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

Abstract

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.