Lecture 01 - Regular Curves

#math

Definition

A regular curve is a $C^{\infty}$ function $r (t) : I \to R^{n}$ such that $r^{'} (t) \neq 0$ .

Locally (and globally since we are working on $R^{n}$ ) we can express the coordinates of $r (t) = (x^{1} (t), . . ., x^{n} (t))$ . We call $r^{'} (t)$ the velocity and $| r^{'} (t) |$ the speed.

Definition

Arc length is defined to be:

\int_{t_{0}}^{t_{1}} | r^{'} (t) | d t = s (t)

We interpret $r (s)$ , parametrized by arclength s, as the trajectory of a particle moving through 3-dimensional space. Since we are interested in curves with nonzero speed everywhere, we can always reparametrize to have unit speed; then the parameter coincides with arc length along the curve, denoted by s. Moreover, a parameterized space curve is a parameterized curve taking values in 3-dimensional Euclidean space.

Example

Let $r (t) = (c o s (e^{t}), s i n (e^{t}))$ , then one can obtain that $s (t) = e^{t} - 1$
We can rearrange the latter formula to obtain $t (s) = l o g (s + 1)$ giving us $r (t (s)) = (c o s (s + 1), s i n (s + 1))$

$r (s)$ is regular even though $r (t)$ may not.

Theorem

Given any regular curve $r (t)$ , one can always reparameterize it by its arc-length.
Then, $t$ can be regarded as a $C^{\infty}$ function of $s$ , and the reparameterize curve $r (s) = r (t (s))$ is regular with:

| \frac{d}{d s} r (s) | = 1, \forall s

Proof

$\frac{d s}{d t} = \frac{d}{d t} \int_{t_{0}}^{t} | r^{'} (τ) d τ | = | r^{'} (t) | > 0$
so one can regard t as a function of s by the inverse function theorem
$\frac{d}{d s} r (s) = \frac{d r}{d t} \frac{d t}{d s} = \frac{r^{'}}{\frac{d s}{d t}}$
$| \frac{d}{d s} r (s) | = \frac{| r^{'} (t) |}{| (\frac{d s}{d t}) |} = \frac{| r^{'} (t) |}{| r^{'} (t) |} = 1$

Curvature

Definition

Let $r (s)$ , be an arc-length parameterized curve in $R^{n}$ , the curvature is defined by:

κ (s) = | r^{″} (s) |

The curvature $κ$ at a point $s$ measures the deviation of the curve from linearity. The reciprocal of the curvature $\frac{1}{κ (s)}$ is called the curvature radius. For example, a circle
with radius $r$ has a constant curvature of $κ (s) = \frac{1}{r}$ whereas a line has a curvature of 0.

Example

Let $r (s) = (R c o s (\frac{s}{R}), R s i n (\frac{s}{R}))$ , then $κ (s) = \frac{1}{R}$ which matches our intuition that the smaller the radius the larger the curvature.

Lemma

Let $r (s)$ , be a curve parameterized by arc length, then the velocity $r^{'} (s)$ and acceleration $r^{″} (s)$ is always orthogonal for any $s \in I$ .

Proof

$| r^{'} (s) | = 1$ giving us that $\frac{d}{d s} | r^{'} (s) |^{2} = 0$
That is $\frac{d}{d s} (r^{'} (s) \cdot r^{'} (s)) = 0 ⟹ r^{'} (s) \cdot r^{″} (s) = 0$ .

Proposition

Given any regular curve $r (t) \in R^{3}$ the curvature as a function of $t$ can be computed by the following formula:

κ (t) = \frac{| r^{'} (t) \times r^{″} (t) |}{| r^{'} (t) |^{3}}