Control Polygon Research Articles

Quasi-quintic trigonometric Bézier curves with two shape parameters

In this work, we propose a family of six new quasi-quintic trigonometric blending functions with two shape parameters. Based on these blending functions, a class of quasi-quintic trigonometric Bezier curve is proposed, which has some properties analogous to the classical quintic Bezier curves. For the same control points, the resulting quasi-quintic trigonometric Bezier curves can be closer to the control polygon than the classical quintic Bezier curves. The shape of the quasi-quintic trigonometric Bezier curves can be flexibly adjusted by altering the values of the two shape parameters without changing their control points. Under the $${C^2}$$ smooth connection conditions, the resulting composite quasi-quintic trigonometric Bezier curves can automatically reach $${C^2} \cap F{C^3}$$ continuity.

NEW CUBIC TIMMER TRIANGULAR PATCHES WITH C¹ AND G¹ CONTINUITY

In this study, a new cubic Timmer triangular patch is constructed by extending the univariate cubic Timmer basis functions. The best scheme that lies towards the control polygon is cubic Timmer curve and surface compared to the other methods. From the best of our knowledge, nobody has extended the univariate cubic Timmer basis to the bivariate triangular patch. The construction of the proposed cubic Timmer triangular patch is based on the main idea of the cubic Ball and cubic Bezier triangular patches construction. Some properties of the new cubic Timmer triangular patch are investigated. Furthermore, the composite cubic Timmer triangular patches with parametric continuity (C1) and geometric continuity (G1) are discussed. Simple error analysis between the triangular patches and one test function is provided for each continuity type. Numerical and graphical results are presented by using Mathematica and MATLAB.

De Boor-like evaluation algorithm for Analysis-suitable T-splines

Analysis-suitable T-splines form a practically useful subset of T-splines. They maintain the design flexibility of T-splines with an efficient and highly localized refinement capability, while preserving the important analysis-suitable mathematical properties of the NURBS basis. The present paper proposes a new evaluation algorithm for analysis-suitable T-splines. The algorithm is based on the control polygon directly and it reduces both time and storage cost comparing with Bézier extraction.

Gauss–Lobatto polygon of Pythagorean hodograph curves

The Gauss–Legendre polygon of Pythagorean hodograph (PH) curves can be used as the rectifying control polygon, which has (i) the end point interpolation property, (ii) the rectifying property, and (iii) the same degree of freedom as the PH curve. These properties make the Gauss–Legendre polygon a nice tool to control the shape of the PH curve. A drawback of the Gauss–Legendre polygon is that it does not determine the end tangent vectors. In this paper, we introduce the Gauss–Lobatto polygon as an alternative to the Gauss–Legendre polygon. Since the Gauss–Lobatto quadrature has the end points as the predetermined nodes, the Gauss–Lobatto polygon naturally determines the end tangent vectors of the PH curve. We analyze the rectifying property of the Gauss–Lobatto polygon for both planar and spatial PH curves. Concerning the degree of freedom, we show that the Gauss–Lobatto polygons of planar PH curves are not the rectifying control polygons. For spatial PH curves, we identify the relation between the degree of a PH curve and the number of edges in the Gauss–Lobatto polygon that makes the Gauss–Lobatto polygon a rectifying control polygon. We also provide the method to compute the spatial septic PH curve with the given Gauss–Lobatto polygon, and the algorithm for the deformation of the spatial septic PH curves.

Hierarchical Functional Maps between Subdivision Surfaces

AbstractWe propose a novel approach for computing correspondences between subdivision surfaces with different control polygons. Our main observation is that the multi‐resolution spectral basis functions that are open used for computing a functional correspondence can be compactly represented on subdivision surfaces, and therefore can be efficiently computed. Furthermore, the reconstruction of a pointwise map from a functional correspondence also greatly benefits from the subdivision structure. Leveraging these observations, we suggest a hierarchical pipeline for functional map inference, allowing us to compute correspondences between surfaces at fine subdivision levels, with hundreds of thousands of polygons, an order of magnitude faster than existing correspondence methods. We demonstrate the applicability of our results by transferring high‐resolution sculpting displacement maps and textures between subdivision models.

Deformation of spatial septic Pythagorean hodograph curves using Gauss–Legendre polygon

The notion of the rectifying control polygon of planar Pythagorean hodograph (PH) curve is extended to the spatial PH curves. We construct the Gauss–Legendre polygon by applying the Gauss–Legendre quadrature to the Pythagorean hodograph. Since the parametric speed of a PH curve is a polynomial, the Gauss–Legendre polygon with enough edges has the rectifying property, that is, the length of the polygon is the same as the length of the spatial PH curve. We also present the method to compute septic PH curves with the given Gauss–Legendre polygon. This method can be used to develop the deformation algorithm for the spatial septic PH curves.

A Computational framework to design optimally loaded supercavitating hydrofoils by differential evolution algorithm and a new viscous lifting line method

The engineering design of a three dimensional submerged hydrofoil operating at very high speeds is obtained leveraging a Differential Evolution (DE) approach. The final goal is to identify the optimal load distribution over the span of a super-cavitating hydrofoil by using a design by optimization approach driven by hydrodynamic analysis of complex, turbulent, multi-phase flows. We achieve this goal by modeling the load distribution over the hydrofoil by means of a B-spline curve, which provides a rigorous parametric description of the hydrofoil operating conditions through the points of the load distribution control polygon. The parametric model includes design variables representing the most relevant hydrofoil shape parameters. We predict hydrodynamic performance by means of a Viscous Lifting Line method specifically conceived for the application targeted in the present study. This computational model accounts for the strong non-linear hydrodynamic characteristics of super-cavitating hydrofoils. We demonstrate the validity of the proposed design by optimization framework for high speed super-cavitating hydrofoils showcasing two design applications, namely a fully submerged hydrofoil operating close to a rigid boundary and a surface-piercing hydrofoil with variable dihedral angle. A statistical analysis of DE algorithm is performed to assess its performance on such an engineering design problem.

Open-Pit Mine Optimization with Maximum Satisfiability

A common casualty of modern open-pit mine optimization is the assurance that the resulting design is actually achievable. Optimized mine plans that consider value and a bare minimum of precedence constraints do not, in general, translate into practical, operational mine designs that can be used in the field. Ultimate pits may come to a sharp point at the bottom. Schedules may require taking small parcels of material from many disparate areas of the pit in a single period, and grade control polygons may be ragged, narrow, and not minable with realistic equipment. In this paper, all of these problems are addressed by encoding these three fundamental open-pit mine optimization problems as maximum satisfiability problems. Maximum satisfiability provides a useful framework for problems that are non-linear and may guarantee the optimality that metaheuristics cannot.

Exact computation for existence of a knot counterexample

Previously, numerical evidence was presented of a self-intersecting Bezier curve having the unknot for its control polygon. This numerical demonstration resolved open questions in scientic visualization, but did not provide a formal proof of self-intersection. An example with a formal existence proof is given, even while the exact self-intersection point remains undetermined.

A New Method to Design Cubic Pythagorean-Hodograph Spline Curves with Control Polygon

A new method to design a cubic Pythagorean-hodograph (PH) spline curve from any given control polygon is proposed. The key idea is to suitably choose a set of auxiliary points associated with the edges of the given control polygon to guarantee the constructed PH spline has $$G^1$$ continuity or curvature continuity. The method facilitates intuitive and efficient construction of open and closed cubic PH spline curves that typically agrees closely with the same friendly interface and properties as B-splines, for example, the convex hull and variation-diminishing properties.

A CAD-BASED CONCEPTUAL METHOD FOR SKULL PROSTHESIS MODELLING

The geometric modeling of a personalized part of the tissue built according to individual morphology is an essential requirement in anatomic prosthesis. A 3D model to fill the missing areas in the skull bone requires a set of information sometimes unavailable. The unknown information can be estimated through a set of rules referenced to a similar yet known set of parameters of the similar CT image. The proposed method is based on the Cubic Bezier Curves descriptors generated by the de Casteljou algorithm in order to generate a control polygon. This control polygon can be compared to a similar CT slice in an image database. The level of similarity is evaluated by a meta-heuristic fitness function. The research shows that it is possible to reduce the amount of points in the analysis from the original edge to an equivalent Bezier curve defined by a minimum set of descriptors. A study case shows the feasibility of method through the interoperability between the prosthesis descriptors and the CAD environment.

Rational Quadratic Bézier Spirals

A quadratic Bezier representation withholds a curve segment with free from loops, cusps and inflection points. Furthermore, this rational form provides extra freedom to generate visually pleasing curves due to the existence of weights. In this paper, we propose sufficient conditions for rational quadratic Bezier curves to possess monotonic increasing/decreasing curvatures by means of monotone curvature tests which are based on the derivative of curvature functions. We have derived a simple interval of the middle weight that assures the construction of a family of rational quadratic Bezier curves to be planar spirals, which is characterized by the turning angle, end curvatures and the chords of control polygon. The proposed formulation can be used by CAD systems for aesthetic product design, highway/railway design and robot trajectory design avoiding unwanted curvature oscillations.

Repeated local operations and associated interpolation properties of dual 2n-point subdivision schemes

In this paper we first derive a recursive relation of the generating functions of a family of dual 2 n -point subdivision schemes. Based on the recursive relation we design repeated local operations for implementing the 2 n -point subdivision schemes. Associated interpolation properties of the limit curve sequence of the dual 2 n -point subdivision schemes when n tends to infinity are then investigated. Based on the repeated local operations, we further prove that the limit curves of the family of the dual 2 n -point subdivision scheme sequence approach a circle that interpolates all initial control points as n approaches infinity, provided that the initial control points form a regular control polygon. Other interpolation properties show that the limit curve interpolates all closed initial control points with odd points or with even points but satisfying an extra condition, and interpolates all newly inserted vertices of an original closed polygon, when n approaches infinity. Some numerical examples are provided to illustrate the validity of our theoretic analyses.

A combined approximating and interpolating ternary 4-point subdivision scheme

In this paper, a new combined approximating and interpolating ternary 4-point subdivision scheme with multiple parameters is proposed. A set of nice properties, such as support, continuity and polynomial generation, are briefly discussed. The new combined scheme not only contains a lot of classical ternary schemes as special cases, but also generates brand-new ternary schemes. Compared to other approximating subdivision schemes, limit curves generated by the new scheme are more consistent with the corresponding control polygons and keep detail features better. Examples are given to show the effectiveness of the scheme. Furthermore, fractal property is analyzed and fractal curves are also given.

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.

A note on Pythagorean hodograph quartic spiral

By using the geometric constraints on the control polygon of a Pythagorean hodograph (PH) quartic curve, we propose a sufficient condition for this curve to have monotone curvature and provide the detailed proof. Based on the results, we discuss the construction of spiral PH quartic curves between two given points and formulate the transition curve of a G2 contact between two circles with one circle inside another circle. In particular, we deduce an attainable range of the distance between the centers of the two circles and summarize the algorithm for implementation. Compared with the construction of a PH quintic curve, the complexity of the solution of the equation for obtaining the transition curves is reduced.

Bézier curves that are close to elastica

We study the problem of identifying those cubic Bézier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are difficult to work with in a digital environment. We seek a sub-class of special Bézier curves as a proxy. We identify an easily computable quantity, which we call the λ-residual eλ, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a Bézier curve has λ-residual below 0.4, which effectively implies that the curve is within 1% of its arc-length to an elastic curve in the L2 norm. Finally we give two projection algorithms that take an input Bézier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles fixed, until it is close to an elastic curve.

Depth Contour Smoothing Based on the Fitting of Multi-Segment Bezier Curves

ABSTRACTTo overcome smoothness inconsistencies in depth contours interpolated from sounding data, this paper proposes a smoothing method based on the fitting of multi-segment Bezier curves considering navigational safety. The basic principle of this method is the division of a generated depth contour into a series of bends categorized as either convex bends or concave bends. Different smoothing strategies are designed for different bends while considering the safety constraint. Convexity-preserving Bezier curves are used to fit convex bends. For concave bends, the line segment of the bend is utilized as a boundary to construct a control polygon within the interior of the bend, after which a Bezier curve is constructed based on the polygon. Experiments with real data are implemented, and the results are compared with those of other methods. The results show that the smoothing procedure for depth contours with angularities is effective and respects the safety constraint; moreover, the shape characteristics are preserved, and high positional accuracy is achieved. In addition, the proposed smoothing method is free of topological inconsistencies.

A combined approach based on Subdivision Surface and Free Form Deformation for smart ship hull form design and variation

ABSTRACTTechniques for shape representation and further modification of a hull surface definitely play a key role in both the design of new buildings and in the optimisation of the existing ones. The simplification of the methods is the goal to reach in order to create useful tools for real applications. A new approach for hull shape modification is proposed. It is based on a combination of the Subdivision Surface technique for hull surface modelling and Free Form Deformation for shape variation. The formal relation between the two methods is established by the Free Form Deformation control volume and the Subdivision Surface control polygon, introducing significant simplification to the definition of the transformation. The new approach is described in detail highlighting its benefits. Its effectiveness is finally proved by an example of application on a real hull shape, where a combination of a local and global modification has been analysed.

P-Bézier and P-Bspline curves – new representations with proximity control

Proximity curves represent a family of curves that are associated with a given parametric curve, defined by control points and basis functions. Proximity curves continuously sweep from this curve to its control polygon depending on a proximity value, that determines the location of an intermediate curve and its fullness (or tension). The proximity value also determines sensitivity, i.e. how strongly the shape is affected by displacing control points. An important feature of proximity curves relates to the insertion of new control points: while in other schemes a new degree of freedom leads to repositioning the existing control points, in our case the new control points are always placed on some chord of the control polygon.Our first proximity curve scheme – called P-curves – has been published recently (Kovács and Várady, 2017), having C∞ continuity and G1 endpoint interpolation. The basis functions were constructed by means of generalized barycentric coordinates, and a somewhat limited algorithm for control point insertion was proposed.Our current paper takes a different approach: the basis functions are calculated by a much simpler algebra that is capable to reproduce standard formulations like Bézier and B-splines curves, and can maintain Cn end constraints. We introduce the Proximity-Bézier, shortly P-Bézier, and P-Bspline curves, including the construction of basis functions and the most important mathematical properties. A general control point insertion algorithm is also described. Several examples are shown to compare the classical and the new representations. A tensor product generalization of the scheme is also demonstrated.