition of trimmed surface

A trimmed surface is usually represented by a parametric surface and a set of trimming curves. Because of the complexity in manipulating trimmed surfaces, many CAD processes and algorithms cannot be applied to trimmed surfaces directly. It is thus desirable to represent a trimmed surface by a group of regular surfaces. In this paper, an algorithm for decomposing a trimmed surface is presented. First, bisectors of the Voronoï diagram developed in the parametric space are used to define an isolated region for every trimming curve. Feature points on the trimming curves are extracted by considering curvatures of the curves. Correspondence between feature points and vertices on the bisectors are established by considering the similarity between the trimming curves and the bisectors. Regions of parametric patches are then identified. Finally, a group of regular surfaces are constructed by interpolating a set of sampled surface points on each of the identified regions.     

CAGD中参数曲面的光滑拼接研究

参数曲面作为CAGD中形状数学描述的标准形式一直受到关注,而参数曲面的光滑拼接作为实现复杂客体几何造型的重要手段一直是该领域的一个热点和难点问题.以不同参数域的参数曲面为线索,对常用的矩形域和三角域参数曲面的GC1光滑拼接的条件进行了分析,同时对这些条件在实际应用中的一些问题进行了讨论,最后对曲面光滑拼接中一些令人关注问题进行了展望.     

The homotopy model: a generalized model for smooth surface generation from cross sectional data

A generalized model, called the homotopy model, is presented to reconstruct surfaces from cross-sectional data of objects using a homotopy to generate surfaces connecting consecutive contours. The homotopy model consists of continuous toroidal graph representation and homotopic generation of surfaces from the representation. It is shown that the homotopy model includes triangulation as a special case and generates smooth parametric surfaces from contour-line definitions using homotopy. The model can be applied to contours represented by parametric curves as well as linear line segments. First, a heuristic method that finds the optimal path on the toroidal graph is presented. Then the toroidal graph is expanded to a continuous version. Finally, homotopy is used for reconstructing parametric surfaces from the toroidal graph representation. A loft surface is also a special case of homotopy, a straight-line homotopy. Homotopy that corresponds to the cardinal spline surface is also introduced. Three-dimensional surface reconstruction of human auditory surface reconstruction of human auditory ossicles illustrates the advantages of the homotopy model over the others.     

Stripification of Free-Form Surfaces With Global Error Bounds for Developable Approximation

Developable surfaces have many desired properties in the manufacturing process. Since most existing CAD systems utilize tensor-product parametric surfaces including B-splines as design primitives, there is a great demand in industry to convert a general free-form parametric surface within a prescribed global error bound into developable patches. In this paper, we propose a practical and efficient solution to approximate a rectangular parametric surface with a small set of C ⁰ -joint developable strips. The key contribution of the proposed algorithm is that, several optimization problems are elegantly solved in a sequence that offers a controllable global error bound on the developable surface approximation. Experimental results are presented to demonstrate the effectiveness and stability of the proposed algorithm.     

Point-Based Rendering of Non-Manifold Surfaces

We are concerned with producing high‐quality images of parametric and implicit surfaces, in particular those with non‐manifold features. We present a point‐based technique for rendering implicit surfaces that uses octree spatial subdivision with a natural interval exclusion test that guarantees that no parts of the surface are missed. This allows us to render non‐manifold implicit surfaces at speeds comparable to parametric surfaces. We also derive criteria that guarantee complete pixel coverage of the surface. The point‐based method allows for hidden surface elimination using a z‐buffer, and shadow casting using a shadow buffer. We illustrate the technique with a number of surfaces, and discuss its advantages and disadvantages.     

Surface algorithms using bounds on derivatives

This paper generalizes three very important algorithms for surfaces which previously only worked well with polynomials. The algorithms are calculation of piecewise linear approximations, min-max boxes, and surface/surface intersections. The class of surfaces that can be handled are all parametric C² surfaces. All the algorithms are related by theorems from approximation theory which give information about the maximum deviation an approximation to a surface can have if bounds on partial derivatives are known. We generalize these theorems to work with parametric geometry, and we also show how to obtain the necessary bounds.     

Controllable C1 continuous blending of time-dependent parametric surfaces

This paper proposes the concept of blending time-dependent varying surfaces, and develops a new method to create a controllable C1 continuous blending surface between primary parametric surfaces whose position and shape change with time. We treat it as a boundary-valued problem defined by the mathematical model of a vectored dynamic fourth-order partial differential equation subjected to time-dependent C1 continuous blending boundary constraints. High performance blending surface generation is achieved through the development of an approximate analytical solution of the mathematical model. We investigate the accuracy and efficiency of the solution, study the effective shape control of the blending surfaces, and apply the obtained solution to tackle surface blending problems. The applications demonstrate that our proposed approach is very effective and efficient in dealing with controllable C1 continuous surface blending between time-dependent varying parametric surfaces.     

On the local shape effect of a moving control point

In this paper we prove that the domain, where a control point of a parametric surface is permitted to move in order to ascertain local convexity at a finite set of parametric points, is a convex polyhedron. This result can be exploited for fine surface design, which is herein illustrated for two test surfaces.     

Corner blending of free-form N-sided holes

Geometric modeling requires constructing blends between surfaces to meet manufacturing specifications, reduce stress concentrations in designs, and enhance aesthetics. Industrial engineers may design parts using different surface types to satisfy design requirements. Surface blending integrates the diverse representations. Because most CAD/CAM software uses parametric representations for curves and surfaces, blending techniques for parametric surfaces are more urgently needed than methods for implicit surfaces. The authors discuss a new method for parametric surface blending. They apply their corner blending technique to three- to six-sided holes and discuss corner blending of three sided holes for three different cases, each presenting different blending challenges     

裁剪面的一类参数变换

给出了一类可以保持几何与拓扑信息一致性的裁剪面的参数变换定理及其算法。首先,确定了参数变换对裁剪面表示信息的影响。然后,根据参数变换后几何与拓扑信息的一致性要求,给出了对裁剪面表示信息进行调整的方法。最后,通过建立参数变换的关系,以裁剪球面为例阐述了这类参数变换的具体实现方法。     

Isophote interpolation

The isophote is an important class of characteristic curves on a parametric surface. Accordingly, as a kind of feature based path generation method, isophote based tool paths has been proposed as an important path pattern for parametric surface machining. In this paper, an approach has been proposed for the isophote interpolation on a parametric surface. The paper has related the arc-length derivatives to time derivatives of parameters along the isophote. The results are then used to derive the parametric interpolation. The isophote curve interpolation is developed based on parametric interpolation. The proposed interpolation guarantees that interpolated points always stay on the parametric surface. An improvement interpolation has been presented to alleviate inclination errors and point deviation from the isophote. Simulations of isophote interpolation have been carried out to verify the effectiveness of the proposed algorithm. The proposed algorithm has applications in real time tool path interpolation for the machining of parametric surfaces.     

Tunnel-free voxelisation of rational Bézier surfaces

To synthesize natural and artificial objects into a hybrid graphics scene represented by a set of voxels, voxelisation of geometric models is necessary. Rational parametric surfaces have been widely used in the representation of free-form surfaces. Voxelisation of these surfaces is therefore of great importance in the development of a voxel-based modeling system. A key issue is to develop a tunnel-free voxelisation algorithm for these continuous surfaces. In this paper, we propose such an algorithm for a rational Bézier surface. We derive the bound of the parametric steps to ensure that the voxelised rational Bézier surface is, by our algorithm, 6-tunnel-free, and we give the mathematical proof of this property. For efficient computation, we employ the forward difference technique in homogeneous form in the implementation of the algorithm. For more general applications, we show that voxelisation of a NURBS surface can be realised by first converting it into a piecewise rational Bézier surface and then voxelising each of the rational Bézier surfaces. We indicate the advantages carrying out this procedure.     

Surface-to-surface intersections

Techniques for computing intersections of algebraic surfaces with piecewise rational polynomial parametric surface patches and intersections of two piecewise rational polynomial parametric surface patches are discussed. The techniques are classified using four categories-lattice evolution methods, marching methods, subdivision methods, and analytic methods-and their principal features are discussed. It is shown that some of these methods also apply to the general parametric surface-intersection problem     

Reliable Collision Detection for Time-Dependent Parametric Surfaces

We improve on an algorithm by Von Herzen, Barr, and Zatz (HBZ) to detect geometric collisions between pairs of time-dependent parametric surfaces. The HBZ algorithm uses upper bounds on the parametric derivatives to guarantee detection of collisions and near misses, thus avoiding the defects of algorithms which sample the surfaces, possibly missing sharp spikes. Unfortunately, the user of the HBZ algorithm must supply not only routines computing the surface functions, but also routines bounding every component in the Jacobian of these surface functions over an arbitrary parametric range. Although they give helpful analyses for several types of surfaces, HBZ admit the need to provide Jacobian bounding routines as a disadvantage.We propose using interval arithmetic to bound functional values over a parametric input range, thus eliminating the need for the Jacobian entirely. Our interval version of the collision detection algorithm assumes neither bounded differentiability nor satisfaction of the Lipschitz criterion. Therefore, our code can detect geometric collisions for the much larger class of surface functions for which bounds on the function values can be computed using interval arithmetic. We contrast our code to that of HBZ and give timing comparisons.     

Smoothing of bicubic parametric surfaces

Bicubic parametric surfaces are often used to represent complex shapes in systems for computer-aided design and manufacture. Such as surface can be defined by a topologically rectangular mesh of cubic parametric splines, a curve which is an approximate mathematical model of the linear elastic beam.Smoothing a bicubic parametric surface can be done by smoothing the curve net that defines it. This paper describes a method for moving datapoints in a curve net to new ‘smoother’ positions. Different techniques to analyse the result of the smoothing are also discussed.     

基于曲面精确表示的距离极值点的计算及在刀具干涉检测中的应用

针对基于曲面精确表示的刚体碰撞检测中裁剪曲面距离极值点的求解问题,提出了 利用平面向量场估计初始曲面距离极值点的方法,避免了曲面过度细分,讨论了距离极值点满足 的微分几何条件,给出了解析曲面/参数曲面、参数曲面/参数曲面、点/参数曲面和曲线/参数曲面 的距离极值点迭代算法。实例验证分析了该算法的高效性和可靠性。