Curves and Surfaces in Three-Dimensional Euclidean Space

  • Mikhail Itskov
Part of the Mathematical Engineering book series (MATHENGIN)


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
$$\begin{aligned} \theta ^i=\theta ^i\left( t\right) , \quad i=1,2,3. \end{aligned}$$

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Continuum MechanicsRWTH Aachen UniversityAachenGermany

Personalised recommendations