Isoparametric Curves Research Articles

Reparameterization of B-spline surface and its application in ship hull modeling

NURBS are widely used in the design of free curves and surfaces and have been the industrial standard for product design in CAD/CAM. Ship hull surfaces are complex doubly-curved surfaces and usually decomposed into patches based on feature curves. Due to the topologically rectangular limitations of tensor-product NURBS surfaces and the inevitable problem of surface merging caused by decomposition, the reconstruction of ship hull still remains a complicated problem. The theoretical research of merging NURBS surfaces is relatively comprehensive, but the difficulty lies in how to merge surfaces defined in different knot vectors at the common boundary curve since control meshes form T-junctions. This paper proposes a methodology of reparameterizing B-spline surfaces, using a fast progressive-iterative approximation (FPIA) method to refit section curves or isoparametric curves with constraints, thus making the topological structure of control meshes regular, uniform and orthogonal. Subsequently, a KCS container ship with bulbous bow and stern is utilized as an illustration for reconstructing with B-splines. The lighting effects and lines plan derived from the hull surface demonstrate that this approach can achieve fair, smooth and watertight surfaces. The approach exhibits strong robustness, versatility and generalization ability, and provides theoretical innovation and practical design methods for representing and reconstructing complex ship hull.

Digital Interactive Design and Robot-Assisted Preparation Experiment of Tooth Veneer Preparation: An in Vitro Proof-of-Concept

In the fixed prosthesis process dental prosthesis treatment, the dentist needs to conceive the morphology of the tooth preparation and make numerous micro-adjustments by manual operation. This study aims to establish a digital design system for accurate designing of veneer preparation and study the trajectory planning of the robot-assisted preparation. According to the standard operation requirements, the projection line segmentation method based on point drawing is proposed for interactive selection to ensure the margin smoothness of the preparation area. The preparation depth is parameterized, and the foldback point correction method based on secondary segmentation is used for repairing the offset preparation area to ensure surface smoothness. The fenestration preparation model was obtained in the digital interactive design software. The model is divided into two feature areas which are interpolated by the B-spline curve and NURBS surface, respectively. The isoparametric curve method is used to plan the preparation trajectory. The experimental and theoretical values of the target feature points from the X, Y and Z coordinate directions were also calculated and analyzed. The experiment is carried out by the physical prototype of the tooth preparation robot. The maximum error of the feature points on the preparation are 0.180 mm, 0.104 mm and 0.142 mm in X, Y and Z directions, respectively. The confidence interval of the system error is stable within 0.30 mm. The error of the feature points verifies the feasibility of the digital interactive design system and the correctness of the trajectory planning of tooth veneer preparation. The system could have the potential to assist dentists in the precise design of the veneer preparation, improve the processing efficiency and ensure the effect of tooth veneer prosthesis.

Equiareal Parameterization of Triangular Bézier Surfaces

Parameterization is the key property of a parametric surface and significantly affects many kinds of applications. To improve the quality of parameterization, equiareal parameterization minimizes the equiareal energy, which is presented as a measure to describe the uniformity of iso-parametric curves. With the help of the binary Möbius transformation, the equiareal parameterization is extended to the triangular Bézier surface on the triangular domain for the first time. The solution of the corresponding nonlinear minimization problem can be equivalently converted into solving a system of bivariate polynomial equations with an order of three. All the exact solutions of the equations can be obtained, and one of them is chosen as the global optimal solution of the minimization problem. Particularly, the coefficients in the system of equations can be explicitly formulated from the control points. Equiareal parameterization keeps the degree, control points, and shape of the triangular Bézier surface unchanged. It improves the distribution of iso-parametric curves only. The iso-parametric curves from the new expression are more uniform than the original one, which is displayed by numerical examples.

Classification of polynomial minimal surfaces

Minimal surfaces are widely applied in Computer-Aided Design and architecture due to their elegant shapes and remarkable geometric properties. On the basis of the theoretical results of Pythagorean hodograph (PH) curves, we provide a complete classification of the polynomial surface of degrees three to five. We point out the existence of a unique cubic minimal surface, one family of quartic polynomial minimal surfaces, and three families of quintic polynomial minimal surfaces up to similarities and linear reparametrization. We give explicit expressions for each family of polynomial minimal surfaces with parameters that can be used to adjust their shapes. We also prove that an isoparametric curve that is a planar PH curve always exists for any polynomial minimal surfaces. Finally we apply the classification to some existing minimal surfaces.

Surfaces Using a Common Geodesic Curve With an Alternative Moving Frame in The 3-Dimensional Lie Group

Our purpose in this research is to use an alternative moving frame in the 3-dimensional Lie group to construct the problem of how to characterize a surface family and derive the conditions from a given common geodesic curve as an isoparametric curve. We also derive the relation about developability along the common geodesic of a ruled surface as a member of the surface family. Finally, we will give some examples to show some applications of the method.

Constructing quadratic birational maps via their complex rational representation

We present a method to construct quadratic birational planar maps, which is based on the observation that any rational planar map with complex rational representation possesses two special syzygies. After establishing the relation between degree one complex rational Bézier curves and quadratic rational Bézier curves, we derive conditions to determine when a quadratic rational planar map has a complex rational representation. Hence, we can make a quadratic planar map be birational by suitably choosing the middle Bézier control points and their corresponding weights. In addition, this paper explores an interesting geometric property of the isoparametric curves of quadratic birational planar maps with complex rational representation, i.e., all isoparametric curves meet at a common point.

Toolpath planning optimization for end milling of free-form surfaces using a clustering algorithm

A two-step approach to address the toolpath planning optimization problem is introduced. The first step divides the surface into zones with similar local geometric properties in order to improve efficiency of tool-path planning algorithms. To do this, an unsupervised clustering algorithm (like K-means) is used on a mesh defined by isoparametric curves. The second step applies a toolpath planning algorithm to each zone, according to its optimal machining direction. This optimal direction is calculated using a blackbox optimization software. The final goal is to enhance gradually the formulation of the optimization problem to obtain better and better results.

Construction of the spacelike constant angle surface family in Minkowski 3-space

A spacelike constant angle surface in $E_1^3$ (Minkowski 3-space) is defined by its unit normal vector field forms a constant angle with some fixed vector direction. In this paper, we use the Frenet frame to express the surface and construct a family of spacelike constant angle surfaces which possess the given curve as isoparametric curve. The normal vector of each surface forms a constant angle with a fixed vector lying in the spacelike cone and in the timelike cone respectively. Finally, we give some representative examples.

A novel method to generate the tooth surface model of face-milled generated spiral bevel gears

Tooth surface model is a fundamental element used in the design, analysis, and manufacturing of spiral bevel gears. Based on the face-milled generated process, the tooth surface of spiral bevel gears is mathematically calculated as a complicated representation related to cutter parameters and machine settings. With the representation, the discretized points and curves can be easily obtained from the parameterization of tooth surface. However, the discretized curves for the face-milled generated spiral bevel gears are usually too non-equidistant to directly generate 3D (three dimensional) and mesh models in commercial softwares. This paper presents a novel method to recalculate tooth surface points to generate new approximately equidistant points and curves. The tooth surface representation is introduced at the beginning. Subsequently, a mapping between the isoparametric curve and new approximately equidistant curve is used to recalculate the tooth surface points. To solve the nonlinear equation established from the mapping, a new global optimization algorithm, named real-coded relaxation genetic algorithm (RRGA), is proposed to stably obtain the solutions. Examples are given to validate the proposed method.

Automatic Generation of Dynamic Skin Deformation for Animated Characters

Since non-automatic rigging requires heavy human involvements, and various automatic rigging algorithms are less efficient in terms of computational efficiency, especially for current curve-based skin deformation methods, identifying the iso-parametric curves and creating the animation skeleton requires tedious and time-consuming manual work. Although several automatic rigging methods have been developed, but they do not aim at curve-based models. To tackle this issue, this paper proposes a new rigging algorithm for automatic generation of dynamic skin deformation to quickly identify iso-parametric curves and create an animation skeleton in a few milliseconds, which can be seamlessly used in curve-based skin deformation methods to make the rigging process fast enough for highly efficient computer animation applications.

Certain Algorithms for Modeling Uncertain Data Using Fuzzy Tensor Product Bézier Surfaces

Real data and measures are usually uncertain and cannot be satisfactorily described by accurate real numbers. The imprecision and vagueness should be modeled and represented in data using the concept of fuzzy numbers. Fuzzy splines are proposed as an integrated approach to uncertainty in mathematical interpolation models. In the context of surface modeling, fuzzy tensor product Bézier surfaces are suitable for representing and simplifying both crisp and imprecise surface data with fuzzy numbers. The framework of this research paper is concerned with various properties of fuzzy tensor product surface patches by means of fuzzy numbers including fuzzy parametric curves, affine invariance, fuzzy tangents, convex hull and fuzzy iso-parametric curves. The fuzzification and defuzzification processes are applied to obtain the crisp Beziér curves and surfaces from fuzzy data points. The degree elevation and de Casteljau’s algorithms for fuzzy Bézier curves and fuzzy tensor product Bézier surfaces are studied in detail with numerical examples.

The generation of axially aligned seams on triangulated pipe faces

This paper presents a method for creating seams on pipe faces which are effective for reducing the distortion of the face flattening and thus promote the generation of high quality meshes on the faces. The method is applicable to groups of connected B-rep triangulated faces whose underlying surfaces resemble Generalised Cylinders. Isoparametric curves of Generalised Cylinder parametrisations are approximated by establishing singularity-free cross-fields starting with estimated principal curvature directions from which the seams are derived.

Discrete CAD Model for Visualization and Meshing

During the design of an object using a CAD (computer aided design) platform, the user can visualize the ongoing model at every moment. Visualization is based on a discrete representation of the model that coexists with the exact analytical representation of the object. Most CAD systems have this discrete representation available, and each of them applies its own construction methodology. This paper presents a new method to build a discrete (“triangulated” with quadrilaterals and triangles) model for CAD surfaces. It presents two major particularities: most elements are aligned with iso-parametric curves and the accuracy of the surface approximation is controlled. In addition, we present a new technique of surface mesh generation that is based on this discrete model. Several examples are presented to confirm the efficacy of this approach.

Tensor-product surface patches with Pythagorean-hodograph isoparametric curves

Tensor-product surface patches with Pythagorean-hodograph isoparametric curves Rida T. Farouki Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616, USA. Francesca Pelosi Dipartimento di Matematica, Universit`a di Roma “Tor Vergata,” Via della Ricerca Scientifica, 00133 Roma, Italy. Maria Lucia Sampoli Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Universit`a di Siena, San Niccol`o, Via Roma 56, 53100 Siena, Italy. Alessandra Sestini Dipartimento di Matematica e Informatica, Universit`a di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy. Abstract The construction of tensor–product surface patches with a family of Pythagorean–hodograph (PH) isoparametric curves is investigated. The simplest non–trivial instances, interpolating four prescribed patch boundary curves, involve degree (5, 4) tensor–product surface patches x(u, v) whose v = constant isoparametric curves are all spatial PH quintics. It is shown that the construction can be reduced to solving a novel type of quadratic quaternion equation, in which the quaternion unknown and its conjugate exhibit left and right coefficients, while the quadratic term has a coefficient interposed between them. A closed– form solution for this type of equation is derived, and conditions for

Surface design based on direct curvature editing

This paper presents a novel method for modifying the shapes of existing uniform bi-cubic B-spline surfaces by interactively editing the curvatures along isoparametric curves. The method allows us to edit the curvatures of the two intersecting isoparametric curves at each knot with specified positions, unit tangents, and unit normals. The user adjusts the radii of circles, representing the radii of curvature in the u and v isoparametric directions directly via a GUI without having to work with control points and knots. Such shape specifications are converted into iterative repositionings of the control points on the basis of geometrical rules. Using these point-based curvature-editing techniques, we successfully embedded log-aesthetic curves into existing surfaces along their isoparametric curves. Moreover, we were able to distribute the cross curvature with log-aesthetic variation along the isoparametric curves. We applied our technique to the design of automobile hood surfaces to demonstrate the effectiveness of our algorithms.

A new iso-parametric machining algorithm for free-form surface

In this paper, a new iso-parametric tool path generation technique has been presented for machining truncated free-form surfaces by means of re-parameterisation. Three methods of re-parameterisation have been discussed, namely, the Coons method, partial differential equation method and the newly developed ‘Boundary Interpolation method.’ Tool paths have been generated based on the iso-parametric curves produced by re-parameterisation. The complete mathematical formulations have been given and subsequently their applications have been studied through some typical cases. Case studies show that the Coons and Laplace partial differential equation have some drawbacks in generating tool paths when the surface boundaries are irregular but the new Boundary Interpolation algorithm gives smooth and efficient iso-parametric tool paths by eliminating the problems like overcrossing (in case of Coons method) and uneven distribution of iso-parametrics (in case of Laplace partial differential equation method).

Boundary-conformed machining for trimmed free-form surfaces based on mesh mapping

Boundary-conformed iso-parametric or streamline-like tool paths are highly desired for machined surfaces requiring extra high finish quality or superior aerodynamic effect in some applications. However, for a trimmed free-form surface, its trimmed boundaries usually do not follow the original iso-parametric curves. As a result, conventional iso-parametric tool path methods are not applied readily on the trimmed surface to generate boundary-conformed tool path, especially on the multiple-trimmed surfaces. In this study, a new method to the problem of boundary-conformed tool path generation on the trimmed surfaces is proposed. The basic idea of the method is to map directly the trimmed surface onto a 2D parametric square domain using mesh mapping technique. Then, the iso-parametric lines on the parametric domain are inversely mapped onto the original trimmed surface to regenerate boundary-conformed tool paths. As the method does not rely on processing on the original parametric domain of the surface patches, it is not limited to the single-trimmed surface and is also well applicable to the multiple-trimmed surfaces and mesh surfaces. The proposed method is implemented and tested on several typical examples to demonstrate its validity and significance.

Tool path generation for algebraically parameterized surface

In CAD/CAM integrated environment, parametrically defined free-form surfaces may require to be truncated to suit the profile of an object to be machined. In such a case the truncated surface needs to be parameterized anew according to the boundaries. Bilinear Coons parameterization is a well known algebraic method which can be used for generating boundary-fitted iso-parametric curves depending on the boundary information. In this paper an iso-parametric tool path generation scheme has been presented based on Coons parameterization technique for free-form surface. The scheme has been tested on some typical surfaces. Limitations in application of this scheme have also been discussed in this paper.

Isoparametric Curve of Quadratic F-Bézier Curve

AbstractIn this thesis, we consider isoparametric curves of quadratic F-Bezier curves. F-Bezier curves unify C-Bezier curves whose basis is aš•
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’iIae . Thus F-Bezier curves are more useful in Geometric Modeling or CAGD(Computer Aided Geometric Design). We derive the relation between the quadratic F-Bezier curves and the quadratic rational Bezier curves. We also obtain the geometric properties of isoparametric curve of the quadraticF-Bezier curves at both end points and prove the continuity of the isoparametric curve. Key words : F-Bezier Curve, Quadratic Rational Bezier Curve, Isoparametric Curve, Q-Bezier Curve, H-Bezier Curve 1 Graduate School of Education, Chosun University, Gwangju 501-759, Korea 2 Department of Mathematics Education, Chosun University, Gwangju 501-759, Korea † Corresponding Author : ahn@chosun.ac.kr(Received : January 2, 2013, Revised : March 20, 2013, Accepted : March 25, 2013) 1. Introduction C-Bezier curve

An algorithm to improve parameterizations of rational Bézier surfaces using rational bilinear reparameterization

The parameterization of rational Bézier surfaces greatly affects rendering and tessellation results. The uniformity and orthogonality of iso-parametric curves are two key properties of the optimal parameterization. The only rational Bézier surfaces with uniform iso-parametric curves are bilinear surfaces, and the only rational Bézier surfaces with uniform and orthogonal iso-parametric curves are rectangles. To improve the uniformity and orthogonality of iso-parametric curves for general rational Bézier surfaces, an optimization algorithm using the rational bilinear reparameterizations is presented, which can produce a better parameterization with the cost of degree elevation. Examples are given to show the performance of our algorithm for rendering and tessellation applications.