Control Polygon Research Articles

A novel generalization of Bézier curve and surface

A new formulation for the representation and designing of curves and surfaces is presented. It is a novel generalization of Bézier curves and surfaces. Firstly, a class of polynomial basis functions with n adjustable shape parameters is present. It is a natural extension to classical Bernstein basis functions. The corresponding Bézier curves and surfaces, the so-called Quasi-Bézier (i.e., Q-Bézier, for short) curves and surfaces, are also constructed and their properties studied. It has been shown that the main advantage compared to the ordinary Bézier curves and surfaces is that after inputting a set of control points and values of newly introduced n shape parameters, the desired curve or surface can be flexibly chosen from a set of curves or surfaces which differ either locally or globally by suitably modifying the values of the shape parameters, when the control polygon is maintained. The Q-Bézier curve and surface inherit the most properties of Bézier curve and surface and can be more approximated to the control polygon. It is visible that the properties of end-points on Q-Bézier curve and surface can be locally controlled by these shape parameters. Some examples are given by figures.

Quasi-Bezier curves integrating localised information

Bezier curves (BC) have become fundamental tools in many challenging and varied applications, ranging from computer-aided geometric design to generic object shape descriptors. A major limitation of the classical Bezier curve, however, is that only global information about its control points (CP) is considered, so there can often be a large gap between the curve and its control polygon, leading to large distortion in shape representation. While strategies such as degree elevation, composite BC, refinement and subdivision reduce this gap, they also increase the number of CP and hence bit-rate, and computational complexity. This paper presents novel contributions to BC theory, with the introduction of quasi-Bezier curves (QBC), which seamlessly integrate localised CP information into the inherent global Bezier framework, with no increase in either the number of CP or order of computational complexity. QBC crucially retains the core properties of the classical BC, such as geometric continuity and affine invariance, and can be embedded into the vertex-based shape coding and shape descriptor framework to enhance rate-distortion performance. The performance of QBC has been empirically tested upon a number of natural and synthetically shaped objects, with both qualitative and quantitative results confirming its consistently superior approximation performance in comparison with both the classical BC and other established BC-based shape descriptor methods.

Preserving computational topology by subdivision of quadratic and cubic Bézier curves

Non-self-intersection is both a topological and a geometric property. It is known that non-self-intersecting regular Bezier curves have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within ℝ3 for a non-self-intersecting, regular C 2 cubic Bezier curve to be ambient isotopic to its control polygon formed after sufficiently many subdivisions. The benefit of using the control polygon as an approximant for scientific visualization is presented in this paper.

Fast Voronoi modeling

The common way to construct Voronoi tessellations is to compare the distances between given reference points using a given distance function. To generalize this distance-function concept we expand an existing approach which defines distance functions by their ``unit circles''. Our new approach allows modeling the ``unit circles'' by a closed Spline curve. Changing the control polygon directly affects the tessellation's appearance. Typically generalized Voronoi diagrams are represented by Voronoi vertices and curves separating the individual tiles. To obtain interactive modeling we extended an existing hardware accelerated rendering approach computing a bitmap-representation using different colors for individual tiles. With our extension, we are able to use our Spline distance representations as input for a growing process. This growing process easily takes into account weighting approaches like multiplicative, additive, and even free functional weighting.

A framework for quad/triangle subdivision surface fitting: Application to mechanical objects

In this paper we present a new framework for subdivision surface approximation of three-dimensional models represented by polygonal meshes. Our approach, particularly suited for mechanical or Computer Aided Design (CAD) parts, produces a mixed quadrangle-triangle control mesh, optimized in terms of face and vertex numbers while remaining independent of the connectivity of the input mesh. Our algorithm begins with a decomposition of the object into surface patches. The main idea is to approximate the region boundaries first and then the interior data. Thus, for each patch, a first step approximates the boundaries with subdivision curves (associated with control polygons) and creates an initial subdivision surface by linking the boundary control points with respect to the lines of curvature of the target surface. Then, a second step optimizes the initial subdivision surface by iteratively moving control points and enriching regions according to the error distribution. The final control mesh defining the whole model is then created assembling every local subdivision control meshes. This control polyhedron is much more compact than the original mesh and visually represents the same shape after several subdivision steps, hence it is particularly suitable for compression and visualization tasks. Experiments conducted on several mechanical models have proven the coherency and the efficiency of our algorithm, compared with existing methods.

An interpolating 4-point [formula omitted] ternary non-stationary subdivision scheme with tension control

In this paper we present a non-stationary 4-point ternary interpolatory subdivision scheme which provides the user with a tension parameter that, when increased within its range of definition, can generate C 2 -continuous limit curves showing considerable variations of shape. As a generalization we additionally propose a locally-controlled C 2 -continuous subdivision scheme, which allows a different tension value to be assigned to every edge of the original control polygon.

Offset-rational sinusoidal spirals in Bézier form

Mineur et al. [Mineur, Y., Lichah, T., Castelain, J.M., Giaume, H., 1998. A shape controlled fitting method for Bézier curves. Computer Aided Geometric Design 15 (9), 879–891] presented a remarkable family of Bézier curves, characterized by a control polygon with constant angle at each vertex and constant ratio between lengths of consecutive edges. We note that such curves belong to a classical family of polynomial sinusoidal spirals that encompasses Tschirnhausen's cubic, and can also be expressed as p-Bézier curves (Bézier curves in polar coordinates). These spirals have rational offsets, obtained simply by adding two p-Bézier curves, namely the spiral and a circle. Such a construction includes the offset to a parabola.

An unconditionally convergent method for computing zeros of splines and polynomials

We present a simple and efficient method for computing zeros of spline functions. The method exploits the close relationship between a spline and its control polygon and is based on repeated knot insertion. Like Newton’s method it is quadratically convergent, but the new method overcomes the principal problem with Newton’s method in that it always converges and no starting value needs to be supplied by the user.

A control polygon scheme for design of planar [formula omitted] PH quintic spline curves

A scheme to specify planar C 2 Pythagorean-hodograph (PH) quintic spline curves by control polygons is proposed, in which the “ordinary” C 2 cubic B-spline curve serves as a reference for the shape of the PH spline. The method facilitates intuitive and efficient constructions of open and closed PH spline curves, that typically agree closely with the corresponding cubic B-spline curves. The C 2 PH quintic spline curve associated with a given control polygon and knot sequence is defined to be the “good” interpolant to the nodal points of the C 2 cubic spline curve with the same B-spline control points, knot sequence, and end conditions—it may be computed to machine precision by just a few Newton–Raphson iterations from a close starting approximation. The relation between the PH spline and its control polygon is invariant under similarity transformations. Multiple knots may be inserted to reduce the order of continuity to C 1 or C 0 at specified points, and by means of double knots the PH splines offer a linear precision and local shape modification capability. Although the non-linear nature of PH splines precludes proofs for certain features of cubic B-splines, such as convex-hull confinement and the variation-diminishing property, this is of little practical significance in view of the close agreement of the two curves in most cases (in fact, the PH spline typically exhibits a somewhat better curvature distribution).

A new three-point approximating [formula omitted] subdivision scheme

A new three-point approximating subdivision scheme is presented which generates C 2 curves. Its limit function has a support on [−3,2]. The smoothness of the new scheme is shown using the Laurent polynomial method, and a comparison with the Hassan et al. [M.F. Hassan, N.A. Dodgson, Ternary and three-point univariate subdivision schemes, in: A. Cohen, J.-L. Merrien, L.L. Schumaker (Eds.), Curve and Surface Fitting: Sant-Malo 2002, Nashboro Press, Brentwood, 2003, pp. 199–208] scheme is depicted through two examples. It can be observed that the method developed generates a curve more consistent with the control polygon.

Constructing and reconstructing characters, words, and sentences by synthesizing writing motions

In this paper, we investigate the problem of generating and manipulating characters, words, and sentences. First, characters are obtained as the trace of the cross-sectional area between a virtual writing plane and a virtual writing device modeled by a circular cone, which is moving in a three-dimensional space. Such a motion is designed as a sum of weighted and time-shifted B-spline functions. A sequence of weight vectors called control points defines the so-called control polygon, and represents the motion and hence, a character. Characters or control polygons designed in this way are then connected in a systematic way to yield words and sentences, which again are represented as a control polygon, either small or large. Then, we consider transforming the typeface of characters or words to the omitted running styles as seen in Japanese calligraphy. Such a scheme is developed by formulating the problem as an optimal function-approximation problem, which minimizes an input energy of writing motions. Finally, a design example of cursive words is included and also the method is implemented for robotic handwriting. Thus, the paper provides a unified framework for generating and manipulating characters, words, and sentences, which are complex planar patterns, in a systematic as well as theoretically sound fashion. Although we describe the method using Japanese Kana characters, the idea and methodology for constructing and reconstructing characters and character strings is general and is not only limited to Japanese characters

Formula omitted] spherical Bézier splines

The classical de Casteljau algorithm for constructing Bézier curves can be generalised to a sphere of arbitrary dimension by replacing line segments with shortest great circle arcs. The resulting spherical Bézier curves are C ∞ and interpolate the endpoints of their control polygons. In the present paper, we address the problem of piecing these curves together into C 2 splines. For this purpose, we compute the endpoint velocities and accelerations of a spherical Bézier curve of arbitrary degree and use the formulae to define control points that give the curve a desired initial velocity and acceleration. In addition, for uniform splines we establish a simple relationship between the control points of neighbouring curve segments that is necessary and sufficient for C 2 continuity. As illustration, we solve an interpolation problem involving sparse data using both the present method and a normalised polynomial interpolant. The normalised spline exhibits large variations in speed and magnitude of acceleration, whilst the spherical Bézier spline is far better behaved. These considerations are important in applications where velocities and accelerations need to moderated or estimated, notably computer animation and rigid body trajectory planning, where interpolation in the 3-sphere is a fundamental task.

Piecewise quartic polynomial curves with a local shape parameter

Piecewise quartic polynomial curves with a local shape parameter are presented in this paper. The given blending function is an extension of the cubic uniform B-splines. The changes of a local shape parameter will only change two curve segments. With the increase of the value of a shape parameter, the curves approach a corresponding control point. The given curves possess satisfying shape-preserving properties. The given curve can also be used to interpolate locally the control points with GC 2 continuity. Thus, the given curves unify the representation of the curves for interpolating and approximating the control polygon. As an application, the piecewise polynomial curves can intersect an ellipse at different knot values by choosing the value of the shape parameter. The given curve can approximate an ellipse from the both sides and can then yield a tight envelope for an ellipse. Some computing examples for curve design are given.

Estimating error bounds for binary subdivision curves/surfaces

We estimate error bounds between binary subdivision curves/surfaces and their control polygons after k-fold subdivision in terms of the maximal differences of the initial control point sequences and constants that depend on the subdivision mask. The bound is independent of the process of subdivision and can be evaluated without recursive subdivision. Our technique is independent of parameterizations therefore it can be easily and efficiently implemented. This is useful and important for pre-computing the error bounds of subdivision curves/surfaces in advance in many engineering applications such as curve/surface intersection, mesh generation, NC machining, surface rendering and so on.

NURBS output based tool path generation for freeform pockets

A robust method is proposed to generate tool paths for NURBS-based machining of arbitrarily shaped freeform pockets with islands. Although the input and output are all of higher-degree NURBS curves, only one simple category of geometric entities, i.e., line segments, is required for initial offsetting and for detecting and removing self-intersecting loops. Furthermore, using those linear non-self-intersecting offsets as the legs of NURBS control polygons, NURBS-format tool paths can be smoothly reconstructed with G1-continuity, no overcutting, no cusps, and global error control. Since all operations involved in computing tool path curves are linear geometric calculations, the method is robust and simple. Examples with integrated rough and finish cutting tool paths of pockets demonstrate the usefulness and effectiveness of this method.

Convergence and [formula omitted] analysis of subdivision schemes on manifolds by proximity

Curve subdivision schemes on manifolds and in Lie groups are constructed from linear subdivision schemes by first representing the rules of affinely invariant linear schemes in terms of repeated affine averages, and then replacing the operation of affine average either by a geodesic average (in the Riemannian sense or in a certain Lie group sense), or by projection of the affine averages onto a surface. The analysis of these schemes is based on their proximity to the linear schemes which they are derived from. We verify that a linear scheme S and its analogous nonlinear scheme T satisfy a proximity condition. We further show that the proximity condition implies the convergence of T and continuity of its limit curves, if S has the same property, and if the distances of consecutive points of the initial control polygon are small enough. Moreover, if S satisfies a smoothness condition which is sufficient for its limit curves to be C 1 , and if T is convergent, then the curves generated by T are also C 1 . Similar analysis of C 2 smoothness is postponed to a forthcoming paper.

Bézier curves with shape parameter

In this paper, Bezier basis with shape parameter is constructed by an integral approach. Based on this basis, we define the Bezier curves with shape parameter. The Bezier basis curves with shape parameter have most properties of Bernstein basis and the Bezier curves. Moreover the shape parameter can adjust the curves' shape with the same control polygon. As the increase of the shape parameter, the Bezier curves with shape parameter approximate to the control polygon. In the last, the Bezier surface with shape parameter is also constructed and it has most properties of Bezier surface.

Sharp bounds on the approximation of a Bézier polynomial by its quasi-control polygon

By connecting the points which are the kind of linear combinations of Bézier control points, a broken line polygon called quasi-control polygon is produced. Using it to approximate Bézier segment, this paper obtains two sharp, quantitative bounds, besides depending on the degree of the polynomial, the bounds depend only on the maximal absolute second differences or the sum of absolute second differences of the control point sequence respectively. The advantage of this method is hardly increasing calculation, the effect of using quasi-control polygon to approximate is better than that of using control polygon to approximate.

A new subdivision based approach for piecewise smooth approximation of 3D polygonal curves

This paper presents an algorithm dealing with the data reduction and the approximation of 3D polygonal curves. Our method is able to approximate efficiently a set of straight 3D segments or points with a piecewise smooth subdivision curve, in a near optimal way in terms of control point number. Our algorithm is a generalization for subdivision rules, including sharp vertex processing, of the Active B-Spline Curve developed by Pottmann et al. We have also developed a theoretically demonstrated approach, analysing curvature properties of B-Splines, which computes a near optimal evaluation of the initial number and positions of control points. Moreover, our original Active Footpoint Parameterization method prevents wrong matching problems occurring particularly for self-intersecting curves. Thus, the stability of the algorithm is highly increased. Our method was tested on different sets of curves and gives satisfying results regarding to approximation error, convergence speed and compression rate. This method is in line with a larger 3D CAD object compression scheme by piecewise subdivision surface approximation. The objective is to fit a subdivision surface on a target patch by first fitting its boundary with a subdivision curve whose control polygon will represent the boundary of the surface control polyhedron.

Ternary univariate curvature-preserving subdivision

We present an interpolating, univariate subdivision scheme which preserves the discrete curvature and tangent direction at each step of subdivision. Since the polygon have a geometric information of some original (in some sense) curve as a discrete curvature, we can expect that the limit curve has the same curvature at each vertex as the control polygon. We estimate the curvature bound of odd vertices and give an error estimate for restoring a curve from sampled vertices on curves.