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

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

Exercises

3.1.

Show that a curve $\varvec{r}\left( s\right)$ is a straight line if $\varkappa \left( s\right) \equiv 0$ for any s.

3.2.

Show that the curves $\varvec{r}\left( s\right)$ and $\varvec{r}'\left( s\right) =\varvec{r}\left( -s\right)$ have the same curvature and torsion.

3.3.

Show that a curve $\varvec{r}\left( s\right)$ characterized by zero torsion $\tau \left( s\right) \equiv 0$ for any s lies in a plane.

3.4.

Evaluate the Christoffel symbols of the second kind, the coefficients of the first and second fundamental forms, the Gaussian and mean curvatures for the cylinder (3.56).

3.5.

Evaluate the Christoffel symbols of the second kind, the coefficients of the first and second fundamental forms, the Gaussian and mean curvatures for the sphere (3.58).

3.6.

For the so-called hyperbolic paraboloidal surface defined by

$$\begin{aligned} \varvec{r}\left( t^1,t^2\right) = t^1\varvec{e}_1+t^2\varvec{e}_2+\frac{t^1 t^2}{c}\varvec{e}_3, \quad c>0, \end{aligned}$$

(3.162)

evaluate the tangent vectors to the coordinate lines, the Christoffel symbols of the second kind, the coefficients of the first and second fundamental forms, the Gaussian and mean curvatures.

3.7.

For a cone of revolution defined by

$$\begin{aligned} \varvec{r}\left( t^1,t^2\right) = ct^2\cos t^1\varvec{e}_1+ct^2\sin t^1\varvec{e}_2+ t^2\varvec{e}_3, \quad c\ne 0, \end{aligned}$$

(3.163)

evaluate the vectors tangent to the coordinate lines, the Christoffel symbols of the second kind, the coefficients of the first and second fundamental forms, the Gaussian and mean curvatures.

3.8.

An elliptic torus is obtained by revolution of an ellipse about a coplanar axis. The rotation axis is also parallel to one of the ellipse axes the lengths of which are denoted by 2a and 2b. The elliptic torus can thus be defined by

$$\begin{aligned} \varvec{r}\left( t^1,t^2\right) = \left( R_0+a\cos t^2\right) \cos t^1\varvec{e}_1+\left( R_0+a\cos t^2\right) \sin t^1\varvec{e}_2+ b \sin t^2\varvec{e}_3. \end{aligned}$$

(3.164)

Evaluate the vectors tangent to the coordinate lines, the Christoffel symbols of the second kind, the coefficients of the first and second fundamental forms, the Gaussian and mean curvatures.

3.9.

Using the results of Exercise 3.8 calculate stresses in a thin wall vessel of the elliptic torus form (3.164) subject to the internal pressure p.

3.10.

Verify relation (3.114).

3.11.

Prove the product rule of differentiation for the covariant derivative of the vector $\varvec{f}^\alpha$ (3.119)$_1$ by using (3.76) and (3.77).

3.12.

Derive relations (3.128) and (3.129) from (3.125) and (3.127) utilizing (3.78), (3.80), (3.119), (3.120) and (2.105–2.107).

3.13.

Write out equilibrium equations (3.143–3.144) of the membrane theory for a cylindrical shell and a spherical shell.