Definition 2.5.1:   Principal Normal

The principal normal along $C$, a curve described by the position vector $\mathbf{R}\left(p\right)$, is the vector $\mathbf{N}\=\mathbf{T}\prime \left(s\right)\/\mathrm{\κ}\=\frac{\mathbf{T}\prime \left(p\right)}{\mathrm{\κ} \mathrm{\ρ}}$.

Example 2.5.1

At $x\=2$ on the graph of $C$, the curve defined by $y\=x^2$, compute N. Graph $C$, along with $\mathbf{N}\left(2\right)$ and $\mathbf{T}\left(2\right)$. Does N point towards the center of curvature?

Example 2.5.2

At $x\=1$ on the graph of $C$, the curve defined by $y\=x^{3\/2}$, compute N. Graph $C$, along with $\mathbf{N}\left(1\right)$ and $\mathbf{T}\left(1\right)$. Does N point towards the center of curvature? Hint: The curvature $C$ was obtained in Example 2.4.3.

Example 2.5.3

At $p\=2 \mathrm{\π}\/3$ on the graph of $C$, the cycloid defined by $x\= \left(p-\sin \left(p\right)\right)\,y\= \left(1-\cos \left(p\right)\right)$, $p\in \left[0\,2 \mathrm{\π}\right]$, compute N. Graph $C$, along with $\mathbf{N}\left(2 \mathrm{\π}\/3\right)$ and $\mathbf{T}\left(2 \mathrm{\π}\/3\right)$. Does N point towards the center of curvature? Hint: The curvature of $C$ was obtained in Example 2.4.4.

Example 2.5.4

At $x\=1$ on the graph of $C$, the catenary defined by $y\=\cosh \left(x\right)$, compute N. Graph $C$, along with $\mathbf{N}\left(1\right)$ and $\mathbf{T}\left(1\right)$. Does N point towards the center of curvature? Hint: The curvature of $C$ was obtained in Example 2.4.5.

Example 2.5.5

At $x\=1\/\sqrt{3}$ on the graph of $C$, the upper semicircle defined by $x^2\+y^2\=1$, compute N. Graph $C$, along with $\mathbf{N}\left(1\/\sqrt{3}\right)$ and $\mathbf{T}\left(1\/\sqrt{3}\right)$. Does N point towards the center of curvature? Hint: The curvature of a circle was obtained in Example 2.4.2.

Example 2.5.6

At $p\=\mathrm{\π}\/3$ on the graph of $C$, the helix defined by $\mathbf{R}\left(p\right)\=\cos \left(p\right) \mathbf{i}\+\sin \left(p\right) \mathbf{j}\+p \mathbf{k}$, compute N. Graph $C$, along with $\mathbf{N}\left(\mathrm{\π}\/3\right)$ and $\mathbf{T}\left(\mathrm{\π}\/3\right)$. Does N point towards the center of curvature? Hint: The curvature of $C$ was obtained in Example 2.4.6.

Example 2.5.7

At $p\=1$ on the graph of $C$, the curve defined by $\mathbf{R}\left(p\right)\=p \mathbf{i}\+3 p^2 \mathbf{j}\+p^3 \mathbf{k}$, compute N. Graph $C$, along with $\mathbf{N}\left(1\right)$ and $\mathbf{T}\left(1\right)$. Does N point towards the center of curvature? Hint: The curvature of $C$ was obtained in Example 2.4.7.

Example 2.5.8

At $p\=\mathrm{\π}\/4$ on the graph of $C$, the curve defined by $\mathbf{R}\left(p\right)\=\ln \left(\cos \left(p\right)\right) \mathbf{i}\+\ln \left(\sin \left(p\right)\right) \mathbf{j}\+\sqrt{2}p \mathbf{k}$, $p\in \left(0\,\mathrm{\π}\/2\right)$, compute N. Graph $C$, along with $\mathbf{N}\left(\mathrm{\π}\/4\right)$ and $\mathbf{T}\left(\mathrm{\π}\/4\right)$. Does N point towards the center of curvature? Hint: The curvature of $C$ was obtained in Example 2.4.8.

Example 2.5.9

At $p\=-1$ on the graph of $C$, the curve defined by $\mathbf{R}\left(p\right)\=\left(3 p-p^3\right) \mathbf{i}\+3 p^2 \mathbf{j}\+\left(3 p\+p^3\right) \mathbf{k}$, compute N. Graph $C$, along with $\mathbf{N}\left(-1\right)$ and $\mathbf{T}\left(-1\right)$. Does N point towards the center of curvature? Hint: The curvature of $C$ was obtained in Example 2.4.9.

Example 2.5.10

Obtain and graph the evolute of $C$, the ellipse $x\=2 \cos \left(p\right)\,y\=\sin \left(p\right)\,p\in \left[0\,2 \mathrm{\π}\right]$.