Lecture 008

Arc Length and Curvature

In $\mathbb{R}^2$, $\text{Arc Length} = \int_{x=a}^{x=b} \sqrt{((\frac{dy}{dx})^2 + (\frac{dy}{dy})^2)} dx = \int_{x=a}^{x=b} \sqrt{(f'(x)^2 + 1)} dx$

• parameterized: $\text{Arc Length} = \int_{t=a}^{t=b} r'(t) dt = \int_{t=a}^{t=b} \sqrt{(y'(t))^2 + (x'(t))^2} dt$

In $\mathbb{R}^3$, if $\overrightarrow{r}(t) = \langle{f(t), g(t), h(t)}\rangle$, then $\begin{align} \text{Arc Length} &= \int_{t=a}^{t=b} \sqrt{(f'(t))^2 + (g'(t))^2 + (h'(t))^2} dt\\ &= \int_{t=a}^{t=b} \|\overrightarrow{r}'(t)\| dt\\ \end{align}$

Arc Length Function: function computes arc length from $t=a$ to $t=t$ given $\overrightarrow{r}(t)$ is $S(t) = \int_a^t \|\overrightarrow{r}'(u)\| du$.

Re-parametrizing by arc length: ensure travel 1 unit time per unit length

1. rearrange $S(t) = \int_a^t \|\overrightarrow{r}'(u)\| du$ to get a new function $t(S)$
2. subsitute all $t(S)$ into original vector function $\overrightarrow{r}(t)$. (getting a vector function in terms of its arc length $S$)
3. arc length of $\overrightarrow{r}(S)$ from $S=a$ to $S=b$ is $b-a$ after re-parametrizing by arc length

Principal Unit Tangent Vector:

• in terms of t: $\overrightarrow{T}(t) = \frac{\overrightarrow{r}'(t)}{\|\overrightarrow{r}'(t)\|} = \frac{\overrightarrow{r}'(t)}{\|\frac{dS}{dt}\|}$

• in terms of S:...

• because it is unit vector $T \perp T'$ because $\|T\| = 1$

// TODO: study curvature Curvature: $\kappa(S) = \|\frac{d \overrightarrow{T}}{dS}\| = \|\overrightarrow{T'}(S)\|$ for $\overrightarrow{r}(S)$ (parameterized by arc length $S$)

• $\kappa(t) = \frac{\|\overrightarrow{T'}(t)\|}{\|\overrightarrow{r'}(t)\|}$

• In $\mathbb{R}^3$: $\kappa(t) = \frac{\|\overrightarrow{r}'(t) \times \overrightarrow{r}''(t)\|}{\|\overrightarrow{r}'(t)\|^3} = \frac{\|\overrightarrow{v} \times \overrightarrow{a}\|}{\|\overrightarrow{v}\|^3}$, which means $\|\overrightarrow{T}'(t)\| = \frac{\|\overrightarrow{r}'(t) \times \overrightarrow{r}''(t)\|}{\|\overrightarrow{r}'(t)\|^2}$

• In $\mathbb{R}^2$: $\kappa(x) = \frac{|\frac{d^2y}{dx^2}|}{(1+(\frac{dy}{dx})^2)^{\frac{3}{2}}}$

• Note that $T' \perp T$ always hold because $\|T\| = 1$ implies that vector $T$ is constrainted in a circle, therefore any change is orthogonal to it.

On Orthogonal:

• $\|\overrightarrow{v}\| = \overrightarrow{v} \cdot \overrightarrow{v} = C_{\text{constant}} \implies \overrightarrow{v} \cdot \overrightarrow{v}' = 0$ (this is the case for all unit vectors)

• $\overrightarrow{v} \cdot \overrightarrow{w} = 0 \implies \|\overrightarrow{v} \times \overrightarrow{w}\| = \|\overrightarrow{v}\|\|\overrightarrow{w}\|$

Principal Unit Normal Vector: $\overrightarrow{N}(t) = \frac{\overrightarrow{T}'(t)}{\|\overrightarrow{T}'(t)\|}$

• always point in the same direction as curvature

Binormal Vector: $\overrightarrow{B}(t) = \overrightarrow{T}(t) \times \overrightarrow{N}(t)$

Acceleration Components on Curve:

• $(\exists a_{\overrightarrow{T}}, a_{\overrightarrow{N}} \in \mathbb{R}) (\overrightarrow{a} = a_{\overrightarrow{T}}\overrightarrow{T} + a_{\overrightarrow{N}}\overrightarrow{N})$

• $a_{\overrightarrow{T}} = \frac{\overrightarrow{T} \cdot \overrightarrow{a}}{\|\overrightarrow{T}\|} = \overrightarrow{T} \cdot \overrightarrow{a}$

• $a_{\overrightarrow{N}} = \frac{\overrightarrow{N} \cdot \overrightarrow{a}}{\|\overrightarrow{N}\|} = \overrightarrow{N} \cdot \overrightarrow{a} = \frac{\|\overrightarrow{v} \times \overrightarrow{a}\|}{\|\overrightarrow{v}\|} = \sqrt{\|a\|^2 - a_{T}^2}$

Theorem: $a_{\overrightarrow{N}} = \overrightarrow{N} \cdot \overrightarrow{a} = \frac{\|\overrightarrow{v} \times \overrightarrow{a}\|}{\|\overrightarrow{v}\|} = \sqrt{\|\overrightarrow{a}\|^2 - a_{\overrightarrow{T}}^2}$ // TODO: why is this true

Kepler's Law of Planetory Motion:

• Law of Ellipse: planetary motion is ellipse

• Equal Area: if area are equal, time equal

• Law of Harmony: $T_{time/year}^2 = D_{average distance from sun/AU}^3$

  • $F = ma(t) = - \frac{GmM}{\|\overrightarrow{v}\|^2} \cdot \frac{r}{\|r\|}$ // TODO: proof

$F_{\text{centripital}} = \frac{mv^2}{r}$ // TODO: find a problem

// WARNING: $\overrightarrow{N}(t) = \frac{\overrightarrow{T}'(t)}{\|\overrightarrow{T}'(t)\|} = \frac{r''(t)}{\|r''(t)\|}$ only when $\|r'\|$ is constant.