normal curvatures

Let us determine the normal curvatures (http://planetmath.org/NormalSection) $\varkappa$ of the surface

$z=z(x,y)$ (1)

in the origin, when (1) has the continuous 1st and 2nd order partial derivatives in a neighbourhood of  $(0,\mathrm{\hspace{0.17em}0})$  and satisfies

$z(0,\mathrm{\hspace{0.17em}0})=z_x^{\prime }(0,\mathrm{\hspace{0.17em}0})=z_y^{\prime }(0,\mathrm{\hspace{0.17em}0})=0.$ (2)

It’s a question of the curvature (http://planetmath.org/CurvaturePlaneCurve) of the intersection curves of the surface (1) and planes containing the $z$-axis, which is the normal of the surface in the origin.

If the angle between the $zx$-plane and a plane $\tau$ containing the $z$-axis is denoted by $\phi$, when the line of intersection of the plane $\tau$ and the $xy$-plane is represented by the equations

then equation of the the normal section curve $C_{\phi }$ is

$$z=z(\varrho \cos \phi ,\varrho \sin \phi ),$$

where $\varrho$ is the abscissa and $z$ the ordinate.  It follows that

$$\frac{dz}{d\varrho }=\frac{\partial z}{\partial x}\cos \phi +\frac{\partial z}{\partial y}\sin \phi ,$$

$$\frac{d^{2}z}{d{\varrho }^2}=\frac{{\partial }^{2}z}{\partial x^2}{\cos }^2\phi +2\frac{{\partial }^{2}z}{\partial x\partial y}\sin \phi \cos \phi +\frac{{\partial }^{2}z}{\partial y^2}{\sin }^2\phi ;$$

thus by (2), in the origin we have

$$\frac{dz}{d\varrho }=\mathrm{\hspace{0.33em}0},\frac{d^{2}z}{d{\varrho }^2}=a{\cos }^2\phi +2b\sin \phi \cos \phi +c{\sin }^2\phi ,$$

where $a$, $b$, $c$ the values of the derivatives $\frac{{\partial }^{2}z}{\partial x^2}$, $\frac{{\partial }^{2}z}{\partial x\partial y}$, $\frac{{\partial }^{2}z}{\partial y^2}$ in the origin.

Using those values, we obtain for the normal curvature of $C_{\phi }$ in the origin the value

$\varkappa (\phi )={\left[{\displaystyle \frac{\frac{d^{2}z}{d{\varrho }^2}}{{\left(1+{\left(\frac{dz}{d\varrho }\right)}^2\right)}^{3/2}}}\right]}_{\varrho =\mathrm{\hspace{0.17em}0}}=a{\cos }^2\phi +2b\sin \phi \cos \phi +c{\sin }^2\phi .$ (3)

This result gets a more illustrative form when we try to express it by using the least and the greatest value of $\varkappa (\phi )$.  Instead to utilize the zeros of the derivative of the sum in (3), it’s simpler first to transfer to the double angle (http://planetmath.org/DoubleAngleIdentity),

$\varkappa (\phi )={\displaystyle \frac{a+c}{2}}+{\displaystyle \frac{a-c}{2}}\cos 2\phi +b\sin 2\phi ,$ (4)

and here to introduce an auxiliary angle $\alpha$  () such that

$$\frac{a-c}{2}:=k\cos 2\alpha ,b:=k\sin 2\alpha .$$

This allows us to write (4) as

$\varkappa (\phi )={\displaystyle \frac{a+c}{2}}+k\cos 2(\phi -\alpha ).$ (5)

From this we see immediately that the curvature attains its greatest and least value ${\displaystyle \frac{a+c}{2}}\pm k$ when  $\phi =\alpha$  and  $\phi =\alpha +\frac{\pi }{2}$.

Accordingly, the corresponding $\tau$, the principal normal planes, are perpendicular to each other; their normal section curves on the surface (1) in the origin are briefly called the principal sections.

The expression (5) of the normal curvature may still be edited.  Let us take a new parameter angle  $\phi -\alpha :=\theta$.  One can write

$$\varkappa (\phi )=\frac{a+c}{2}({\cos }^2\theta +{\sin }^2\theta )+k({\cos }^2\theta -{\sin }^2\theta )=\left(\frac{a+c}{2}+k\right){\cos }^2\theta +\left(\frac{a+c}{2}-k\right){\sin }^2\theta :={\varkappa }_{\theta }.$$

So the final result, the so-called Euler’s theorem (http://planetmath.org/SecondFundamentalForm), can be expressed in the form

${\varkappa }_{\theta }={\varkappa }_1{\cos }^2\theta +{\varkappa }_2{\sin }^2\theta .$ (6)

Here, the principal curvatures ${\varkappa }_1$ and ${\varkappa }_2$ are the greatest and the least value of the normal curvature, respectively, and $\theta$ is the angle between (http://planetmath.org/AngleBetweenTwoPlanes) the normal section plane corresponding ${\varkappa }_1$ and the normal section plane corresponding ${\varkappa }_{\theta }$. As it becomes clear in the parent entry (http://planetmath.org/NormalSection), the result (6) is true not only in the origin but at any point on a surface when the given function has the continuous 1st and 2nd derivatives in some neighbourhood of the point.

Title	normal curvatures
Canonical name	NormalCurvatures
Date of creation	2013-03-22 17:26:27
Last modified on	2013-03-22 17:26:27
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Topic
Classification	msc 53A05
Classification	msc 26B05
Classification	msc 26A24
Related topic	SecondFundamentalForm
Related topic	MeusniersTheorem
Related topic	ErnstLindelof
Defines	principal normal plane
Defines	principal section
Defines	principal curvature
Defines	Euler’s theorem