Isoparametric Curves Research Articles

Polynomial Representation of NURBS and Its Application to High Frequency Scattering Prediction

This article presents a method that uses physical optics (PO) techniques to compute the monostatic radar cross section (RCS) of electrically large conducting objects modeled by non-uniform rational B-spline (NURBS) surfaces. At the beginning, a new algorithm to convert recursive B-spline basis function into piecewise polynomials in power form is presented. Then, algorithm computes the polynomial representation of B-spline basis functions and NURBS surface geometric parameters are obtained. The PO integral over NURBS surfaces of an electrically large conducting object is used to predict the object's RCS. The NURBS surface is divided into small piecewise polynomial parametric patches by isoparametric curves, and the PO integral expression over the parametric domain of each polynomial parametric patch is reduced to an analytical expression which permits an accurate and effective computation of the PO integral by using a modified Ludwig's algorithm. The RCS of the object can be obtained by adding up the PO integral contribution of each polynomial parametric patch. The effectiveness of this method is verified by numerical examples.

Isoparametric and Spiral Toolpath for Free-form Surfaces Machining

By analysis of the validity of tool path, tool path planning is divided into two processes, the choice of curves on surface and the design of effective and rational path arrangement. On this basis, a spiral topology toolpath for continuous cutfing is developed in order to reduce the number of interruptions and rapid traversals during high-speed machining (HSM) process. Based on the isoparametric toolpath, the contour-parallel isoparametric path is firstly created, and then a spiral path is generated as a kind of diagonal curve between the isoparametric offset curves. Experimental results demonstrate that the proposed method is straightforward and avoids the self-intersection of the toolpath, especially suitable for HSM.

Curves with rational chord-length parametrization

It has been recently proved that rational quadratic circles in standard Bézier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limaçon of Pascal. The extension to 3D rational curves is also tackled.

A geometric approach to boundary-conformed toolpath generation

Isoparametric or streamline-like toolpath generation is an important method for CNC surface machining. For a trimmed surface, the generation of such a toolpath is not straightforward because by simply following the isoparametric curves of the parametric surface to generate the toolpath, the resulting toolpath may no longer conform to the trimmed boundary of the surface. Various methods have been proposed to solve this problem. Some methods work well for single surface machining, but may not be feasible for multi-patch surface machining. A new geometric approach to the problem of multi-patch machining is proposed in this paper. The new method works by generating bisectors to partition the region into smaller subregions, and generates the toolpath by offsetting the subregion boundary by using a special offset function. As this new method does not rely on processing on the parametric domain of the surface patches, it can be used for both NURBS and subdivision surfaces. Examples are given to demonstrate the capability of the new method, and to show that the new method compares favorably with existing techniques.

CAD-based sampling for CMM inspection of models with sculptured features

This paper addresses sampling models for trimmed sculptured surfaces, and multiple patches with boundary conditions. A CAD-based sampling system is developed and implemented in this work. The sculptured features are sampled along their isoparametric curves. These curves are then used to re-construct the model geometry using the skinning of cross section curves. We refer to the re-constructed model as the substitute geometry. The problem is to determine the sample curve locations such that the substitute geometry satisfies certain geometric conditions. These are, the form error, and the continuity of the substitute geometry across the boundaries of adjacent surfaces. Three criteria are integrated to determine the sample locations: the surface curvature change, the substitute geometry deviation from the CAD model, and the significance of trimmed portions of the surface. A boundary representation-based methodology for the sampling of trimmed surfaces is developed and implemented. This methodology is extended to handle n-sided surfaces obtained through filling n-sided regions with quadrilateral surface patches, and models that may include multiple surface patches. Furthermore, a tool to assess the sampling plans based on the continuity of the substitute geometry across boundaries of adjacent surface patches is developed. The developed algorithms, their implementations, and case studies are presented in this paper.

Generation of Discrete Bicubic G1 B-Spline Ship Hullform Surfaces from a Given Curve Network Using Virtual Iso-Parametric Curves

We propose a method that automatically generates discrete bicubic G1 continuous B-spline surfaces that interpolate the curve network of a ship hullform. First, the curves in the network are classified into two types: boundary curves and “reference curves”. The boundary curves correspond to a set of rectangular (or triangular) topological type that can be represented with tensor-product (or degenerate) B-spline surface patches. Next, in the interior of the patches, surface fitting points and cross boundary derivatives are estimated from the reference curves by constructing “virtual” iso-parametric curves. Finally, a discrete G1 continuous B-spline surface is generated by a surface fitting algorithm. Several smooth ship hullform surfaces generated from curve networks corresponding to actual ship hullforms demonstrate the quality of the method.

Interference-free tool path generation for 5-axis sculptured surface machining using rational Bézier motions of a flat-end cutter

This paper presents a new approach that uses rational Bézier motions to generate 5-axis tool path for sculptured surface machining (finish cut) with a flat-end cutter. By using dual quaternion to represent a spatial displacement, the representation of kinematic motions for the cutter bottom circle of the flat-end cutter is formulated. Based on that, a new approach for tool path generation using rational Bézier cutter motions is described, in which key issues such as interference avoidance and surface accuracy requirement are addressed. First, a set of cutter contact points on an iso-parametric curve of the designed surface is obtained based on a given fitting tolerance. The associated cutter locations (CLs) are then obtained by finding the suitable cutter orientations that avoid any gouging. The conversion from the CLs to dual quaternion representation is carried out and the rational Bézier dual quaternion curve for cutter motion is generated. The entire tool path is therefore established based on the cutter undergoing the rational Bézier motion. Next, the whole tool path is checked to find (1) if there is any interference between the cutter and the designed surface, and (2) whether the deviation between the surface generated by the cutter motion and the designed surface is larger than the given tolerance. The problematic CLs, which cause either gouging or accuracy problem, are then modified. The process of tool path checking → CLs modification → tool path regeneration continues until the whole tool path is interference-free and satisfies the accuracy requirement. Furthermore, a more accurate representation of the effective cutting shape is proposed, which is used to evaluate the scallop height between adjacent tool paths. A method for constructing the adjacent tool path has been developed by considering the allowable scallop height. Finally, computer implementation and an illustrative example are presented to demonstrate the efficacy of the approach.

A composite B-spline method for cutter path generation on free-form surfaces

This paper presents a composite B-spline method for cutter path generation on free-form surfaces on a point-by-point basis, with appropriate radial offset and interference checking along the cutter path. It is aimed at developing a more efficient and accurate free-form surface machining method for an open-architecture CNC system with B-spline interpolation capability. In this method, the cutter paths over the free-form surface are composed of a sequence of B-spline segments. The length of each spline segment is determined by the desired deviation, within some given tolerance, from the curve of the surface over which the cutter path is to be generated. If the spline segment does not satisfy the specified dimensional tolerance, control points of the spline segment are adjusted to modify the curve shape according to the required accuracy. As a result, the deviation between the actual cutter paths and the desired surface curves will always fall within the allowable tolerance. In addition, by adjusting the position of the control points, the adjacent spline segments are made at least C1 geometrically continuous. The distance between adjacent cutter paths in the side step direction is determined by checking the scallop height between them. To make the cutting process more efficient, a modified adaptive iso-parametric curve generation approach is employed. The resulting cutter paths are more compact and accurate compared with those from commonly used cutter path generation techniques.

The B�zier Tangential Surface System: a Robust Dual Representation of Tangent Space

This paper develops a robust dual representation for the tangent space of a rational surface. This dual representation of tangent space is a very useful tool for visibility analysis. Visibility constructs that are directly derivable from the dual representation of this paper include silhouettes, bitangent developables and kernels.It is known that the tangent space of a surface can be represented by a surface in dual space, which we call a tangential surface. Unfortunately, a tangential surface is usually infinite. Therefore, for robust computation, the points at infinity must be clipped from a tangential surface. This clipping requires two complementary refinements, the first to allow clipping and the second to do the clipping. First, three cooperating tangential surfaces are used to model the entire tangent space robustly, each defined within a box. Second, the points at infinity on each tangential surface are clipped away while preserving everything that lies within the box. This clipping only involves subdivision along isoparametric curves, a considerably simpler process than exact trimming to the box. The isoparametric values for this clipping are computed as local extrema from an analysis using Sederberg's piecewise algebraic curves.A construction of the tangential surface of a parametric surface is outlined, and it is shown how the tangential surface of a Bezier surface can be expressed as a rational Bezier surface.

Parametric representation of a surface pencil with a common spatial geodesic

In this paper, we study the problem of constructing a family of surfaces from a given spatial geodesic curve. We derive a parametric representation for a surface pencil whose members share the same geodesic curve as an isoparametric curve. By utilizing the Frenet trihedron frame along the given geodesic, we express the surface pencil as a linear combination of the components of this local coordinate frame, and derive the necessary and sufficient conditions for the coefficients to satisfy both the geodesic and the isoparametric requirements. We illustrate and verify the method by finding exact surface pencil formulations for some simple surfaces, such as surfaces of revolution and ruled surfaces. Finally, we demonstrate the use of this method in a garment design application.

Boundary-conformed toolpath generation for trimmed free-form surfaces via Coons reparametrization

In the process of CAD/CAM, predefined parametric surfaces are often subject to the operation of trimming or extending. In such a case, the original iso-parametric curves are no longer boundary-conformed. Consequently, they are not anymore suitable for being used as natural CNC toolpaths. In this paper a method to regenerate boundary-conformed iso-parametric toolpaths on altered free-form surfaces is presented. The main idea of this method is to fit the 2D parametric domain of the given trimmed surface with a 2D Coons patch. Then boundary-conformed toolpaths on the altered surface can be regenerated naturally via the iso-parametric curves of the Coons patch via a two-step mapping. The application of this method is successfully illustrated on a typical trimmed surface, a turbine blade. The applicability and limitations of this method are also discussed.

Geodesic curvature preservation in surface flattening through constrained global optimization

In this paper, two methods for generating a planar development of a three-dimensional (3D) surface are proposed. The first method is based on the solution of a global optimization problem without constraints taking into consideration the geodesic curvature of the surface isoparametric curves. The second method is based on the solution of a global optimization problem subject to constraints which are used to control the local accuracy in the derived planar development. Contrarily to previous approaches the proposed second method does not depend on the surface parameterization, since the formed energy function and the proposed constraints are based only on the triangulation of the object surface. Several experiments illustrate that when the 3D surface is approximated with satisfying accuracy, modifications in the triangulation do not have any significant influence to the result of the proposed method. The proposed second method can be used without any modifications with trimmed surfaces, while the generated planar developments can also be used for minimally distorted texture mapping. Finally, the effectiveness of the proposed second method is illustrated through an indicative application in shoe designing.

Web-based digital medical images

The author presents a parallel stripes artistic rendering scheme inspired by Victor Vasarely's art work. The scheme derives stripes as isoparametric curves of a warped surface. The article attempts to create an automated system to produce similar illusive drawings from 3D geometry. Toward this end, we first establish a scheme that would prohibit the crossing of warped curves, a scheme that is based on injective or one-to-one planar warping. This implementation and all the examples were created with the aid of the IRIT solid modeling system, developed at the Technion, Israel Institute of Technology.

Web-based digital medical images

The author presents a parallel stripes artistic rendering scheme inspired by Victor Vasarely's art work. The scheme derives stripes as isoparametric curves of a warped surface. The article attempts to create an automated system to produce similar illusive drawings from 3D geometry. Toward this end, we first establish a scheme that would prohibit the crossing of warped curves, a scheme that is based on injective or one-to-one planar warping. This implementation and all the examples were created with the aid of the IRIT solid modeling system, developed at the Technion, Israel Institute of Technology.

Seismic zoning for initial- and total-cost minimization

We consider two design criteria to study seismic zoning. In the first, codes require that structures be designed for some specified values. Zoning is then optimal when it minimizes the expected present value of the initial costs of all structures to be built in the region being zoned. In the second criterion, it is designed so that the present value of the total cost is minimized, including initial and maintenance costs as well as losses due to damage and failure. We will call these criteria zoning for the initial- and total-cost minimization, respectively. It is shown that under certain conditions, the boundaries coincide with isoparametric curves and the problem may be solved in one dimension. We also deal with problems not reducible to a single dimension. Different methods are proposed to solve the various kinds of problems. The work ends with some illustrative examples. Copyright © 2000 John Wiley & Sons, Ltd.

Self-intersection elimination in metamorphosis of two-dimensional curves

:[0, 1]×\(\)3→\(\)3, where H(t, r) for t=0 and t=1 are two given planar curves C1(r) and C2(r). The first t parameter defines the time of fixing the intermediate metamorphosis curve. The locus of H(t, r) coincides with the ruled surface between C1(r) and C2(r), but each isoparametric curve of H(t, r) is self-intersection free. The second algorithm suits morphing operations of planar curves. First, it constructs the best correspondence of the relative parameterizations of the initial and final curves. Then it eliminates the remaining self-intersections and flips back the domains that self-intersect.

Deformable B-Solids and Implicit Snakes for 3D Localization and Tracking of SPAMM MRI Data

To date, MRI–SPAMM data from different image slices have been analyzed independently. In this paper, we propose an approach for 3D tag localization and tracking of SPAMM data by a novel deformable B-solid. The solid is defined in terms of a 3D tensor product B-spline. The isoparametric curves of the B-spline solid have special importance. These are termed implicit snakes as they deform under image forces from tag lines in different image slices. The localization and tracking of tag lines is performed under constraints of continuity and smoothness of the B-solid. To track motion from boundaries and restrict image forces to the myocardium, a volumetric model is employed as a pair of coupled endocardial and epicardial B-spline surfaces. To recover deformations of the left ventricle (LV), an energy-minimization problem is posed where both tag and LV boundary data are used. The framework has been implemented on tag data from short axis (SA) cardiac images, as well as SA LV boundaries, and is currently being extended to include long axis data.

画上線による自由曲面形状の操作 面上点列による操作法

Surface modelling systems are indispensable tools for designing aesthetic shapes of mechanical products. Contemporary CAD systems provide modelling functions for generating and modifying free form surfaces. They can also perform surface evaluation by rendering surfaces or drawing characteristic curves, such as iso-parametric curves, contour curves, highlight curves. The aim of this research is to propose a new and intuitive surface modification method based on those characteristic curves. With this method, designers can modify surfaces by directly manipulating those characteristic curves. The formalization of the problem is simple, but results in an inverse problem. This problem is solved by applying the standard regularization theory. A computer program is developed to numerically evaluate the method and applied to several examples. It is demonstrated that the method can be used as a basic surface modelling capability.

Line art rendering via a coverage of isoparametric curves

A line art nonphotorealistic rendering scheme of scenes composed of freeform surfaces is presented. A freeform surface coverage is constructed using a set of isoparametric curves. The density of the isoparametric curves is set to be a function of the illumination of the surface determined using a simple shading model, or of regions of special importance such as silhouettes. The outcome is one way of achieving an aesthetic and attractive line art rendering that employs isoparametric curve based drawings that is suitable for printing publication.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Toolpath generation for freeform surface models

The generation of optimal NC code to drive milling machines for models defined by freeform trimmed surfaces is a difficult problem. In practice, two main approaches are used to generate toolpaths for surfaces, neither of which is optimal, in general. The first exploits the parametric representation, and generates isocurves that are uniformly distributed across the parametric domain. This approach is not optimal if the surface mapping into Euclidean space is not isometric. The second approach contours the models by intersecting the surfaces with planes equally spaced in Euclidean space, resulting in a piecewise-linear toolpath approximation which is nonadaptive to the local surface geometry. Further, the toolpath generated by contouring is suitable for 3-axis milling, but is inappropriate for 5-axis milling. In the paper, an algorithm developed to extract isocurves for rendering adaptively is modified and enhanced to generate milling toolpaths for models consisting of trimmed surfaces, and it can be used in both -3 and 5-axis milling. The resulting toolpaths do not gouge locally, and they combine the advantages of both prior approaches. The output toolpath is appealing, since it is composed of isoparametric curves, and is therefore compact, exact, and easy to process. Further, it is more optimal than the previous methods in that the resulting toolpath is shorter, and it provides a direct quantitative bound on the resulting scallop height. This algorithm has been used to compute gouge-avoiding toolpaths for the automatic milling of freeform surfaces, without the introduction of auxiliary check and drive surfaces being required.