Blending Surfaces Research Articles

A vertex-first parametric algorithm for polyhedron blending

This paper enhances the conventional parametric algorithms for polyhedron blending, by strategically inverting the edges-first approach to vertex-first, so that matching the vertex blending surface (using a triangular or tensor product Bézier surface, or an S -patch) with the edge blending surfaces (generated by Hartmann method) becomes essentially easier. Based on a study of cross boundary derivatives (those of S -patches are deduced herein), G g -continuity between all the above surfaces and the primary planar faces is achieved by a novel trick as a first step: assigning the vertex, some edge points and some face points to be the proper control points. This still leaves enough free parameters usable for changing the blending configuration. The new algorithm is illustrated with two practical examples involving miscellaneous vertices up to 6-edge convex–concave.

Construction of flexible blending parametric surfaces via curves

The main purpose of this paper is to provide a method that allows to solve the blending problem of two parametric surfaces. The blending surface is constructed with a collection of space curves defined by point pairs on the blending boundaries of given primary surfaces. Bézier and C-cubic curves are used to interpolate the blending boundaries. The blending surface is G n continuously connected to the primary surfaces.

Triangulation of molecular surfaces

Given a molecule, which consists of a set of atoms, a molecular surface is defined for a spherical probe approximating a solvent molecule. Molecular surface is used for both the visualization of the molecule and the computation of various molecular properties such as the area and volume of a protein, which are important for studying problems such as protein docking and folding. In this paper, we present an O ( n ) time algorithm, in the worst case, for triangulating molecular surface based on the combinatorial information provided by the β -shape of the molecule with n atoms. The proposed algorithm takes advantage of the concise representation of topology among atoms stored in the β -shape. A molecular surface consists of two parts: a blending surface consisting of blending patches and a (solvent) contact surface consisting of (solvent) contact patches. For each blending patch, the algorithm uses compact masks for the construction of a triangular mesh in O ( c ′ ) time in the worst case, where c ′ is the number of point evaluations on the blending patch. For each contact patch, the algorithm uses a template, for each atom type, for the triangulation of the boundary of the atom. Then, the triangular mesh is trimmed off by hyperplanes where each hyperplane corresponds to an arc of the boundary of the contact patch. The triangulation of a contact patch takes O ( c ″ ) time in the worst case, where c ″ is the number of point evaluations on the boundary of an atom. Since there are at most O ( n ) patches, the worst case time complexity is O ( n ) . The proposed algorithm also handles internal voids and guarantees the watertightness of the produced triangular mesh of a molecular surface. In addition, the level-of-detail is easily achieved as a by-product of the proposed scheme. The proposed algorithm is fully implemented and statistics from experiments are also collected.

Three dimensional modeling and finite element simulation of a generic end mill

The geometry of cutting flutes and the surfaces of end mills is one of the crucial parameters affecting the quality of the machining in the case of end milling. These are usually represented by two-dimensional models. This paper describes in detail the methodology to model the geometry of a flat end mill in terms of three-dimensional parameters. The geometric definition of the end mill is developed in terms of surface patches; flutes as helicoidal surfaces, the shank as a surface of revolution and the blending surfaces as bicubic Bezier and biparametric sweep surfaces. The proposed model defines the end mill in terms of three-dimensional rotational angles rather than the conventional two dimensional angles. To validate the methodology, the flat end milling cutter is directly rendered in OpenGL environment in terms of three-dimensional parameters. Further, an interface is developed that directly pulls the proposed three-dimensional model defined with the help of parametric equations into a commercial CAD modeling environment. This facilitates a wide range of downstream technological applications. The modeled tool is used for finite element simulations to study the cutting flutes under static and transient dynamic load conditions. The results of stress distribution (von mises stress), translational displacement and deformation are presented for static and transient dynamic analysis for the end mill cutter flute and its body. The method described in this paper offers a simple and intuitive way of generating high-quality end mill models for use in machining process simulations. It can be easily extended to generate other tools without relying on analytical or numerical formulations.

A Study on Extending Boundary-Representation for Robust Blending (2nd report)

A solid model representing a product requires filleting or blending, which rounds a specified edge or vertex, in order to make a production model for manufacturing purpose such as mould die model. Many research algorithms for blending have been proposed and commercial CAD/CAM system vendors have implemented blend functions for their systems, however, the practicing users of CAD/CAM systems in the manufacturing industry are suffering from the inability of current blending functions, such as some “holes” in the solid models because of the failure of the blending. It is regarded that the current major solid model structure, the Boundary-Representation (B-rep) lacks the sufficient structure for making robust blending surfaces. In this research, several additional conditions for the B-rep that includes “extended surfaces” that make the blending algorithm simple and robust are proposed. In this report, it is shown that the proposed conditions are effective in the general termination case of edge blending. It is also shown that proposed approach can be applicable to the face-edge blending problem and global blending problem. Several examples are shown to verify the effectiveness of the proposal.

A Study on Extending Boundary-Representation for Robust Blending (1st report)

A solid model representing a product requires filleting or blending, which rounds a specified edge or vertex, in order to make a production model for manufacturing purpose such as mould die model. Many research algorithms for blending have been proposed and commercial CAD/CAM system vendors have implemented blend functions for their systems, however, the practicing users of CAD/CAM systems in the manufacturing industry are suffering from the inability of current blending functions, such as some “holes” in the solid models because of the failure of the blending. We regard that the current major solid model structure, the Boundary-Representation (B-rep) lacks the sufficient structure for making robust blending surfaces. In our research, we propose several additional conditions for the B-rep that includes “extended surfaces” that make the blending algorithm simple and robust. In this report, we show that our proposed conditions are effective in the case of three-edge vertex blending. Several examples are shown to verify the effectiveness of our proposal.

Deformation-based freeform feature reconstruction in reverse engineering

For reconstructing a freeform feature from point cloud, a deformation-based method is proposed in this paper. The freeform feature consists of a secondary surface and a blending surface. The secondary surface plays a role in substituting a local region of a given primary surface. The blending surface acts as a bridge to smoothly connect the unchanged region of the primary surface with the secondary surface. The secondary surface is generated by surface deformation subjected to line constraints, i.e., character lines and limiting lines, not designed by conventional methods. The lines are used to represent the underlying information of the freeform feature in point cloud, where the character lines depict the feature’s shape, and the limiting lines determine its location and orientation. The configuration of the character lines and the extraction of the limiting lines are discussed in detail. The blending surface is designed by the traditional modeling method, whose intrinsic parameters are recovered from point cloud through a series of steps, namely, point cloud slicing, circle fitting and regression analysis. The proposed method is used not only to effectively and efficiently reconstruct the freeform feature, but also to modify it by manipulating the line constraints. Typical examples are given to verify our method.

Computation of rotation minimizing frames

Due to its minimal twist, the rotation minimizing frame (RMF) is widely used in computer graphics, including sweep or blending surface modeling, motion design and control in computer animation and robotics, streamline visualization, and tool path planning in CAD/CAM. We present a novel simple and efficient method for accurate and stable computation of RMF of a curve in 3D. This method, called the double reflection method , uses two reflections to compute each frame from its preceding one to yield a sequence of frames to approximate an exact RMF. The double reflection method has the fourth order global approximation error, thus it is much more accurate than the two currently prevailing methods with the second order approximation error—the projection method by Klok and the rotation method by Bloomenthal, while all these methods have nearly the same per-frame computational cost. Furthermore, the double reflection method is much simpler and faster than using the standard fourth order Runge-Kutta method to integrate the defining ODE of the RMF, though they have the same accuracy. We also investigate further properties and extensions of the double reflection method, and discuss the variational principles in design moving frames with boundary conditions, based on RMF.

Procedural Representation Method of Blending Surface

A solid model representing a product requires filleting or blending, which rounds a specified edge or vertex, in order to make a production model for manufacturing purpose such as mould die model. As a surface representation, current major solid modeling CAD systems use a tensor product surface such as B-spline or NURBS surface. Representing blend surfaces using tensor product surfaces, however, causes several problems in the case such as, blends for meandering shapes or blends degenerate at one side of a surface. We propose a new blend algorithm which makes blend process robust. In this report, we show a procedural blend surface representation that intersections of offset surfaces are represented in procedural curves and basic blend surfaces values, point positions and differentials, are interrogated through that curves. Several examples are shown to verify that procedural blend surfaces have higher quality compared to tensor product surfaces.

Generating [formula omitted] parametric blending surfaces based on partial reparameterization of base surfaces

A method for generating parametric G n blending surfaces based on reparameterization of partial surface patches of the base surfaces is presented. In the method, partial regions, called reparameterized local base surfaces, near the contact curves on both base surfaces are reparameterized first. Then a parametric G n blending surface is generated by a linear combination of the two reparameterized local base surface patches by using Erich Hartmann’s method. Here, contact curves can be arbitrarily defined by users according to practical requirements. Some methods for determining the reparameterized local base surfaces are given, and thereafter the influences of the reparameterized local base surfaces on the blending surface are discussed in detail. In addition, in order to achieve the fairness of the blending surface, a shape optimization method for minimizing the strain energy of the surface is proposed. Finally, some examples, including a G 2 blending surface generated with the method between an aerofoil and a missile body, are given to demonstrate the effectiveness of the proposed approach.

Smooth blending of subdivision surfaces

A technique for merging freeform objects modeled with subdivision surfaces is proposed. The method takes into consideration the possible distortion of the surfaces while the smoothness of the blending surface connecting the objects is maintained. A blend region between the given object surfaces is located. The topology of the surfaces in the vicinity of the blend region is refined and the boundary curves of the blend region are smoothed to reduce possible distortion in the blending surface. A blend curve between the two surfaces is used for connecting the given surfaces. Vertices on the blend curve are regular whereas extraordinary vertices are restricted to locate along the boundary of the blend region. Locations of the blend vertices are adjusted such that the curvature of the surfaces along the boundaries of the blend region is minimized. This ensures extraordinary vertices on the boundaries to lie on a relatively flat region, and hence, minimizes the distortion of the surfaces in the merging process.

Transition surface design for blank holder in multi-point forming

During the last decades, the manufacturing mode and level of industry have been improving from mass production manufacturing to responsive and flexible manufacturing. Multi-point forming (MPF) is a flexible manufacturing technique, which is made up of a set of adjustable base elements controlled by MPF-CAD/CAM software. It uses discrete punches to express a continuous 3-D surface, by which the workpiece's surface under manufacturing is formed. With respect to the characteristics of MPF, two methods for NURBS surface extension are presented in this paper to design the blending surface between the surfaces of the blank holder and the workpiece. Two design approaches of transition surface are proposed: one is a flexible surface extension, and the other is a bridge surface extension. These two algorithms, through which the transition surface designed can reach G2 continuity, are both effective and reliable. Applications show that high-quality products are manufactured in MPF by this transition surface design.

Basic lines, axes and geometric modeling on implicit algebraic surfaces

We introduce some new concepts such as basic lines and divide the blending problems into two classes, Class I blending and Class II blending. We further develop the results [Li, A software system on blending of pipe surfaces, Master Thesis, Jilin University, June, 2000; Wu et al., Blending of implicit algebraic surfaces, Proceedings of the ASCM 1995, Beijing, China, August 18–20, 1995, pp. 125–131; Wu and Han, Newton form formulae for the n-way blending of quadratic surfaces, Proceedings of IJCC Workshop on Digital Engineering, August 21–22, 2003; Wu and Zhou, On blending of several quadratic algebraic surfaces, Comput. Aided Geom. Design 17(9) (2000) 759–766; Wu, Zhou and Feng, Blending two quadratic algebraic surfaces with cubic surfaces, Proceedings of ASCM, 1996, Kobe, Japan, August 20–22, 1996, pp. 73–79] and analyze the geometric condition for the Class I blending problem by using characteristic roots. We obtain the lowest degree of Class II blending according to the different position of the axes of the primary surfaces. And we derive the efficient parameterization method for a kind of cubic blending surfaces on the basis of the results obtained by Berry and Patterson [Implicitization and parameterization of nonsingular cubic surfaces, Comput. Aided Geom. Design 19 (2001) 723–738].

Mesh blending

A new method for smoothly connecting different patches on triangle meshes with arbitrary connectivity, called mesh blending, is presented. A major feature of mesh blending is to move vertices of the blending region to a virtual blending surface by choosing an appropriate parameterization of those vertices. Once blending is completed, the parameterization optimization is performed to perfect the final meshes. Combining mesh blending with multiresolution techniques, an effective blending technique for meshes is obtained. Our method has several advantages: (1) the user can intuitively control the blending result using different blending radii, (2) the shape of cross-section curves can be adjusted to flexibly design complex models, and (3) the resulting mesh has the same connectivity as the original mesh. In this paper, some examples about smoothing, sharpening, and mesh editing show the efficiency of the method.

Blending Canal Surfaces Based on PH Curves

In this paper, a new method for blending two canal surfaces is proposed. The blending surface is itself a generalized canal surface, the spine curve of which is a PH (Pythagorean-Hodograph) curve. The blending surface possesses an attractive property -- its representation is rational. The method is extensible to blend general surfaces as long as the blending boundaries are well-defined.

Mesh Fusion Based on Variational Implicit Surfaces

3D object fusion provides an easy way to generate novel models from two or more existing geometric models by using 3D cutting and pasting operations. As a new kind of geometric modeling tool, it is now attracting much more attention. In this paper, a new montage mesh fusion method based on variational interpolating implicit surfaces is proposed. The approach first calculates mesh regions to be fused by cutting original meshes with boundless planes, then constructs a variational implicit surface by interpolating mesh regions to be fused and polygonalizes the implicit surface to obtain a blending surface between/among original meshes to be fused. Smooth mesh fusion is finally obtained by cutting away unwanted portions of the blending surface and performing topology merging operations. Compared with current methods that fuse mesh objects by directly connecting mesh regions to be fused, this approach has the advantages of no restriction on topologies of the mesh regions to be fused, and thus enabling the fusion among multiple objects, fast running speed with robustness, and easiness to use. The approach shows a promising future in applications for geometry modeling as well as computer animation.

PDE blending surfaces with C2 continuity

In this paper, we propose to use a general sixth-order partial differential equation (PDE) to solve the problem of C 2 continuous surface blending. Good accuracy and high efficiency are obtained by constructing a compound solution function, which is able to both satisfy the boundary conditions exactly and minimise the error of the PDE. This method can cope with much more complex surface-blending problems than other published analytical PDE methods. Comparison with the existing methods indicates that our method is capable of generating blending surfaces almost as fast and accurately as the closed-form method and it is more efficient and accurate than other extant PDE-based methods.

Analytical C2 smooth blending surfaces

Two factors are important in the generation of blending surfaces for interactive graphical and CAD applications, computational speed and the degree of smoothness. Most surface-blending methods blend surfaces with tangent continuity. However, curvature continuity has recently become increasingly important in various applications. In this paper, we present a method that is able to achieve curvature continuity based on the use of partial differential equations (PDE). The blending surfaces are generated as the solution to a sixth-order PDE with one vector-valued parameter. To achieve interactive performance, we propose an effective analytical method for the resolution of this sixth-order PDE.

Blending surface generation using a fast and accurate analytical solution of a fourth-order PDE with three shape control parameters

In this paper, we propose to use a fourth-order partial differential equation (PDE) to solve a class of surface-blending problems. This equation has three vector-valued shape control parameters. It incorporates all the previously published forms of fourth-order PDEs for surface blending and can generate a larger class of blending surfaces than existing equations. To apply the proposed PDE to the solution of various blending problems, we have developed a fast and accurate resolution method. Our method modifies Navier’s solution for the elastic bending deformation of thin plates by making it satisfy the boundary conditions exactly. A comparison between our method, the closed-form solution method, and other existing analytical methods indicates that the developed method is able to generate blending surfaces almost as quickly and accurately as the closed-form solution method, far more efficiently and accurately than the numerical methods and other existing analytical methods. Having investigated the effects that the vector-valued shape parameters and the force function of the proposed equation have on the blending surface, we have found that they have a significant influence on its shape. They provide flexible user handles that surface designers can use to adjust the blending surface to acquire the desired shape. The developed method was employed in the investigation of surface-blending problems where the primary surfaces were expressed in parametric, implicit, and explicit forms.

Generating parametric G 2 non-four-sided blending surfaces

A method of generating n-sided (n=3, 5, 6) G2 blending surfaces (except that the 3-sided blending surface is C0 continuous at the three vertexes and the 5-sided blending surface is C0 continuous at a vertex) is presented in this paper. When this method is used to generate n-sided (n=3, 5, 6) G2 blending surfaces between the base surfaces, the base surfaces are expressed as n-sided (n=3, 5, 6) by reparameterising the base surfaces; then the n-sided (n=3, 5, 6) G2 blending surfaces are convex combinations of the base surfaces expressed as n-sided (n=3, 5, 6) surfaces. When the contact curves are arbitrary curves in the base surfaces, the regions near the contact curves in the base surfaces are reparameterised and expressed as n-sided (n=3, 5, 6) surfaces; then the n-sided (n=3, 5, 6) G2 blending surfaces are convex combinations of the reparameterised local base surfaces expressed as n-sided (n=3, 5, 6) surface styles. The shape of the n-sided (n=3, 5, 6) blending surfaces can be adjusted by changing the size of the reparameterised local base surfaces and the combination weights. An example of a 5-sided surface generated between an aerofoil, the body and the fringe surface of a missile is given.