An Action of A Regular Curve on $\mathbb{R}^{3}$ and Matlab Applications

Bulent Karakas, Senay Baydas


We define an action set of a regular curve not passing origin using a normed projection. If $\alpha(t)$ is a regular curve not passing origin, then the curve $\beta(t)=\frac{\alpha(t)}{\|\alpha(t)\|}$ is on unit sphere. $\beta(t) $ is called normed projection of $\alpha(t)$ \cite{3}. Every point $b(t)\in\beta(t)$ defines an orthogonal matrix using Cayley's Formula. So we define an action set $R_{\alpha }(t)\subset SO(3)$ of $\alpha(t)$. We study in this article some important relations $\alpha(t)$ and $R_{\alpha }(P)$, orbit of point $P\in R^{3}$. At the end we give some applications in Matlab.


Action set; Normed projection; Regular curve

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I. Arslan, H.H. Hacısalihoglu, On the spherical representatives of a curve, Int. J. Contemp. Math. Sciences 4 (34) (2009), 1665-1670.

˙I. Arslan Güven and S. Kaya Nurkan, The relation among bishop spherical indicatrix curves, International Mathematical Forum 6 (25) (2011), 1209-1215.

S. Baydas and S. Isleyen, A normed projection mapping on unit sphere, YYU, Journal of Science (2011) (in Press)

R. Encheva and G. Georgiev, Shapes of space curves, Journal for Geometry and Graphics 7 (2) (2003), 145-155.

K.S. Fu, R.C. Gonzalez and C.S.G. Lee, Robotics, McGraw-Hill Book Company, New York, 1987, p. 580.

J.M. McCarthy, An Introduction to Theoretical Kinematics, The MIT Press, Cambridge, 1990.

S. Yılmaz, E. Özyılmaz and M. Turgut, New spherical indicatrices and their characterizations, An. St. Univ. Ovidius Constanta 18 (2) (2010), 337-354.


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