Curves and Surfaces in Three-Dimensional Euclidean Space

Abstract

A curve in three-dimensional space is defined by a vector function $$\varvec{r}=\varvec{r}\left( t\right) , \quad \varvec{r}\in \mathbb {E}^3,$$ where the real variable t belongs to some interval: \(t_1\le t \le t_2\). Henceforth, we assume that the function \(\varvec{r}\left( t\right) \) is sufficiently differentiable and $$\begin{aligned} \frac{\mathrm{d}\varvec{r}}{\mathrm{d}t}\ne \varvec{ 0 } \end{aligned}$$ over the whole definition domain. Specifying an arbitrary coordinate system ( 2.16) as $$\begin{aligned} \theta ^i=\theta ^i\left( \varvec{r}\right) , \quad i=1,2,3, \end{aligned}$$ the curve (3.1) can alternatively be defined by $$\begin{aligned} \theta ^i=\theta ^i\left( t\right) , \quad i=1,2,3. \end{aligned}$$