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Kande Dickson Kinyua - One of the best experts on this subject based on the ideXlab platform.

  • Differential Geometry: An Introduction to the Theory of Curves
    International Journal of Theoretical and Applied Mathematics, 2017
    Co-Authors: Kuria Joseph Gikonyo, Kande Dickson Kinyua
    Abstract:

    Differential geometry is a discipline of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry. The theory of plane, curves and surfaces in the Euclidean space formed the basis for development of differential geometry during the 18th and the 19th century. The core idea of both differential geometry and modern geometrical dynamics lies under the concept of manifold. A manifold is an abstract mathematical space, which locally resembles the spaces described by Euclidean geometry, but which globally may have a more complicated structure. The purpose of this paper is to give an elaborate introduction to the theory of curves, and those are, in general, curved. Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by applying the concept of differential and integral calculus. The curves are represented in parametrized form and then their geometric properties and various quantities associated with them, such as curvature and arc length expressed via derivatives and integrals using the idea of vector calculus.

Isabel Vogt - One of the best experts on this subject based on the ideXlab platform.

Rida T Farouki - One of the best experts on this subject based on the ideXlab platform.

  • rational pythagorean hodograph space curves
    Computer Aided Geometric Design, 2011
    Co-Authors: Rida T Farouki
    Abstract:

    A method for constructing rational Pythagorean-hodograph (PH) curves in R^3 is proposed, based on prescribing a field of rational unit tangent vectors. This tangent field, together with its first derivative, defines the orientation of the curve osculating planes. Augmenting this orientation information with a rational support function, that specifies the distance of each osculating plane from the origin, then completely defines a one-parameter family of osculating planes, whose envelope is a developable ruled surface. The rational PH space curve is identified as the edge of regression (or cuspidal edge) of this developable surface. Such curves have rational parametric speed, and also rational adapted frames that satisfy the same conditions as polynomial PH curves in order to be rotation-minimizing with respect to the tangent. The key properties of such rational PH space curves are derived and illustrated by examples, and simple algorithms for their practical construction by geometric Hermite interpolation are also proposed.

  • pythagorean hodograph space curves
    Advances in Computational Mathematics, 1994
    Co-Authors: Rida T Farouki, Takis Sakkalis
    Abstract:

    We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′2(t)+y′2(t)+z′2(t)≡σ2(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ3 is inherently more complicated than that of the corresponding set in ℝ2. We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bezier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ3), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.

  • Pythagorean-hodograph space curves
    Advances in Computational Mathematics, 1994
    Co-Authors: Rida T Farouki, Takis Sakkalis
    Abstract:

    We investigate the properties of polynomial space curves r(t) ={ x(t), y(t), z(t) } whose hodographs (derivatives) satisfy the Pythagorean condition x ′^2( t )+ y ′^2( t )+ z ′^2( t )≡σ^2( t ) for some real polynomial σ( t ). The algebraic structure of the complete set of regular Pythagorean-hodograph curves in ℝ^3 is inherently more complicated than that of the corresponding set in ℝ^2. We derive a characterization for all cubic Pythagoreanhodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ^3), but they have no “simple” all-encompassing characterization. We focus on a subset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) the canal surfaces based on such curves as spines have precise rational parameterizations.

Takis Sakkalis - One of the best experts on this subject based on the ideXlab platform.

  • pythagorean hodograph space curves
    Advances in Computational Mathematics, 1994
    Co-Authors: Rida T Farouki, Takis Sakkalis
    Abstract:

    We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′2(t)+y′2(t)+z′2(t)≡σ2(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ3 is inherently more complicated than that of the corresponding set in ℝ2. We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bezier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ3), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.

  • Pythagorean-hodograph space curves
    Advances in Computational Mathematics, 1994
    Co-Authors: Rida T Farouki, Takis Sakkalis
    Abstract:

    We investigate the properties of polynomial space curves r(t) ={ x(t), y(t), z(t) } whose hodographs (derivatives) satisfy the Pythagorean condition x ′^2( t )+ y ′^2( t )+ z ′^2( t )≡σ^2( t ) for some real polynomial σ( t ). The algebraic structure of the complete set of regular Pythagorean-hodograph curves in ℝ^3 is inherently more complicated than that of the corresponding set in ℝ^2. We derive a characterization for all cubic Pythagoreanhodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ^3), but they have no “simple” all-encompassing characterization. We focus on a subset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) the canal surfaces based on such curves as spines have precise rational parameterizations.

Kuria Joseph Gikonyo - One of the best experts on this subject based on the ideXlab platform.

  • Differential Geometry: An Introduction to the Theory of Curves
    International Journal of Theoretical and Applied Mathematics, 2017
    Co-Authors: Kuria Joseph Gikonyo, Kande Dickson Kinyua
    Abstract:

    Differential geometry is a discipline of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry. The theory of plane, curves and surfaces in the Euclidean space formed the basis for development of differential geometry during the 18th and the 19th century. The core idea of both differential geometry and modern geometrical dynamics lies under the concept of manifold. A manifold is an abstract mathematical space, which locally resembles the spaces described by Euclidean geometry, but which globally may have a more complicated structure. The purpose of this paper is to give an elaborate introduction to the theory of curves, and those are, in general, curved. Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by applying the concept of differential and integral calculus. The curves are represented in parametrized form and then their geometric properties and various quantities associated with them, such as curvature and arc length expressed via derivatives and integrals using the idea of vector calculus.