Control Polygon Research Articles

Approximation of a Continuous Curve by its Bernstein-Bézier Operator

In this paper we consider integer and rational parametric Bezier curves and study the distance between the curve and its control polygon. To measure the distance we use first and second order moduli of smoothness of vector-valued function. We consider also NURBS curves with equidistant knots. Some direct approximation theorems will be presented.

Subdivision depth computation for n-ary subdivision curves/surfaces

This paper deals with subdivision depth computation technique for n-ary subdivision curves/surfaces. This technique also includes error bound evaluation technique for n-ary subdivision curves/surfaces with their control polygon. Both techniques provide error control tools in subdivision schemes.

Analysis of B-Spline Curve Using Discrete Fourier Transform

We apply the Discrete Fourier Transform to the construction of B-Spline curves to gain more insight into their structure. As a B-Spline curve is determined by its control polygon, this analysis is intimately linked to the Fourier analysis of the control polygon. To do this we apply Fast Fourier transform (FFT) algorithm to the structure of B-Spline curve and its rational form. We get inner structure of original B-Spline curve in the transform domain again in the form of B-Spline curve, having control polygon as regular or star polygon. Using the technique mentioned in the paper we get the same curve without change of shape in the transformed case of polygon points. We also extend the idea for the interval form of B-Spline Curve.

Designing objects with a spline

Splines play an important role in approximation theory as well as in computer-aided design and geometric modelling. Specifically, designing of objects is a fundamental problem that is mainly being attempted to be resolved using splines. Many authors have worked in the area and various techniques exist in the current literature. B-splines and non-uniform B-splines (NURBS) are quite popular in literature. However, there also exist some other splines to compete. The paper is devoted towards a new spline method that is a generalization of the Ball's cubic spline method on one hand, and also serves as an effective alternate to the NURBS of degree three on the other hand. Together with a class of shape parameters in its description, it introduces a class of cubic polynomial blending functions. These blending functions play a role of basis functions that ultimately describe the newly proposed spline method. Piecewise polynomial curves with a family of shape parameters are constructed from these blending functions. The generated curves have a second-order geometric continuity similar to those of weighted Nu-splines. The proposed method also provides a strong alternate to the NURBS of degree three as far as shape control is concerned. If the values of the shape parameters are changed, the approaching degree of the curves to their control polygon is adjusted accordingly and the curves are manipulated to approximate the cubic B-spline curve from its both sides. In comparison with the existing techniques, the degree of the spline method is ideally three and the domain of the shape parameters is broader than those in NURBS. The scheme has been implemented practically to demonstrate its effectiveness.This curve design method, in addition to enjoying the good features of Ball's cubic splines, possesses interesting shape design features too. The proposed spline method possesses two families of shape parameters. One family of parameters is associated with intervals and the other with points to have variety of shape control.

Consolidated sharp bounds for Bézier curve approximation with cutdown polygon and corner cutting polygon

This article presents consolidated sharp bounds for Bézier curve approximation using various approximation polygons. The sharp distance bounds between a Bézier curve and its cutdown polygon, which is introduced recently, are first obtained. The result is a further extension and consolidation over recent results. A polygon named corner cutting polygon is further constructed by using the well-known de Casteljau central corner cutting algorithm. The sharp bounds for Bézier curve approximation using the corner cutting polygon are also obtained. Based on similar methods, various sharp bounds for Bézier curve approximation using control polygons and quasi control polygons are further addressed and extended to parametric Bézier curves in multi-dimensional spaces.

High-speed contouring control with NURBS-based C2 PH spline curves

This paper presents a C2 Pythagorean hodograph (PH) spline curve constructed by the non-uniform rational B-spline (NURBS) curve for high-speed contouring control. With the knot vector, weights, and control points given, the C2 PH spline curve is defined to be a “good” interpolant for Hermit data obtained from a NURBS curve of degree 3 specified by the same control polygon, weights, and the knot vector. To this end, the first- and second-order derivatives are evaluated at the nodal points on the NURBS curve. These boundary conditions are imposed on the PH segments of degree 9 to preserve continuity between the connecting segments. The S-curve motion planning architecture with variable feed rate for a planar NURBS-based C2 PH spline curve is also developed in this paper. In particular, C1 cubic feed acceleration/deceleration is imposed on the first and last PH segments. Several NURBS-based C2 PH spline curve-following tasks were conducted to verify the effectiveness of the proposed interpolation algorithm. Experimental results show that the proposed interpolator is not only feasible for machining the complicated parametric curves represented in the NURBS-based C2 PH spline form but also yields satisfactory contouring performance under variable feed rate control.

On control polygons of quartic Pythagorean–hodograph curves

In this paper, we study a necessary and sufficient condition for a planar quartic Bézier curve to possess a Pythagorean–hodograph (PH). Based on the definition of PH curve and complex representation of planar curve, we deduce geometric conditions in terms of the legs of the control polygon which guarantee the PH property. We also discuss the problem of G 1 Hermite interpolation by planar PH quartics.

Periodic Bézier curves

We construct closed trigonometric curves in a Bézier-like fashion. A closed control polygon defines the curves, and the control points exert a push-pull effect on the curve. The representation of circles and derived curves turns out to be surprisingly simple. Fourier and Bézier coefficients of a curve relate via Discrete Fourier Transform (DFT). As a consequence, DFT also applies to several operations, including parameter shift, successive differentiation and degree-elevation. This Bézier model is a particular instance of a general periodic scheme, where radial basis functions are generated as translates of a symmetric function. In addition to Bézier-like approximation, such a periodic scheme subsumes trigonometric Lagrange interpolation. The change of basis between Bézier and Lagrange proceeds via DFT too, which can be applied to sample the curve at regularly spaced parameter values. The Bézier curve defined by certain control points is a low-pass filtered version of the Lagrange curve interpolating the same set of points.

Complex rational Bézier curves

We extend the rational Bézier model for planar curves, by allowing complex weights. The key idea to endow this representation with a geometric meaning is the use of weight points (Farin points), which are no longer constrained to lie on the line segment defined by two successive control points. Instead of a control polygon, we have now the complex quotient of two control polygons, that is, a control curve made up of circular arcs, whose shape is controlled by the weight points. In the complex version of the rational de Casteljau algorithm, ratios become complex magnitudes, and repeated interpolation on lines is substituted by interpolation on circles. Invariance with respect to perspective transformations is replaced with invariance with respect to Möbius transformations. Therefore, not only enjoy these complex curves the customary linear precision, but also circular precision, i.e., if all control points and weight points lie on a circle, then an arc on the circle is reproduced. This complex model furnishes an intrinsically simpler representation for several remarkable curves, whose degree is halved with respect to the customary Bézier form. Circles are linear instead of quadratic, whereas remarkable epitrochoids (including the cardioid), Joukowski profiles and the Lemniscate of Bernoulli, are quadratic instead of quartic.

A cyclic basis for closed curve and surface modeling

We define a cyclic basis for the vectorspace of truncated Fourier series. The basis has several nice properties, such as positivity, summing to 1, that are often required in computer aided design, and that are used by designers in order to control curves by manipulating control points. Our curves have cyclic symmetry, i.e. the control points can be cyclically arranged and the curve does not change when the control points are cyclically permuted. We provide an explicit formula for the elevation of the degree from n to n+r, r>=1 and prove that the control polygon of the degree elevated curve converges to the curve itself if r tends to infinity. Variation diminishing property of the curve is also verified. The proposed basis functions are suitable for the description of closed curves and surfaces with C^~ continuity at all of their points.

Modelling of cubic Bézier curves by electronic circuits

In this paper, analog techniques have been utilized to translate the mathematical formulation of arbitrary functions into electronic circuits which embody instantaneous DC relationships between sets of voltages having a limited dynamic range of variation. Suitable building blocks and electronic circuits have been designed to implement some of the basic functional operations. These functional operations were used to embody an analog modelling technique for the generation of free-form curves based on Bezier curve generation method. To draw a specified curve, the control parameters represented by a group of variable resistors are set at specified positions. These positions correspond to values of the control polygon vertices coordinates that uniquely define the curve shape. To change curve shape, the only requirement is to vary the control parameters represented by the variable resistors, and therefore, the user can use this electronic generation method as a shape designer. Using this analog modelling technique, one can generate curves at speed much more comparable to the alternative software modelling on a digital computer.

Generating fair, continuous splines by blending conics

We present a flexible, stable and locally adjustable conic blending method for the sparse points interpolation, which can avoid the potential pitfalls and unacceptable cases appeared in the circular blending method. Two interpolatory schemes are proposed to generate fair, C^2 continuous splines. One of both schemes is compatible with the NURBS curves. The blended spline curve is consistent with the interpolation control polygon and can be locally adjusted by free parameters. Experimental examples and analysis are presented to show the efficiency of the proposed method.

Formula omitted]-splines

Both the 4-point and the uniform cubic B -spline subdivisions double the number of vertices of a closed-loop polygon k P and produce sequences of vertices f j and b j respectively. We study the J -spline subdivision scheme J s , introduced by Maillot and Stam, which blends these two methods to produce vertices of the form v j = ( 1 − s ) f j + s b j . Iterative applications of J s yield a family of limit curves, the shape of which is parameterized by s. They include four-point subdivision curves ( J 0 ), uniform cubic B -spline curves ( J 1 ), and uniform quintic B -spline curves ( J 1.5 ). We show that the limit curve is at least C 1 when − 1.7 ≤ s ≤ 5.8 , C 2 when 0 < s < 4 , C 3 when 1 < s ≤ 2.8 , and C 4 when s = 3 / 2 , even though a 4-point yields only C 1 curves and a cubic B -spline yields only C 2 curves. We generalize the J s scheme to a two-parameter family J a , b and propose data-dependent and data-independent solutions for computing values of parameters a and b that minimize various objective functions (distance to the control vertices, deviation from the control polygon, change in surface area, and popping when switching levels of subdivision in multi-resolution rendering). We extend the J -spline subdivision to open curves and to a smooth surface subdivision scheme for quad-meshes with arbitrary connectivity.

Four-point curve subdivision based on iterated chordal and centripetal parameterizations

Dubuc's interpolatory four-point scheme inserts a new point by fitting a cubic polynomial to neighbouring points over uniformly spaced parameter values. In this paper we replace uniform parameter values by chordal and centripetal ones. Since we update the parameterization at each refinement level, both schemes are non-linear. Because of this data-dependent parameterization, the schemes are only invariant under solid body and isotropic scaling transformations, but not under general affine transformations. We prove convergence of the two schemes and bound the distance between the limit curve and the initial control polygon. Numerical examples indicate that the limit curves are smooth and that the centripetal one is tighter, as suggested by the distance bounds. Similar to cubic spline interpolation, the use of centripetal parameter values for highly non-uniform initial data yields better results than the use of uniform or chordal ones.

A simple and efficient approximation of a Bézier piece by its cutdown polygon

A new concept named the cutdown polygon is introduced for a Bézier piece approximation. The cutdown polygon is constructed by directly connecting some selected control points of the Bézier piece. Based on the proposed approximation, we obtained two explicit quantitative bounds. Apart from the degree of the polynomial, the two bounds depend only on the maximal absolute second differences or the sum of absolute second differences of the sequence of control points, respectively. Compared with control polygon approximation, the key advantage of using the cutdown polygon is that the edge number can be dramatically reduced to about one half of that using control polygon approximation or less while maintaining the same estimate bound.

Computing intersections of planar spline curves using knot insertion

We present a new method for computing intersections of two parametric B-spline curves. We use an intersection of the control polygons as an approximation for an intersection of the curves in combination with knot insertion. The resulting algorithm is asymptotically Newton-like, but without the need of a starting value. Like Newton's method, it converges quadratically at transversal intersections, the analogue to simple roots. It is a generalization of an algorithm developed by two of the authors for computing zeros of spline functions.

Heavy metal content and response of the blood system of Siberian dace Leuciscus leuciscus baicalensis (Cypriniformes, Cyprinidae) under anthropogenic impact

Chemical, hematological, and immunological characteristics of the Siberian dace Leuciscus baicalensis from different areas of Lake Baikal basin were first determined. It was established that the most considerable changes in hemopoesis, cellular, and humoral links of immunity were in individuals with a high content of heavy metals in the liver from populations inhabiting the Selenga River downstream the Ulan Ude Industrial Center and the Selenga Pulp-and-Cardboard Plant. Changes in immunohematological responses of L. leusiscus baicalensis from Cherkalov Bay of Lake Baikal proceeded within the adaptive possibilities of the species and indicated the development of compensatory reactions in response to the accumulation of heavy metals in the body. The results obtained make it possible to assert that Chivurkuiskii Bay of Baikal may serve as a control polygon during monitoring of immunobiological states of fish populations from Baikal basin.

New method of curve modeling based on continuity for extension-Bézier curves

The extension of Bezier curves with shape control parameters has become the hot spot in computer aided geometric design. A new method of curve modeling based on continuity for Extension-Bezier curves was presented in this paper. This method first constructed a class polynomial curves called cubic extension (CE) Bezier curves with three shape parameter α, β, γ. CE-Bezier curves not only inherit the outstanding properties of cubic Bezier curve, but also are adjustable in shape and fit close to the control polygon. In order to overcome shortcomings that the circular arc and ellipse could not be accurately represented by CE-Bezier curves, this method used the continuity condition of CE-Bezier curves and C-Bezier curves to resolve the representation of circular arc and ellipse in CE-Bezier curve modeling. In addition, some applications of this method in curve and surface design were also discussed. The modeling examples illustrate that this method is valuable for computer aided geometric design.

Dynamic Bezier curves for variable rate-distortion

Bezier curves (BC) are important tools in a wide range of diverse and challenging applications, from computer-aided design to generic object shape descriptors. A major constraint of the classical BC is that only global information concerning control points (CP) is considered, consequently there may be a sizeable gap between the BC and its control polygon (CtrlPoly), leading to a large distortion in shape representation. While BC variants like degree elevation, composite BC and refinement and subdivision narrow this gap, they increase the number of CP and thereby both the required bit-rate and computational complexity. In addition, while quasi-Bezier curves (QBC) close the gap without increasing the number of CP, they reduce the underlying distortion by only a fixed amount. This paper presents a novel contribution to BC theory, with the introduction of a dynamic Bezier curve (DBC) model, which embeds variable localised CP information into the inherently global Bezier framework, by strategically moving BC points towards the CtrlPoly. A shifting parameter (SP) is defined that enables curves lying within the region between the BC and CtrlPoly to be generated, with no commensurate increase in CP. DBC provides a flexible rate-distortion (RD) criterion for shape coding applications, with a theoretical model for determining the optimal SP value for any admissible distortion being formulated. Crucially DBC retains core properties of the classical BC, including the convex hull and affine invariance, and can be seamlessly integrated into both the vertex-based shape coding and shape descriptor frameworks to improve their RD performance. DBC has been empirically tested upon a number of natural and synthetically shaped objects, with qualitative and quantitative results confirming its consistently superior shape approximation performance, compared with the classical BC, QBC and other established BC-based shape descriptor techniques.

The cubic trigonometric Bézier curve with two shape parameters

A cubic trigonometric Bézier curve analogous to the cubic Bézier curve, with two shape parameters, is presented in this work. The shape of the curve can be adjusted by altering the values of shape parameters while the control polygon is kept unchanged. With the shape parameters, the cubic trigonometric Bézier curves can be made close to the cubic Bézier curves or closer to the given control polygon than the cubic Bézier curves. The ellipses can be represented exactly using cubic trigonometric Bézier curves.