Investigating a Quadratic Bezier Curve Due to N-C-W and N-Bishop Frames

Samanci, Hatice Kusak; İncesu, Muhsin

doi:10.47000/tjmcs.704794

Cited by 8 publications

(9 citation statements)

References 8 publications

(17 reference statements)

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“…Proof. From Samanci and Incesu, 39 if the Gaussian curvature of a ruled surface X(u,v) R 3 is zero, then the ruled surface is developable ruled surface or flat ruled surface. So from 33, if the Gaussian curvature K(t,v) of the ruled surface X(t, ν) is zero, then…”

Section: Resultsmentioning

confidence: 99%

“…According to this, α is not the striction curve of X ( t , ν ). So from this, the striction curve of X ( t , ν ) can be written as

σ ((), t) = ϖ ((), t) + ϱ ((), t) δ ((), t),

where

ϱ ((), t) = - \frac{(〈〉,, ϖ^{'} ((), t), δ^{'} ((), t))}{{(‖‖, δ^{'} ((), t))}^{2}} .

39 Here, the magnitude of ‖ δ ′( t )‖ can be calculated as follows:

(‖‖, δ^{'}) = (‖‖, \frac{ϒ \times ({normalϒ}^{'} \times ϒ)}{{‖normalϒ‖}^{3}}) = \frac{1}{{(‖‖, ϒ)}^{3}} \sqrt{〈ϒ \times ({normalϒ}^{'} \times ϒ), ϒ \times ({normalϒ}^{'} \times ϒ)〉},

\frac{1}{{(‖‖, ϒ)}^{3}} \sqrt{(〈〉,, ϒ, ϒ) (〈〉,, ({normalϒ}^{'} \times ϒ), ({normalϒ}^{'} \times ϒ))} = \frac{1}{{(‖‖, ϒ)}^{3}} (‖‖, ϒ) (‖‖, ϒ^{'} \times normalϒ) = \frac{(‖‖, ϒ^{'} \times normalϒ)}{{(‖‖, ϒ)}^{2}} .

…”

Section: Resultsmentioning

confidence: 99%

“…Numerous studies have been conducted on Bézier curves and surfaces, for example, Farin, 14 Farouki, 15 Hoschek, 16 Tiller and Hanson, 17 Potmann 18 Gray, 19 Chen et al, 20 Pottmann and Paukowitsch, 21 Krasauskas, 22 Reyes, 23 Mainar et al, 24 Jüttler and Wagner, 25 Q.J. Ge and Sirchia, 26 Mavroidis and Roth, 27 and Cholewa et al 28 Some other works 29–49 are given recently.…”

Section: Introductionmentioning

confidence: 99%

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The new characterization of ruled surfaces corresponding dual Bézier curves

İncesu

2021

Math Methods in App Sciences

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Let be a Bézier curve in D3, that is, the control points of are the vectors in D3 = {u = (u1, u2, u3) : ui ∈ D,i = 1,2,3} which is the set of dual vectors whose each coordinate components are dual numbers defined as a+εa* : a,a* ∈ R,ε ≠ 0; ε2 = 0. So the spherical projection of is a spherical curve denoted by on the unit sphere in D3 and every point of corresponds to a directed line in real space R3 by Study transformation. In this study, the ruled surface X(t, ν) corresponding to this projection curve of dual Bézier curve is stated in terms of the parametric equation of the real and dual part of a given dual Bézier curve . Also, in this study, some fundamental characteristics such as the striction curve, the Gaussian curvature, and the distribution parameter of the ruled surface X(t, ν) corresponding to this projection curve of the dual Bézier curve are investigated. These concepts at any point are stated in terms of the control points of the given dual Bézier curve .

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Section: Resultsmentioning

confidence: 99%

“…According to this, α is not the striction curve of X ( t , ν ). So from this, the striction curve of X ( t , ν ) can be written as

σ ((), t) = ϖ ((), t) + ϱ ((), t) δ ((), t),

where

ϱ ((), t) = - \frac{(〈〉,, ϖ^{'} ((), t), δ^{'} ((), t))}{{(‖‖, δ^{'} ((), t))}^{2}} .

39 Here, the magnitude of ‖ δ ′( t )‖ can be calculated as follows:

(‖‖, δ^{'}) = (‖‖, \frac{ϒ \times ({normalϒ}^{'} \times ϒ)}{{‖normalϒ‖}^{3}}) = \frac{1}{{(‖‖, ϒ)}^{3}} \sqrt{〈ϒ \times ({normalϒ}^{'} \times ϒ), ϒ \times ({normalϒ}^{'} \times ϒ)〉},

\frac{1}{{(‖‖, ϒ)}^{3}} \sqrt{(〈〉,, ϒ, ϒ) (〈〉,, ({normalϒ}^{'} \times ϒ), ({normalϒ}^{'} \times ϒ))} = \frac{1}{{(‖‖, ϒ)}^{3}} (‖‖, ϒ) (‖‖, ϒ^{'} \times normalϒ) = \frac{(‖‖, ϒ^{'} \times normalϒ)}{{(‖‖, ϒ)}^{2}} .

…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The new characterization of ruled surfaces corresponding dual Bézier curves

İncesu

2021

Math Methods in App Sciences

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“…Frenet apparatus of both the n th degree Bézier curves have been examined in E 3 , in [6]. The Bishop frame and the alternative frame have been associated with the Bézier curves in [7] and [8], respectively. The matrix forms of the cubic Bézier curve and its involute have been examined in [9] and [10], respectively.…”

Section: Introductionmentioning

confidence: 99%

How to Find a Bezier Curve in $\mathbf{E}^{3}$

Şenyurt

Kilicoglu²

2022

Communications in Advanced Mathematical Sciences

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How to find any n th order B ézier curve if we know its first, second, and third derivatives?" Hence we have examined the way to find the B ézier curve based on the control points with matrix form, while derivatives are given in E 3 . Further, we examined the control points of a cubic B ézier curve with given derivatives as an example. In this study first we have examined how to find any n th order Bezier curve with known its first, second and third derivatives, which are inherently, the (n − 1) th order, the (n − 2) th and the (n − 3) th Bezier curves in respective order. There is a lot of the number of B ézier curves with known the derivatives with control points. Hence to find a B ézier curve we have to choose any control point of any derivation İn this study we have chosen two special points which are the initial point P 0 and the endpoint P n .

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“…In [10], the Bertrand pairs have also been examined by B-Spline curves. Last but not least, the Bézier curves have been associated to the alternative frame in [11].…”

Section: Introductionmentioning

confidence: 99%