The new characterization of ruled surfaces corresponding dual Bézier curves

Abstract

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 .

The Analysis of Some Spherical Mechanism Movements and Joint Designs by The New SLERP Interpolations

“…Jafari et al 2014) analyzed the Bezier curves by utilizing the spherical linear interpolation [12]. In addition, Kuşak Samancı et al(2015), Karadağ et al(2019), Incesu (2021) obtained some properties of the Bezier curves [13][14][15]. On the other hand, the mechanisms and machines theory was firstly launched into analyzing in Russia in 1800 and then spread around the world.…”

Magnetic flux ruled surfaces

“…6 On the other hand, ruled surfaces are constructed by a one-parameter family of lines and have been studied extensively in differential geometry. [7][8][9][10][11][12] Since they are simply generated, these surfaces find application in a variety of fields such as computer aided design, architectural design, electric discharge machining, and simulation of rigid bodies.…”