Rational G2 splines

We develop a class of rational, G²-connected splines of degree 3 that allow modeling multiple basic shapes, such as segments of conics and circle arcs in particular, in one structure. ? This can be used, for example, to have portions of a control polygon exactly reproduce segments of the shapes while other portions blend between these primary shapes. We also show how to reparameterize the splines to obtain parametrically C² transitions.     

Modeling with rational biquadratic splines

We develop a rational biquadratic G¹ analogue of the non-uniform C¹ B-spline paradigm. These G¹ splines can exactly reproduce parts of multiple basic shapes, such as cyclides and quadrics, and combine them into one smoothly-connected structure. This enables a design process that starts with basic shapes, re-represents them in spline form and uses the spline form to provide shape handles for localized free-form modification that can preserve, in the large, the initial fair, basic shapes.     

An involute spiral that matches Hermite data in the plane

A construction is given for a planar rational Pythagorean hodograph spiral, which interpolates any two-point G² Hermite data that a spiral can match. When the curvature at one of the points is zero, the construction gives the unique interpolant that is an involute of a rational Pythagorean hodograph curve of the form cubic over linear. Otherwise, the spiral comprises an involute of a Tschirnhausen cubic together with at most two circular arcs. The construction is by explicit formulas in the first case, and requires the solution of a quadratic equation in the second case.     

Curves with rational Frenet-Serret motion

This paper examines a special type of rational curves called rational Frenet-Serret (RF) curves distinguished by the property that the motion of their Frenet-Serret frame is rational. It is shown that a rational curve is an RF curve if and only if it has rational speed and rational curvature. The paper derives a general representation formula for RF curves suitable for geometric design and provides a geometric survey of special RF curves. The special case of a cubic helix is examined thoroughly. Additionally the paper discusses several applications including examples for the design of rational sweeping surfaces, rational pipe surfaces and rational transition surfaces joining sweeps with G¹ continuity.     

G^2 Two-Point Hermite Rational Cubic Interpolation

We consider the problem of G^2 two-point Hermite interpolation by a rational cubic. Given two points with tangent vectors and curvatures, the necessary and sufficient conditions are placed on the weights of the rational cubic curve which ensures that (i) if the data suggest a C -shaped curve, the rational cubic interpolates a C -shaped curve without loops, cusps, or inflections, and (ii) if the data suggest an S -shaped curve, the rational cubic interpolates an S -shaped curve with a single inflection, no loops and no cusps.     

Shape preservation of scientific data through rational fractal splines

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of $\mathcal C ^1$ - rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form $\frac{p_i(x)}{q_i(x)},$ where $p_i(x)$ and $q_i(x)$ are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in $\mathcal C ^2$ is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results.     

Error analysis of the derivative of rational interpolation with a linear denominator

This paper deals with the approximation properties of the derivatives of rational cubic interpolation with a linear denominator. Error expressions of the derivatives of interpolating functions are derived, convergence is established and the optimal error coefficient c _i is proved to be symmetric about the parameters of the rational interpolation. The unified integral form of the error of the second derivative in all subintervals is obtained. A simple expression of the jump of the second derivative at the knots and the conditions for the interpolating function to be C ² in the interpolating interval are given.     

Some remarks on Bézier curves

Parametric polynomial curves in Bézier-Bernstein representation are considered as prohections of rational norm curves of degree n in n-space; from this point of view the singularities of a planar Bézier cubic are determined and expressed by its affine invariants. Secondly, for an arbitrary pair of adjacent parametric curves in homogeneous coordinates, the general conditions for geometric continuity of any order k, G^k, are established.This result generalizes the corresponding conditions in the non-homogeneous (affine) case, recently obtained by [Goodman '84]. Some applications are given for Bézier curves. In particular, for γ-splines [Boehm '85], the existence of a global rational parameter that makes it to a C² parametric curve is shown. Furthermore, for two adjacent rational Bézier curves the complete conditions for G³ are stated using the projective properties of the control points only.     

How many different rational parametric cubic curves are there? 3.The catalog

The author is interested in rational parametric cubic curves, particularly in finding out how many essentially different shapes the given equation can make. Types of homogeneous cubic polynomials are considered     

Quaternion Julia Set Shape Optimization

We present the first 3D algorithm capable of answering the question: what would a Mandelbrot‐like set in the shape of a bunny look like? More concretely, can we find an iterated quaternion rational map whose potential field contains an isocontour with a desired shape? We show that it is possible to answer this question by casting it as a shape optimization that discovers novel, highly complex shapes. The problem can be written as an energy minimization, the optimization can be made practical by using an efficient method for gradient evaluation, and convergence can be accelerated by using a variety of multi‐resolution strategies. The resulting shapes are not invariant under common operations such as translation, and instead undergo intricate, non‐linear transformations.     

Ratioquadrics: an alternative model for superquadrics

This paper presents a new family of 2D curves and its extension to 3D surfaces, respectively, calledrationconics andratioquadrics that have been designed as alternatives to the well-known superconics and superquadrics. This new model is intended as an improvement to the original one on three main points: first, it involves lower computation cost and provides better numerical robustness; second, it offers higher order continuities (C ¹/G ² orC ²/G ² instead ofC ⁰/G ⁰); and third, it provides a greater variety of shapes for the resulting curves and surfaces. All these improvements are obtained by replacing the signed power function involved in the formulation of superconics and superquadrics by linear or quadratic rational polynomials.     

Branching object generation and animation system with cubic hermite interpolation

This paper presents geometric modelling methods for branching object generation and animation with cubic Hermite curves. The procedure for creating a branching object involves two steps. First, the skeleton of an object is constructed. The skeleton consists of geometrical and topological information about the segments of the object. Secondly, cubic Hermite curves are used to deform and join consecutive segments in the skeleton in order to form the body of the object with C¹ continuity. Cubic Hermite curves are also used to deform partial segments which allow an object to grow with desirable continuity in space and time. The techniques are easily implemented on any computer and allow the user to create a branching object using objects which have diverse shapes as segments.     

Fast Parallel Construction of Smooth Surfaces from Meshes with Tri/Quad/Pent Facets

Polyhedral meshes consisting of triangles, quads, and pentagons and polar configurations cover all major sampling and modeling scenarios. We give an algorithm for efficient local, parallel conversion of such meshes to an everywhere smooth surface consisting of low‐degree polynomial pieces. Quadrilateral facets with 4‐valent vertices are ‘regular’ and are mapped to bi‐cubic patches so that adjacent bi‐cubics join C² as for cubic tensor‐product splines. The algorithm can be implemented in the vertex and geometry shaders of the GPU pipeline and does not use the fragment shader. Its implementation in DirectX 10 achieves conversion plus rendering at 659 frames per second with 42.5 million triangles per second on input of a model of 1300 facets of which 60% are not regular.     

Order Statistics in the Farey Sequences in Sublinear Time and Counting Primitive Lattice Points in Polygons

We present the first sublinear-time algorithms for computing order statistics in the Farey sequence and for the related problem of ranking. Our algorithms achieve a running times of nearly O(n ^2/3), which is a significant improvement over the previous algorithms taking time O(n). We also initiate the study of a more general problem: counting primitive lattice points inside planar shapes. For rational polygons containing the origin, we obtain a running time proportional to D ^6/7, where D is the diameter of the polygon. This work represents a merging of 19 and 21, with additional extensions.     

G1 continuous approximate curves on NURBS surfaces

Curves on surfaces play an important role in computer aided geometric design. In this paper, we present a parabola approximation method based on the cubic reparameterization of rational Bézier surfaces, which generates G¹ continuous approximate curves lying completely on the surfaces by using iso-parameter curves of the reparameterized surfaces. The Hausdorff distance between the approximate curve and the exact curve is controlled under the user-specified tolerance. Examples are given to show the performance of our algorithm.     

Independently controllable dual‐band filter using single quad‐mode silver‐loaded dielectric resonator

This article proposes a novel bandpass filter with two controllable passbands using a single quad‐mode silver‐loaded dielectric resonator (DR). The silver plane is inserted in the middle of the cubic DR and two degenerate pairs are used to build the two passbands. Because of the distinct E‐field distributions, the silver plane has significant effect on the degenerate pair (TE^x₁₁₂ and TE^y₁₁₂), whereas another one (TE^x₁₁₁ and TE^y₁₁₁) remains unchanged. With the aid of the silver plane, both center frequencies and bandwidths of the two bands can be controlled independently. To verify the proposed idea, a prototype dual‐band BPF is designed and fabricated. Good agreement between simulated and measured results can be observed.     

Translational Covering of Closed Planar Cubic B-Spline Curves

Spline curves are useful in a variety of geometric modeling and graphics applications and covering problems abound in practical settings. This work defines a class of covering decision problems for shapes bounded by spline curves. As a first step in addressing these problems, this paper treats translational spline covering for planar, uniform, cubic B‐splines. Inner and outer polygonal approximations to the spline regions are generated using enclosures that are inside two different types of piecewise‐linear envelopes. Our recent polygonal covering technique is then applied to seek translations of the covering shapes that allow them to fully cover the target shape. A feasible solution to the polygonal instance provides a feasible solution to the spline instance. We use our recent proof that 2D translational polygonal covering is NP‐hard to establish NP‐hardness of our planar translational spline covering problem. Our polygonal approximation strategy creates approximations that are tight, yet the number of vertices is only a linear function of the number of control points. Using recent results on B‐spline curve envelopes, we bound the distance from the spline curve to its approximation. We balance the two competing objectives of tightness vs. number of points in the approximation, which is crucial given the NP‐hardness of the spline problem. Examples of the results of our spline covering work are provided for instances containing as many as six covering shapes, including both convex and nonconvex regions. Our implementation uses the LEDA and CGAL C++ libraries of geometric data structures and algorithms.     

The hybrid quintic Bézier tetrahedron

The focus of this paper is the construction of a new tetrahedral patch, which allows an explicit representation but does not require a split of the domain tetrahedra. The patch will be represented as a rational convex combination of a quintic Bézier tetrahedron, where the four inner Bézier points are duplicated. It can be regarded as generalization of the hybrid cubic triangular Bézier patch. The patch is used to represent a C¹ trivariate scattered data interpolant to data sampled in a volume.