Control point based exact description of curves and surfaces, in extended Chebyshev spaces

Róth, Ágoston

doi:10.1016/j.cagd.2015.09.005

Cited by 5 publications

(8 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let a = 6, b = 4 √ 2, c = 2 and µ = 3. The Cartesian coordinates of a Dupin-Cyclide are given by [24] (17)…”

Section: Examples and Comparisonsmentioning

confidence: 99%

See 1 more Smart Citation

Dynamic Evaluation of Exponential Polynomial Curves and Surfaces via Basis Transformation

Yang¹,

Hong²

2019

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

It is shown in "SIAM J. Sci. Comput. 39 (2017):B424-B441" that free-form curves used in computer aided geometric design can usually be represented as the solutions of linear differential systems and points and derivatives on the curves can be evaluated dynamically by solving the differential systems numerically. In this paper we present an even more robust and efficient algorithm for dynamic evaluation of exponential polynomial curves and surfaces. Based on properties that spaces spanned by general exponential polynomials are translation invariant and polynomial spaces are invariant with respect to a linear transformation of the parameter, the transformation matrices between bases with or without translated or linearly transformed parameters are explicitly computed. Points on curves or surfaces with equal or changing parameter steps can then be evaluated dynamically from a start point using a pre-computed matrix. Like former dynamic evaluation algorithms, the newly proposed approach needs only arithmetic operations for evaluating exponential polynomial curves and surfaces. Unlike conventional numerical methods that solve a linear differential system, the new method can give robust and accurate evaluation results for any chosen parameter steps. Basis transformation technique also enables dynamic evaluation of polynomial curves with changing parameter steps using a constant matrix, which reduces time costs significantly than computing each point individually by classical algorithms.

show abstract

“…Let a = 6, b = 4 √ 2, c = 2 and µ = 3. The Cartesian coordinates of a Dupin-Cyclide are given by [24] (17)…”

Section: Examples and Comparisonsmentioning

confidence: 99%

“…Besides polynomial curves and surfaces, typical curves and surfaces such as ellipses, cycloids, involutes, helices, etc. can be represented by exponential polynomials exactly [17,21,24,32]. By choosing a proper parameter interval, normalized B-bases that are useful for optimal shape design can be obtained from the exponential polynomials [5,18,25,27].…”

mentioning

confidence: 99%

Dynamic Evaluation of Exponential Polynomial Curves and Surfaces via Basis Transformation

Yang¹,

Hong²

2019

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…Using linear combinations of the initial algebraic or transcendental functions, a system of normalized totally positive (NTP) bases can be constructed for a known function space. A change of basis does not change the vector space spanned by the original bases and the free-form curves defined by NTP bases follow the shapes of their control polygons very well [26,28]. Besides the description of free-form curves using control polygons, a fair planar curve can be represented using ordinary bases when it is designed based on an intrinsic expression of planar curves [35].…”

Section: Free-form Curvesmentioning

confidence: 99%

Dynamic Evaluation of Free-Form Curves and Surfaces

Yang¹,

Hong²

2017

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Free-form curves and surfaces represented by control points together with well-defined basis functions are widely used in computer aided geometric design. Efficient evaluation of points and derivatives of free-form curves and surfaces plays an important role in interactive rendering or CNC machining. In this paper we show that free-form curves with properly defined basis functions are the solutions of linear differential systems. By employing typical numerical methods for solving the differential systems, points and derivatives of free-form curves and surfaces can be computed in a dynamical way. Compared with traditional methods for evaluating free-form curves and surfaces there are two advantages of the proposed technique. First, the proposed method is universal and efficient for evaluating a large class of free-form curves and surfaces. Second, the evaluation needs only arithmetic operations even when the free-form curves and surfaces are defined using some transcendental functions.

show abstract

“…In [Róth, 2015a] we have already constructed the matrix of the general basis transformation that maps the normalized B-basis B α,β n to the ordinary basis F α,β n of the EC space S α,β n , where β − α ∈ 0, n . Namely, we have the next theorem.…”

Section: General Basis Transformationmentioning

confidence: 99%

An OpenGL and C++ based function library for curve and surface modeling in a large class of extended Chebyshev spaces

Róth

2017

Preprint

Self Cite

View full text Add to dashboard Cite

We propose a platform-independent multi-threaded function library that provides data structures to generate, differentiate and render both the ordinary basis and the normalized B-basis of a user-specified extended Chebyshev (EC) space that comprises the constants and can be identified with the solution space of a constantcoefficient homogeneous linear differential equation defined on a sufficiently small interval. Using the obtained normalized B-bases, our library can also generate, (partially) differentiate, modify and visualize a large family of socalled B-curves and tensor product B-surfaces. Moreover, the library also implements methods that can be used to perform dimension elevation, to subdivide B-curves and B-surfaces by means of de Casteljau-like B-algorithms, and to generate basis transformations for the B-representation of arbitrary integral curves and surfaces that are described in traditional parametric form by means of the ordinary bases of the underlying EC spaces. Independently of the algebraic, exponential, trigonometric or mixed type of the applied EC space, the proposed library is numerically stable and efficient up to a reasonable dimension number and may be useful for academics and engineers in the fields of Approximation Theory, Computer Aided Geometric Design, Computer Graphics, Isogeometric and Numerical Analysis. Keywords extended Chebyshev spaces • constant-coefficient homogeneous linear differential equations • normalized B-basis • B-curve/surface modeling • order (dimension) elevation • subdivision (B-algorithm) • basis transformation • control point based exact description • OpenGL • OpenMP Mathematics Subject Classification (2000) 65D17 • 68U07

show abstract

Control point based exact description of curves and surfaces, in extended Chebyshev spaces

Cited by 5 publications

References 31 publications

Dynamic Evaluation of Exponential Polynomial Curves and Surfaces via Basis Transformation

Dynamic Evaluation of Exponential Polynomial Curves and Surfaces via Basis Transformation

Dynamic Evaluation of Free-Form Curves and Surfaces

An OpenGL and C++ based function library for curve and surface modeling in a large class of extended Chebyshev spaces

Contact Info

Product

Resources

About