Parametric Polynomial Curves Research Articles

Extending the Hough transform to recognize and approximate space curves in 3D models

Feature curves are space curves identified by color or curvature variations in a shape, which are crucial for human perception (Biederman, 1995). Detecting these characteristic lines in 3D digital models becomes important for recognition and representation processes. For recognizing plane curves in images, the Hough transform (HT) provided a very good solution to the problem. It selects the best-fitting curve in a dictionary of families of curves through a voting procedure that makes it robust to noise and missing parts. Since 3D digital models are often obtained by scanning real objects and may have many defects, the HT has been extended to recognize and approximate space curves in 3D models that correspond to relevant featuresThis work overviews three HT-based different approaches for identifying and approximating spatial profiles of points extracted from point clouds or meshes. A first attempt at this extension involved projecting the spatial points onto the regression plane, thus reducing the problem to planar recognition and using families of plane curves. A second approach has been proposed to recognize spatial profiles that cannot be projected onto the regression plane, using two types of space curve families. Unfortunately, the main drawback of methods based on traditional HT is that it requires prior knowledge of which family of curves to look for.To overcome this limitation, a third method has been developed that provides a piecewise space curve approximation using specific parametric polynomial curves. Additionally, free-form curves that a parametric or implicit form cannot express can be represented using this technique.In the paper, we also analyze the pros and cons of the various approaches and how they managed and reduced the HT's computational cost, given the large number of parameters introduced when families of space curves are considered.

Computing nodes for plane data points by constructing cubic polynomial with constraints

To construct a parametric polynomial curve for interpolating a set of data points, the interpolation accuracy and shape of the constructed curve are influenced by two principal factors: parameterization of the data points (computing a node for each data point) and interpolation method. A new method of computing nodes for a set of data points was proposed. In this paper, the functional relationship between data points and corresponding nodes in cubic polynomials was established. Using this functional relationship, a functional cubic polynomial with one degree of freedom can pass through four adjacent data points. The degree of the freedom can be represented by two adjacent node intervals can be obtained by minimizing the cubic terms of the cubic polynomial. Since each node is computed in different node spaces, a method for constructing a quadratic curve is presented, which transforms all the quadratic curves into a unified form to compute nodes. Nodes computed using the new method exhibit quadratic polynomial precision, i.e., if the set of data point is taken from a quadratic polynomial F(t), the nodes by the new method are used to construct a interpolation curve, an interpolation method reproducing quadratic polynomial gives quadratic polynomial F(t). The primary advantage of the proposed method is that the constructed curve has a shape described by data points. Another advantage of the new method is that the nodes computed by it have affine invariance. The experimental results indicate that the curve constructed by the nodes using the new method has a better interpolation accuracy and shape compared to that constructed using other methods.

Reconstruction and Preservation of Feature Curves in 3D Point Cloud Processing

Given a 3D point cloud, we propose a method for suitably resampling the cloud while reconstructing and preserving the feature curves to which some points are identified to belong. The first phase of our strategy enriches the cloud by approximating the curvilinear profiles outlined by the feature points with piece-wise polynomial parametric space curves through the use of the Hough transform. The second phase describes how the removal of a point or its insertion can be performed without affecting the approximated profiles and preserving the enriched structure of the cloud. The combination of the two steps provides multiple possibilities for processing a point cloud by varying its size or improving its density homogeneity without affecting the retrieved feature curves. The various capabilities of our approach are investigated to produce simplification, refinement, and resampling techniques whose effectiveness is evaluated through experiments and comparisons.

Adaptive Bézier Degree Reduction and Splitting for Computationally Efficient Motion Planning

As a parametric polynomial curve family, Bézier curves are widely used in safe and smooth motion design of intelligent robotic systems from flying drones to autonomous vehicles to robotic manipulators. In such motion planning settings, the critical features of high-order Bézier curves such as curve length, distance-to-collision, maximum curvature/velocity/acceleration are either numerically computed at a high computational cost or inexactly approximated by discrete samples. To address these issues, in this article we present a novel computationally efficient approach for adaptive approximation of high-order Bézier curves by multiple low-order Bézier segments at any desired level of accuracy that is specified in terms of a Bézier metric. Accordingly, we introduce a new Bézier degree reduction method, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">parameterwise matching reduction</i> , which approximates Bézier curves more accurately compared to the standard least squares and Taylor reduction methods. We also propose a new Bézier metric, called the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">maximum control-point distance</i> , that can be computed analytically, has a strong equivalence relation with other existing Bézier metrics, and defines a geometric relative bound between Bézier curves. We provide extensive numerical evidence to demonstrate the effectiveness of our proposed Bézier approximation approach. As a rule of thumb, based on the degree-one matching reduction error, we conclude that an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ n^{\rm{th}} $</tex-math></inline-formula> -order Bézier curve can be accurately approximated by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3(n-1)$</tex-math></inline-formula> quadratic and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$6(n-1)$</tex-math></inline-formula> linear Bézier segments, which is fundamental for Bézier discretization.

On planar polynomial geometric interpolation

In the paper, the planar polynomial geometric interpolation of data points is revisited. Simple sufficient geometric conditions that imply the existence of the interpolant are derived in general. They require data points to be convex in a certain discrete sense. Since the geometric interpolation is based precisely on the known data only, one may consider it as the parametric counterpart to the polynomial function interpolation. The established result confirms the Höllig–Koch conjecture on the existence and the approximation order in the planar case for parametric polynomial curves of any degree stated quite a while ago.

Irregular Symmetrical Object Designed By Using Lambda Miu B-Spline Degree Four

In Computer Aided Geometry Design (CAGD), B-splines curves are piecewise polynomial parametric curves that play an important role. CAGD involves the interpolation and approximation curves and surfaces. CAGD has been widely used which brings good impact of computers to industries in manufacturing. There are many improved methods in the B-spline curve such as extended cubic B-spline, trigonometric B-spline, quasi trigonometric B-spline, and λμ-B-spline. Each of the methods has its behaviour and advantage. In this paper, λμ-B-spline was used to be implemented in generating irregular symmetrical objects. λμ-B-spline has a shape parameter that can change the global shape by manipulating the value of the shape parameter. The bottle has been chosen as an irregular symmetrical object. The 2-dimensional symmetrical curves of Bottle design were formed by using λμ-B-spline degree 4. The curves designed are dependent on the shape parameter which can be adjusted. Then, the curves generated were revolved using the Sweep Surface method to form 3-dimensional objects. Every object has its volume and this research focused on the numerical method which was Simpson’s 3/8 to compute the volume. The volumes obtained were compared to the actual volume to determine the best shape parameter used. The results show that the λμ-B-spline curve with a shape parameter of 1 is the best shape parameter in designing symmetrical irregular objects with the desired volume.

New shape control tools for rational Bézier curve design

Bézier curves are indispensable for geometric modelling and computer graphics. They have numerous favourable properties and provide the user with intuitive tools for editing the shape of a parametric polynomial curve. Even more control and flexibility can be achieved by associating a shape parameter with each control point and considering rational Bézier curves, which comes with the additional advantage of being able to represent all conic sections exactly. In this paper, we explore the editing possibilities that arise from expressing a rational Bézier curve in barycentric form. In particular, we show how to convert back and forth between the Bézier and the barycentric form, we discuss the effects of modifying the constituents (nodes, interpolation points, weights) of the barycentric form, and we study the connection between point insertion in the barycentric form with degree elevation of the Bézier form. Moreover, we analyse the favourable performance of the barycentric form for evaluating the curve.

On optimal polynomial geometric interpolation of circular arcs according to the Hausdorff distance

The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and has not been studied yet in the literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This includes practically interesting cases of parabolic G0, cubic G1, quartic G2 and quintic G3 interpolation. Several numerical examples are presented which confirm theoretical results.

$\boldsymbol{G^2[C^1]}$ Hermite interpolation using septic PH curves

In this paper, we examine the problem of $G^2[C^1]$ Hermite interpolation using septic Pythagorean hodograph (PH) curves. PH curves are a special class of polynomial parametric curves, which have a polynomial arc length function and rational offsets, and are thus widely used in computer-aided design. According to different factorizations of their first derivative in complex form, septic PH curves are classified into three classes. The curves in the first class are all regular, and their construction under any $G^2[C^1]$ condition has already been studied. Therefore, in this paper we focus on the remaining two classes. The number of septic PH curves in the second class is even and no more than six. The existence of septic PH curves in the third class is dependent on the initial Hermite data, and users may specify a real parameter to determine the resultant curve. In addition, we provide the approximation of arcs with septic PH curves as examples demonstrating the application of our results.

Identification of two classes of planar septic Pythagorean hodograph curves

Polynomial Pythagorean hodograph (PH) curves have a polynomial arc length function and rational offsets, which distinguish PH curves from general polynomial parametric curves. However, these algebraic properties can hardly be used directly for the identification of PH curves. In order to determine whether or not a given septic planar polynomial curve is a PH curve, Zheng et al. (2016) studied a class of septic curves’ geometric properties and proposed an efficient algorithm. In this paper, we further complete their work on Bézier control polygons of septic PH curves. We point out that there are three classes of septic PH curves according to different factorizations of their derivatives. Except the first class which has been studied, geometric characteristics of the other two classes are proposed. By introducing auxiliary points, the results are in terms of angles and lengths of legs of their Bézier control polygons. Moreover, we give feasible methods for the construction of auxiliary points, including various degenerate cases.

Assessment and Performance Evaluation of a Wind Turbine Power Output

Estimation errors have constantly been a technology bother for wind power management, often time with deviations of actual power curve (APC) from the turbine power curve (TPC). Power output dispersion for an operational 800 kW turbine was analyzed using three averaging tine steps of 1-min, 5-min, and 15-min. The error between the APC and TPC in kWh was about 25% on average, irrespective of the time of the day, although higher magnitudes of error were observed during low wind speeds and poor wind conditions. The 15-min averaged time series proved more suitable for grid management and energy load scheduling, but the error margin was still a major concern. An effective power curve (EPC) based on the polynomial parametric wind turbine power curve modeling technique was calibrated using turbine and site-specific performance data. The EPC reduced estimation error to about 3% in the aforementioned time series during very good wind conditions. By integrating statistical wind speed forecasting methods and site-specific EPCs, wind power forecasting and management can be significantly improved without compromising grid stability.

A general framework for the optimal approximation of circular arcs by parametric polynomial curves

We propose a general framework for geometric approximation of circular arcs by parametric polynomial curves. The approach is based on constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the B\'ezier form is derived and a detailed analysis provided for some low degree cases which might be important in practice. At least for these cases the solutions can be, in principal, written in a closed form, and provide the best known approximants according to the radial distance. A general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given.

On parametric polynomial circle approximation

In the paper, the uniform approximation of a circle arc (or a whole circle) by a parametric polynomial curve is considered. The approximant is obtained in a closed form. It depends on a parameter that should satisfy a particular equation, and it takes only a couple of tangent method steps to compute it. For low degree curves, the parameter is provided exactly. The distance between a circle arc and its approximant asymptotically decreases faster than exponentially as a function of polynomial degree. Additionally, it is shown that the approximant could be applied for a fast evaluation of trigonometric functions too.

A curve based hull form variation with geometric constraints of area and centroid

Modifying an existing geometric shape to obtain another is a common process in automobile, aircraft, and shipbuilding design. In particular, shipyard design departments rely on hull form variation, which involves designing a new ship from a well-made existing ship, as an efficient and effective process. However, this process is inefficient as it is complicated and unintuitive, thereby reliant on highly skilled expertise. An efficient approach to perform the variation with the given geometric constraints of area and centroid is proposed. To modify an existing hull form shape, a boundary curve of the shape is selected as a design variable. A parametric piecewise polynomial curve satisfying new geometric requirements is constructed and superimposed on the top of the chosen boundary curve to yield the desired shape. The main process of the variation preserves the original shape as much as possible, which means the new shape can be obtained simply and efficiently. The proposed algorithm can be readily extended to similar modification processes involving an existing geometric shape by adopting different geometric requirements.

Uniform approximation of a circle by a parametric polynomial curve

In the paper two new approaches for construction of parametric polynomial approximants of a unit circle are presented. The obtained approximants have better approximation properties than those given by other methods, i.e., smaller radial error, symmetry, and exponential error decay.

Using Genetic Algorithms for Optimizing the PPC in the Highway Horizontal Alignment Design

AbstractVarious studies have emphasized the interesting advantages related to the use of new transition curves for improving the geometric design of highway horizontal alignments. In a previous paper, one of the writers proposed a polynomial curve, called a polynomial parametric curve (PPC), proving its efficiency in solving several design problems characterized by a very complex geometry (egg-shaped transition, transition between reversing circular curves, semidirect and inner-loop connections, and so on). The PPC also showed considerable advantages from a dynamic perspective, as evidenced by the analysis of the main dynamic variables related to motion (as well as rate of change of radial acceleration, steering speed, roll speed, and so on). In this paper, an optimization procedure using genetic algorithms (GAs) for selecting the different parameters of the PPC has been proposed. In particular, a specific algorithm defines the parameter values in order to minimize an appropriate fitness function. Besides...

Planar [formula omitted] Hermite interpolation with uniform and non-uniform TC-biarcs

Pythagorean hodograph curves (shortly PH curves), introduced in Farouki and Sakkalis (1990), form an important subclass of polynomial parametric curves and currently represent standard objects in geometric modelling. In this paper, we focus on Tschirnhausen cubic as the only one Pythagorean hodograph cubic and we study planar C1 Hermite interpolation with two arcs of Tschirnhausen cubic joined with C1 continuity (the so-called TC-biarc). We extend results presented in Farouki and Peters (1996) in several ways. We study an asymptotical behaviour of the conversion of an arbitrary planar curve with well defined tangent vectors everywhere to a C1 PH cubic spline curve and we prove that the approximation order is 3. Further, we analyze the shape of TC-biarcs and provide a sufficient condition for input data guaranteeing TC-biarc without local and pairwise self-intersections. Finally, we generalize the basic uniform method to the non-uniform case, which introduces a free shape parameter, and we formulate an algorithm for a suitable choice of this shape parameter such that the corresponding non-uniform TC-biarc is without local and pairwise self-intersections (if such a parameter exists).

A Polynomial Parametric Curve (PPC‐CURVE) for the Design of Horizontal Geometry of Highways

Abstract: The use of transition curves in the road design is a solution to make the gradual evolution of curvature and, at the same time, to improve the comfort level of drivers and provide a good visual perception of the curve. Clothoid is the most widely used transition curve in road design so far, because it ensures the continuity of the curvature with the other geometric elements of the alignment.However, several researches allow the use of polynomial functions as an alternative to the clothoid. Such use is permitted in accordance with the verification of allowable vehicle–road dynamics.Polynomial solutions of transition curves can be a valuable alternative for the traditional solutions (first transition curve, circular arc, second transition curve).A fifth‐degree polynomial parametric curve (PPC‐curve) for the design of highway alignment is proposed in this article. An analysis of the theoretical aspects to solve more complex geometrical problems recurring in practical highway geometric design is carried out. With regard to this problem, a shape parameter giving flexibility to the polynomial solution in relation to project needs has also been introduced.To implement the procedure, an original computer program has been developed. Numerical applications have been performed for comparison with the traditional solutions.

Curve-induced distortion of polynomial space curves, flat-mapped extension modeling, and their impact on ANCF thin-plate finite elements

The Absolute Nodal Coordinate Formulation (ANCF) is a relatively new nonlinear finite element type that uses Hermite splines for shape functions. In this investigation, the ANCF is examined as a possible tool for use in modeling the media in flexible media transport systems, such as printers, copy machines, and roll-to-roll systems. However, it is demonstrated using an example of a thin plate-type ANCF finite element that these elements can suffer from significant membrane locking, which can be problematic for paper or paper-like media. One source of this locking is identified to be a property of all parametric curves that are composed of polynomials. The property is that for parametric polynomial curves, changes in the state of curvature of the curve cause changes in the distribution of points along the curve. This property is labeled Curve-Induced Distortion (CID) by the authors of this paper. CID can cause axial and membrane strain distortion in elements, causing them to be overly stiff. A new solution method is proposed to directly counteract CID in finite elements that use cubic Hermite curves for shape functions, specifically for modeling problems in which bending occurs primarily around one axis, such as paper in printing and media transport machinery. This method is labeled Flat-Mapped Extension Modeling (FMEM). FMEM is a mixed field method that uses a 1D Hermite polynomial kinematically linked to the 3D Hermite curve to represent the axial displacement field. FMEM significantly reduces the effect of CID in the ANCF element tested here. This investigation demonstrates using a single ANCF plate element type that the ANCF’s accuracy can be significantly improved by FMEM with only a small increase in computational cost. It is shown with this plate-element example that without correcting CID, the ANCF element tested is computationally much slower than contemporary methods like the co-rotational formulation for similar accuracy. But with FMEM, the ANCF is significantly faster than the co-rotational formulation for similar accuracy.

An Evolution-Based Approach for Approximate Parameterization of Implicitly Defined Curves by Polynomial Parametric Spline Curves

We propose a novel approach for the approximate parameterization of an implicitly defined curve in the plane by polynomial parametric spline curves. The method generates the parameterization of the curve (which may consist of several open and closed branches) without using any a priori information about its topology. If needed the topology of the approximate parameterization can be certified against the initial curve in a simple way.