Computer Aided Geometric Design最新文献

Moving planes have been widely recognized as a potent algebraic tool in various fundamental problems of geometric modeling, including implicitization, intersection computation, singularity calculation, and point inversion of parametric surfaces. For instance, a matrix representation that inherits the key properties of a parametric surface is constructed from a set of moving planes. In this paper, we present an efficient approach to computing such a set of moving planes that follow the given rational parametric surface. Our method is based on the calculation of Dixon resultant matrices, which allows for the computation of moving planes with simpler coefficients, improved efficiency and superior numerical stability when compared to the direct way of solving a linear system of equations for the same purpose. We also demonstrate the performance of our algorithm through experimental examples when applied to implicitization, surface intersection, singularity computation as well as inversion formula computation.

The multirational blossom of order k and degree −n of a k-times differentiable function $f\left(t\right)$ is defined as a multivariate function $f(u_1,\dots ,u_k/v_1,\dots ,v_{k+n})$ characterized by four axioms: bisymmetry in the u and v parameters, multiaffine in the u parameters, satisfies a cancellation property and reduces to $f\left(t\right)$ along the diagonal. The existence of a multirational blossom was established in Goldman (1999a) by providing an explicit formula for this blossom in terms of divided differences. Here we show that these four axioms uniquely characterize the multirational blossom. We go on to introduce a homogeneous version of the multirational blossom. We then show that for differentiable functions derivatives can be computed in terms of this homogeneous multirational blossom. We also use the homogeneous multirational blossom to convert between the Taylor bases and the negative degree Bernstein bases.

Computing the intersection curves of two quadrics (QSIC) is a fundamental problem in computer aided design system, where the topology analysis of the QSIC is an essential task to realize the robust computation. In this paper, a new method is proposed to classify the topology of the QSIC in three dimensional projective space by using a set of discriminants associated with the two quadrics. The new topology classification method presents explicit representations and can be applied even when the coefficients of the two quadrics contain symbols. Some examples are provided to illustrate the usage of the new classification method.

We study the smoothness of envelopes generated by motions of rotational rigid bodies in the context of 5-axis Computer Numerically Controlled (CNC) machining. A moving cutting tool, conceptualized as a rotational solid, forms a surface, called envelope, that delimits a part of 3D space where the tool engages the material block. The smoothness of the resulting envelope depends both on the smoothness of the motion and smoothness of the tool. While the motions of the tool are typically required to be at least $C^2$, the tools are frequently only $C^0$ continuous, which results in discontinuous envelopes. In this work, we classify a family of instantaneous motions that, in spite of only $C^0$ continuous shape of the tool, result in $C^0$ continuous envelopes. We show that such motions are flexible enough to follow a free-form surface, preserving tangential contact between the tool and surface along two points, therefore having applications in shape slot milling or in a semi-finishing stage of 5-axis flank machining. We also show that $C^1$ tools and motions still can generate smooth envelopes.

5-axis CNC (Computer Numerical Control) milling is widely used to create complex part geometries in the industrial area. The cutting tool creates the swept volume as it moves along the defined path. The swept volume is subtracted from the initial stock for the machining simulation. Although a swept volume consists of three parts such as ingress, egress, and swept envelope, the number of faces of the swept volume is higher than three. In this work, parametric equations of the faces of the swept volume were obtained for a flat-end mill in four steps. The model is based on the decomposition of the tool into circles along the tool orientation vector since a flat-end mill can be modeled as a cylinder. The fundamental principle is that each circle is tangent to the swept envelope of the cylinder. Locations of the grazing points with respect to a local coordinate system were determined by applying the Envelope Theory to the parameterized circle of the cylinder. Then, each face of the swept volume was represented based on the equation of the parameterized circle. The model was verified by using an alternative analytical model. As a result, boundaries of the swept volume were represented fully analytically for a flat-end mill in 5-axis CNC milling in an efficient way.

This paper summarizes interesting results on systematic backward and forward error analyses performed for corner cutting algorithms providing evaluation of univariate and multivariate functions defined in terms of Bernstein and Bernstein related bases. Relevant results on the conditioning of the bases are also recalled. Finally, the paper surveys important advances, lately obtained, for the design of algorithms adapted to the structure of totally positive matrices, allowing the resolution of interpolation and approximation problems with Bernstein-type bases achieving computations to high relative accuracy.

The quintics are the lowest–order planar Pythagorean–hodograph (PH) curves suitable for free–form design, since they can exhibit inflections. A quintic PH curve $r\left(t\right)$ may be constructed from a complex quadratic pre–image polynomial $w\left(t\right)$ by integration of $r^{\prime }\left(t\right)=w^2\left(t\right)$, and it thus incorporates (modulo translations) six real parameters — the real and imaginary parts of the coefficients of $w\left(t\right)$. Within this 6–dimensional space of planar PH quintics, a 5–dimensional hypersurface separates the inflectional and non–inflectional curves. Points of the hypersurface identify exceptional curves that possess a tangent–continuous point of infinite curvature, corresponding to the fact that the parabolic locus specified by the quadratic pre–image polynomial $w\left(t\right)$ passes through the origin of the complex plane. Correspondingly, extreme curvatures and tight loops on $r\left(t\right)$ are incurred by a close proximity of $w\left(t\right)$ to the origin of the complex plane. These observations provide useful insight into the disparate shapes of the four distinct PH quintic solutions to the first–order Hermite interpolation problem.

In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. In the framework of similarity geometry, those curves are characterized as invariant curves under the integrable flow on plane curves governed by the Burgers equation. They also admit a variational formulation leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formulation, we propose a discretization of these curves and the associated variational principle which preserves the underlying integrable structure. We finally present algorithms for generating discrete log-aesthetic curves for given $G^1$ data based on similarity geometry. Our method is able to generate S-shaped discrete curves with an inflection as well as C-shaped curves according to the boundary condition. The resulting discrete curves are regarded as self-adaptive discretization and thus high-quality even with the small number of points. Through the continuous representation, those discrete curves provide a flexible tool for the generation of high-quality shapes.

UV mapping is a classical problem in computer graphics aiming at computing a planar parameterization of the input mesh with the lowest possible distortion while minimizing the seams length. Recent works propose optimization methods for solving these two joint problems at the same time with variational models, but they tend to be slower than other cutting methods. We present a new variational approach for this problem inspired by the Ambrosio-Tortorelli functional, which is easier to optimize than already existing methods. This functional has widely been used in image and geometry processing for anisotropic denoising and segmentation applications. The key feature of this functional is to model both regions where smoothing is applied, and the loci of discontinuities corresponding to the cuts. Our approach relies on this principle to model both the low distortion objective of the UV map, and the minimization of the seams length (sequences of mesh edges). Our method significantly reduces the distortion in a faster way than state-of-the art methods, with comparable seam quality. We also demonstrate the versatility of the approach when external constraints on the parameterization are provided (packing constraints, seam visibility).

Curvature analysis is frequently employed in shape interrogation. Umbilical points are of particular interest in curvature analysis because of their identical normal curvatures in all tangential directions. Locating the umbilical points is the basis of geometric analysis. On the one hand, as singularities, umbilical points severely hinder the analysis (e.g., in nets of curvature lines). On the other hand, they provide qualitative information about the intrinsic shape of a surface and are therefore desirable quantities in some applications. In this study, we develop a straightforward and effective method to detect generic umbilical points on triangular meshes. This method is applicable to any type of admissible parametrization. We propose two local parametrization schemes–orthogonal projection and conformal transformation–to be used with the proposed method. Furthermore, we systematically analyze our method and prove its convergence behavior. The algorithm used in our approach is flexible and straightforward to implement for triangular meshes of arbitrary topology.