Minimal rational parametrizations of canal surfaces

It is well known that canal surfaces defined by a rational spine curve and a rational radius function are rational. The aim of the present paper is to construct a rational parameterization of low degree. The author uses the generalized stereographic projection in order to transform the problem to a parameterization problem for ruled surfaces. Two problems are discussed: parameterization with boundary conditions (design of canal surfaces with two curves on it, as is the case for rolling ball blends) and parameterization without boundary conditions.     

Almost rotation-minimizing rational parametrization of canal surfaces

Almost rotation minimizing parametrization of the canal surface is given. The basic building block of our approach is the curve approximation scheme that enables us to construct a curve, called the parameter curve, on the canal surface that produces, when projected and rescaled, an almost parallel normal vector field on the spine curve. Its construction relies on our earlier methods and results on relating the geometry of the canal surface to the Lorentzian geometry of the Minkowski geometry via the Clifford algebra formalism. We then iteratively construct patches out of the parameter curves via suitable interpolation procedure. Furthermore, its rotational deviation, i.e., the angle deviation from the parallel (no rotation) frame along the spine curve, can be controlled with the use of the rotation deviation estimate of the parameter curves. Our numerical experiment shows that the rotation deviation is minuscule. When compared with other earlier results including our earlier one, our result fares extremely favorably. To facilitate the implement, a practitioners' summary is given in the appendix.     

Blends of canal surfaces from polyhedral medial transform representations

We present a new method for constructing G¹ blending surfaces between an arbitrary number of canal surfaces. The topological relation of the canal surfaces is specified via a convex polyhedron and the design technique is based on a generalization of the medial surface transform. The resulting blend surface consists of trimmed envelopes of one- and two-parameter families of spheres. Blending the medial surface transform instead of the surface itself is shown to be a powerful and elegant approach for blend surface generation. The performance of our approach is demonstrated by several examples.     

Wavelet BEM on molecular surfaces: solvent excluded surfaces

The present paper is concerned with the rapid solution of a boundary integral equation for the apparent surface charge which arises from solvation continuum models. In order to apply the wavelet Galerkin scheme the molecular surface needs to be represented as a parametric surface consisting of smooth four-sided patches. We develop an algorithm which decomposes a solvent excluded surface into a set of globally continuous four-sided NURBS patches. Numerical experiments are carried out to demonstrate the feasibility and scope of the present approach.     

Self-intersections of offsets of quadratic surfaces: Part II, implicit surfaces

The paper investigates self-intersections of offsets of implicit quadratic surfaces. The quadratic surfaces are the simplest curved objects, referred to as quadrics, and are widely used in mechanical design. In an earlier paper, we have investigated the self-intersections of offsets of explicit quadratic surfaces, such as elliptic paraboloid, hyperbolic paraboloid and parabolic cylinder, since not only are they used in mechanical design, but also any regular surface can be locally approximated by such explicit quadratic surfaces. In this paper, we investigate the rest of the quadrics whose offsets may degenerate, i.e. the implicit quadratic-surfaces (ellipsoid, hyperboloid, elliptic cone, elliptic cylinder and hyperbolic cylinder). We found that self-intersection curves of offsets of all the implicit quadratic surfaces are planar implicit conics and their corresponding curve on the progenitor surface can be expressed as the intersection curve between an ellipsoid, whose semi-axes are proportional to the offset distance, and the implicit quadratic surfaces themselves.     

Self-intersections of offsets of quadratic surfaces: Part I, explicit surfaces

Although offset surfaces are widely used in various engineering applications, their degenerating mechanism is not well known in a quantitative manner. Offset surfaces are functionally more complex than their progenitor surfaces and may degenerate even if the progenitor surfaces are regular. Self-intersections of the offsets of regular surfaces may be induced by concave regions of surface where the positive offset distance exceeds the maximum absolute value of the negative minimum principal curvature or the absolute value of the negative offset distance exceeds the maximum value of the positive maximum principal curvature. It is well known that any regular surface can be locally approximated in the neighborhood of a pointp by the explicit quadratic surface of the form r(x,y)=[x,y1/2(x²+y²)]^T to the second order where – and – are the principal curvatures at pointp. Therefore investigations of the selfintersecting mechanisms of the offsets of explicit quadratic surfaces due to differential geometry properties lead to an understanding of the self-intersecting mechanisms of offsets of regular parametric surfaces. In this paper, we develop the equations of the self-intersection curves of an offset of an explicit quadratic surface. We also develop an algorithm to detect and trace a small loop of a self-intersection curve of an offset of a regular parametric surface based on our analysis of offsets of explicit quadratic surfaces. Examples illustrate our method.     

Conchoid surfaces of rational ruled surfaces

The conchoid surface G of a given surface F with respect to a point O is roughly speaking the surface obtained by increasing the radius function of F with respect to O by a constant d. This paper studies real rational ruled surfaces in this context and proves that their conchoid surfaces possess real rational parameterizations, independently of the position of O. Thus any rational ruled surface F admits a rational radius function r(u,v) with respect to any point in space. Besides the general skew ruled surfaces and examples of low algebraic degree we study ruled surfaces generated by rational motions.     

Isogeometric segmentation: Construction of cutting surfaces

The objective of Isogeometric Segmentation is to generate a decomposition of a solid, given in boundary representation, into a collection of a relatively small number of base solids, which can easily be subdivided into topological hexahedra. This can be achieved by repeatedly splitting the solid. In each splitting step, one chooses a cutting loop, which is a cycle of curves around the boundary of the solid, and constructs a cutting surface that splits the solid into two simpler ones. When only hexahedra or pre-defined base solids are left this process terminates.The construction of the cutting surface must ensure that two essential properties are fulfilled: the boundary curves of the surface interpolate the previously constructed cutting loop and the surface neither intersects itself nor the boundary of the solid. A novel method for generating the cutting surface is presented in this paper. The method combines two steps: First we generate an implicit guiding surface, which is subsequently approximated by a trimmed spline surface in the second step.     

Illustrative visualization: interrogating triangulated surfaces

Geometrical modeling is a crucial aspect of simulations involving manufactured objects and is usually performed using free-form surfaces. However, to simulate the flow through or about a manufactured object or to simulate structural integrity, the free-form surfaces must be tessellated into triangulated surfaces. To concurrently visualize the simulation results and the quality of the surfaces, we present two novel visualization algorithms for triangulated surfaces as opposed to the traditional freeform surfaces. The proposed algorithms are for curvature estimation based on local surface fitting with cubic triangular Bézier patches and for reflection-line computation.     

Q-warping: Direct computation of quadratic reference surfaces

We consider the problem of wrapping around an object, of which two views are available, a reference surface and recovering the resulting parametric flow using direct computations (via spatio-temporal derivatives). The well known examples are affine flow models and eight-parameter flow models-both describing a flow field of a planar reference surface. We extend those classic flow models to deal with a quadric reference surface and work out the explicit parametric form of the flow field. As a result we derive a simple warping algorithm that maps between two views and leaves a residual flow proportional to the 3D deviation of the surface from a virtual quadric surface. The applications include image morphing, model building, image stabilization, and disparate view correspondence     

Correspondenceless stereo and motion: planar surfaces

It is shown that a binocular observer can recover the depth and three-dimensional motion of a rigid planar patch without using any correspondences between the left and right image frames (static) or between the successive dynamic frames (dynamic). Uniqueness and robustness issues are studied with respect to this problem and experimental results are given from the application of the theory to real images     

Visualizing chaos: Lyapunov surfaces and volumes

The use of a graphic technique to visualize chaos for a dual-parameter one-dimensional map is described. It involves plotting the Lyapunov exponent with both height and color in a three-dimensional map, as a function of the two parameters. Color is determined using a geographic lookup table. A graphics supercomputer can rotate the map in real time. The technique demonstrates graphically interesting behavior in chaotic systems     

Horizon mapping: shadows for bump-mapped surfaces

Bump mapping produces realistic shading by perturbing normal vectors to a surface, but does not show the shadows that the bumps cast on nearby parts of the same surface. In this paper, these shadows are found from precomputed tables of horizon angles, listing, for each position entry, the elevation of the horizon in a sampled collection of directions. These tables are made for bumps on a standard flat surface, and then a transformation is developed so that the same tables can be used for an arbitrary curved parametrized surface patch. This necessitates a new method for scaling the bump size to the patch size. Incremental calculations can be used in a scan line algorithm for polygonal surface approximations. The errors in the bump shadows are discussed, as well as their anti-aliasing. (An earlier version of this article appeared as Max [10].)     

IRIS: illustrative rendering for integral surfaces

Integral surfaces are ideal tools to illustrate vector fields and fluid flow structures. However, these surfaces can be visually complex and exhibit difficult geometric properties, owing to strong stretching, shearing and folding of the flow from which they are derived. Many techniques for non-photorealistic rendering have been presented previously. It is, however, unclear how these techniques can be applied to integral surfaces. In this paper, we examine how transparency and texturing techniques can be used with integral surfaces to convey both shape and directional information. We present a rendering pipeline that combines these techniques aimed at faithfully and accurately representing integral surfaces while improving visualization insight. The presented pipeline is implemented directly on the GPU, providing real-time interaction for all rendering modes, and does not require expensive preprocessing of integral surfaces after computation.     

One-machine n-part-type optimal setup scheduling: analyticalcharacterization of switching surfaces

The authors consider optimal setup scheduling of a single reliable machine. Production flow of n different part types and the setup process are described by differential equations. Setup change rates are control variables. Necessary conditions on optimal setup changes are characterized analytically, and optimal setup change times are derived for a given setup change sequence. The linearization of optimal setup switching surfaces is derived, indicating the existence of attractors observed in numerical optimal solutions. The approach developed in this paper establishes a strong basis for studying multimachine production systems and for constructing tractable near-optimal numerical solution techniques     

Flow charts: visualization of vector fields on arbitrary surfaces

We introduce a novel flow visualization method called Flow Charts, which uses a texture atlas approach for the visualization of flows defined over curved surfaces. In this scheme the surface and its associated flow are segmented into overlapping patches which are then parameterized and packed in the texture domain. This scheme allows accurate particle advection across multiple charts in the texture domain, providing a flexible framework that supports various flow visualization techniques. The use of surface parameterization enables flow visualization techniques requiring the global view of the surface over long time spans, such as Unsteady Flow LIC (UFLIC), particle-based Unsteady Flow Advection-Convolution (UFAC), or dye advection. It also prevents visual artifacts normally associated with view-dependent methods. Represented as textures, Flow Charts can be naturally integrated into GPU flow visualization techniques for interactive performance.