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The Calculus of Curves

In the last lecture, the derivatives of a vector that varied continuously with a parameter, $\vec{r}(t)$ , were considered. It is natural to think of $\vec{r}(t)$ as a curve in whatever space the vector $\vec{r}$ is defined. In this way, a curve is represented by $N$ coordinates as a single value takes on a range of numbers.

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Curves, Tangents, Surfaces

Because the derivative of a curve with respect to its parameter is a tangent vector, the unit tangent can be defined immediately:

$$\hat{u} = \frac{\ensuremath{\frac{d{\vec{r}}}{d{t}}}}{\norm {\ensuremath{\frac{d{\vec{r}}}{d{t}}}}}$$ (12-1)

It is convenient to find a new parameter, $s(t)$ , that would make the denominator in Eq. 12-1 equal to one. This parameter, $s(t)$ , is the arc-length:

$$\begin{split}s(t) & = \int_{t_o}^{t} ds \ & = \int_{t_o}^{t}... ...}}})\cdot(\ensuremath{\frac{d{\vec{r}}}{d{t}}})} dt \end{split}$$ (12-2)

and with $s$ instead of $t$ ,

$$\hat{u}(s) = \ensuremath{\frac{d{\vec{r}}}{d{s}}}$$ (12-3)

This is natural because $\norm {\vec{r}}$ and $s$ have the same units (i.e., meters and meters, foots and feet, etc) instead of, for instance, time, $t$ , that makes ${d\vec{r}}/{dt}$ a velocity and involving two different kinds of units (e.g., furlongs and hours).

With the arc-length $s$ , the magnitude of the curvature is particularly simple,

$$\kappa(s) = \norm {\ensuremath{\frac{d{\hat{u}}}{d{s}}}} = \norm {\ensuremath{\frac{d^2{\vec{r}}}{d{s}^2}}}$$ (12-4)

as is its interpretation--the curvature is a measure of how rapidly the unit tangent is changing direction.

Furthermore, the rate at which the unit tangent changes direction is a vector that must be normal to the tangent (because $d(\hat{u} \cdot \hat{u} = 1) = 0$ ) and therefore the unit normal is defined by:

$$\hat{p}(s) = \frac{1}{\kappa(s)} \ensuremath{\frac{d{\hat{u}}}{d{s}}}$$ (12-5)

There two unit vectors that are locally normal to the unit tangent vector $\hat{u'}(s)$ and the curve unit normal $\hat{p}(s) \times \hat{u}$ and $\hat{u}(s) \times \hat{p}$ . This last choice is called the unit binormal, $\hat{b} \equiv \hat{u}(s) \times \hat{p}$ and the three vectors together form a nice little moving orthogonal axis pinned to the curve. Furthermore, there are convenient relations between the vectors and differential geometric quantities called the Frenet equations.

Using Arc-Length as a Curve's Parameter

However, it should be pointed out that--although re-parameterizing a curve in terms of its arc-length makes for simple analysis of a curve--achieving this re-parameterization is not necessarily simple.

Consider the steps required to represent a curve $\vec{r}(t)$ in terms of its arc-length:

integration

The integral in Eq. 12-2 may or may not have a closed form for $s(t)$ .

If it does then we can perform the following operation:

inversion

$s(t)$ is typically a complicated function that is not easy to invert, i.e., solve for $t$ in terms of $s$ to get a $t(s)$ that can be substituted into $\vec{r}(t(s)) = \vec{r}(s)$ .

These difficulties usually result in treating the problem symbolically and the resorting to numerical methods of achieving the integration and inversion steps.

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Calculating arclength