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

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:

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

On Orthogonal:

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

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

Acceleration Components on Curve:

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:

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.

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