Answer to Question #117667 in Differential Geometry | Topology for Nouhad

Question

Let E be a Euclidian vector space, let (i,j) be its standard basis. Note C
the circle centered at the origin with radius a, where a is a real positive number. ⃗⃗
1. Let δ = (a cos as )i+(a sin as )j. Show that ([0, 2πa], δ) is a unit speed parametrization of C

Answer 1

First, the equation of a circle centered at the origin $p.O\left(0;0\right)$ and radius $R=a>0$ has the form

$$x^2+y^2=a^2$$

For example, $a=3$

Secondly, from physics we know the relationship between the velocity vector and the radius vector

$$\overrightarrow{r}=x(t)\cdot\overrightarrow{i}+y(t)\cdot\overrightarrow{j}\longrightarrow\\[0.3cm] \overrightarrow{v}=\frac{d\overrightarrow{r}}{dt}=\frac{dx(t)}{dt}\cdot\overrightarrow{i}+\frac{dy(t)}{dt}\cdot\overrightarrow{j}\\[0.3cm] \overrightarrow{v}=v_x(t)\cdot\overrightarrow{i}+v_y(t)\cdot\overrightarrow{j}\longrightarrow\\[0.3cm] \left|\overrightarrow{v}\right|=\sqrt{v_x^2+v_y^2}$$

In our case, the natural variable $t-$ time was replaced by the general variable $s$ .

$$\overrightarrow{v}\equiv\overrightarrow{\delta}(s)=\underbrace{\left(a\cos as\right)}_{\delta_x(s)}\cdot\overrightarrow{i}+\underbrace{\left(a\sin as\right)}_{\delta_y(s)}\cdot\overrightarrow{i}\longrightarrow\\[0.3cm] \left|\overrightarrow{\delta}(s)\right|=\sqrt{\delta_x^2(s)+\delta_y^2(s)}=\sqrt{\left(a\cos as\right)^2+\left(a\sin as\right)^2}=\\[0.3cm] =\sqrt{a^2\left(\cos^2as+\sin^2as\right)}=a\cdot\sqrt{1}=a$$

Conclusion,

$$\left|\overrightarrow{\delta(s)}\right|=\left\{\begin{array}{l} \text{is a unit speed},\quad\text{if}\quad a=1\\[0.3cm] \text{is not a unit speed},\quad\text{if}\quad a\neq1 \end{array}\right.$$