Helical Curves on Surfaces for Computer-Aided Geometric Design and Manufacturing

Puig-Pey, Jaime; Gálvez, Antonio; Iglesias, Andrés

doi:10.1007/978-3-540-24709-8_81

Cited by 24 publications

(21 citation statements)

References 8 publications

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“…For example, we used the ODEs solver function ode45 of Matlab [30], which is based on an adaptive step-by-step technique combining 4th-and 5th-order Runge-Kutta methods for controlling the error and the step. This method also provides user with a good control of both absolute error (a threshold below which the estimated error value of ith solution component is unimportant) or relative error (a measure of the error relative to the size of each solution component) as described in [32]. Greater attention should be paid to the case that the surface is composed of several patches.…”

Section: Numerical Integrationmentioning

confidence: 98%

Intersection of a ruled surface with a free-form surface

Wang

Zhang

2007

Numer Algor

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This paper presents a simple method for computing the intersection curve of a ruled surface and a free-form surface. The basic idea is to reduce the problem of surface intersection to the one of projecting an appropriate curve such as a directrix of the ruled surface, along its indicatrix curve (direction vector field of its generating lines), onto the free-form surface; the projection curve is just the intersection curve. With techniques in classical differential geometry, we derive the differential equations of the intersection curve in the cases of parametrically and implicitly defined free-form surfaces. The intersection curve naturally inherits the parameter of the chosen directrix. Moreover, it is independent of the base surface geometry and its parameterization, and is obtained by numerically solving the initial-value problem for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for parametric case or in 3D space for implicit case. Some experimental examples are also given to demonstrate that the presented method is effective and potentially useful in computer aided design and computer graphics.

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Section: Numerical Integrationmentioning

confidence: 98%

Intersection of a ruled surface with a free-form surface

Wang

Zhang

2007

Numer Algor

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“…Applying our method, the relatively normal-slant helix on M which makes the constant angle θ = π 3 with the chosen direction d = (0, 0, 1) and starting from the inital point P = (1, 0, 1) is given in Figure 1. Another application is given in Figure 2 in which we obtain two relatively normal-slant helices lying on the surface (x 2 + y 2 )z 2 + x 2 +y 2 4 − 1 4 = 0 with the initial point −1 √ 13 , 0, − √ 3 (in each figure the general helices are obtained by the method given in [20]). Proof.…”

Section: Examplesmentioning

confidence: 99%

“…Puig-Pey et al introduce the method for generating the general helix for both implicit and parametric surfaces [20].…”

Section: Introductionmentioning

confidence: 99%

Relatively normal-slant helices lying on a surface and their characterizations

Macit¹

2016

HJMS

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In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame {T, V, U} along the curve, where T is the unit tangent vector field of the curve, U is the surface normal restricted to the curve and V = U × T. We define a new curve on a surface by using the Darboux frame. This new curve whose vector field V makes a constant angle with a fixed direction is called as relatively normal-slant helix. We give some characterizations for such curves and obtain their axis. Besides we give some relations between some special curves (general helices, integral curves, etc.) and relatively normal-slant helices. Moreover, when a regular surface is given by its implicit or parametric equation, we introduce the method for generating the relatively normal-slant helix with the chosen direction and constant angle on the given surface.

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“…The well-known characterizations obtained for a single curve have allowed to define some special curves such as helix, slant helix, plane curve, spherical curve, etc. [1,11,13,20,23] and these special curves, especially helices, are used in many applications [2,9,10,19]. Similarly, by considering two curves, some special curve pairs such as involute-evolute curves, Bertrand curves, Mannheim curves have been defined and studied so far [4,12,14,15,18,21,22].…”

Section: Introductionmentioning

confidence: 99%

Associated Curves of Frenet curves in Three Dimensional Compact Lie Group

Kızıltuğ¹,

Önder²

2015

Miskolc Math Notes

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Abstract. General definition of associated curves of a Frenet curve is given in a three dimensional compact Lie group G. The principal normal direction curve and principal normal donor curve are introduced and some characterizations for these curves are obtained in G. Later, the relationships between a principal normal direction curve and some special curves such as helix, slant helix or curve with a special torsion are obtained.

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Helical Curves on Surfaces for Computer-Aided Geometric Design and Manufacturing

Cited by 24 publications

References 8 publications

Intersection of a ruled surface with a free-form surface

Intersection of a ruled surface with a free-form surface

Relatively normal-slant helices lying on a surface and their characterizations

Associated Curves of Frenet curves in Three Dimensional Compact Lie Group

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