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Chapter 2: Space Curves
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Section 2.5: Principal Normal
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Essentials


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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 $\mathbf{N}\=\mathbf{T}\prime \left(s\right)\/\mathrm{\κ}\=\frac{\mathbf{T}\prime \left(p\right)}{\mathrm{\κ}\mathrm{rho;}}$.



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In Definition 2.5.1, $\mathbf{T}\mathit{\prime}\left(s\right)$ refers to the derivative of the unit tangent vector $\mathbf{T}\left(s\right)$ taken with respect to arc length $s$; $\mathbf{T}\prime \left(p\right)$ refers to the derivative of $\mathbf{T}\left(p\right)$ taken with respect to $p$. Consequently, the second equality in the definition follows from the chain rule for differentiation. Think of the first equality as the definition; the second, as a means of computing N.

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Since T is a unit vector, $\mathbf{T}\prime$ is necessarily orthogonal to it. (See Example 2.3.5.) Since $\mathrm{\κ}$ is the magnitude of $\mathbf{T}\prime \left(s\right)$, N is necessarily a unit vector that is therefore orthogonal to T. The plane spanned by T and N is called the osculating plane, the plane in which the circle of curvature resides. The principal normal N points towards the center of curvature, that is, towards the center of the circle of curvature.

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For a plane curve, there are just two directions orthogonal to the tangent vector T. The principal normal is the one pointing towards the center of curvature. Moreover, if $\mathbf{T}\=a\mathbf{i}plus;b\mathbf{j}$, then N will be either $\pm \(b\mathbf{i}a\mathbf{j}$) . The vector with the plus sign will point to the right of T; with the minus sign, to the left. Hence, in the plane it is possible to obtain N directly from T without any additional calculations, provided the shape of the curve is known. (From the shape of the curve it should be possible to determine to which side lies the center of curvature.)

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If $\mathbf{R}\left(p\right)$ is the positionvector description of a curve $C$, the center of curvature for $C$ is given by $\mathbf{R}\+\mathbf{N}\/\mathrm{\κ}\=\mathbf{R}\+r\mathbf{N}$, where $r\=1\/\mathrm{\κ}$ is the radius of curvature. For a plane curve, the trajectory traced by the center of curvature as the circle of curvature rolls along the curve is called the evolute for the curve C.



Examples


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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 $p\=2\mathrm{pi;}sol;3$ on the graph of $C$, the cycloid defined by $x\=\left(p\mathrm{sin}\left(p\right)\right)comma;yequals;\left(1\mathrm{cos}\left(p\right)\right)$, $p\in \left[0\,2\mathrm{pi;}\right]$, compute N. Graph $C$, along with $\mathbf{N}\left(2\mathrm{pi;}sol;3\right)$ and $\mathbf{T}\left(2\mathrm{pi;}sol;3\right)$. 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 $y\=\mathrm{cosh}\left(x\right)$, 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 $\mathbf{R}\left(p\right)\=\mathrm{cos}\left(p\right)\mathbf{i}plus;\mathrm{sin}\left(p\right)\mathbf{j}plus;p\mathbf{k}$, 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 $\mathbf{R}\left(p\right)\=p\mathbf{i}plus;3{p}^{2}\mathbf{j}plus;{p}^{3}\mathbf{k}$, 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 $\mathbf{R}\left(p\right)\=\mathrm{ln}\left(\mathrm{cos}\left(p\right)\right)\mathbf{i}plus;\mathrm{ln}\left(\mathrm{sin}\left(p\right)\right)\mathbf{j}plus;\sqrt{2}p\mathbf{k}$, $p\in \left(0\,\mathrm{\π}\/2\right)$, 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 $\mathbf{R}\left(p\right)\=\left(3p{p}^{3}\right)\mathbf{i}plus;3{p}^{2}\mathbf{j}plus;\left(3pplus;{p}^{3}\right)\mathbf{k}$, 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 $x\=2\mathrm{cos}\left(p\right)comma;yequals;\mathrm{sin}\left(p\right)comma;p\in \left[0comma;2\mathrm{pi;}\right]$.



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