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3.4 Principal curvatures

As we can see from (3.30) the normal curvature at a point depends on the direction of $\lambda = \frac{dv}{du}$ . Now we will seek the directions in which the extrema of principal curvature occur following Struik [412]. The extreme values of $\kappa_n$ can be obtained by evaluating $\frac{d\kappa_n}{d\lambda} = 0$ of (3.30), which gives:

$\displaystyle (E + 2F\lambda + G \lambda^2)(N\lambda + M) - (L + 2M\lambda + N\lambda^2)(G\lambda + F) = 0\;,$

(3.37)

and hence

$\displaystyle \kappa_n = \frac{L + 2M\lambda + N\lambda^2}{E + 2F\lambda + G\lambda^2} = \frac{M + N\lambda }{F + G\lambda }\;.$

(3.38)

Furthermore since

$\displaystyle E + 2F\lambda + G\lambda^2 = (E + F\lambda) + \lambda(F + G\lambda)\;,$
$\displaystyle L + 2M\lambda + N\lambda^2 = (L + M\lambda) + \lambda(M + N\lambda)\;,$

(3.37) can be reduced to

$\displaystyle (E + F\lambda)(M + N\lambda ) = (L + M\lambda)(F + G\lambda )\;,$

(3.39)

and hence

$\displaystyle \kappa_n = \frac{L + 2M\lambda + N\lambda^2}{E + 2F\lambda + G\la... ...} = \frac{M + N\lambda }{F + G\lambda } = \frac{L + M\lambda}{E + F\lambda }\;.$

(3.40)

Therefore, the extreme values of $\kappa_n$ satisfy the two simultaneous equations

$\displaystyle (L - \kappa_n E)du + (M - \kappa_n F)dv = 0\;,$
$\displaystyle (M - \kappa_n F)du + (N - \kappa_n G)dv = 0\;.$			(3.41)

These equations form a homogeneous linear system of equations for , , which will have a nontrivial solution if and only if

$\displaystyle \left\vert \begin{matrix} L - \kappa_n E & M - \kappa_n F \cr M - \kappa_n F & N - \kappa_n G \end{matrix} \right\vert = 0\;,$

(3.42)

where $\vert \; \;\vert$ denotes the determinant of a matrix, or expanding

$\displaystyle (EG - F^2)\kappa_n^2 - (EN + GL -2FM)\kappa_n + (LN - M^2) =0\;.$

(3.43)

The discriminant of this quadratic equation in $\kappa_n$ can be re-formulated as

$\displaystyle D=4\left(\frac{EG- F^2}{E^2}\right)(EM-FL)^2 + \left(EN - GL - \frac{2F}{E} (EM-FL)\right)^2\;,%\nonumber \\$

(3.44)

after some algebraic manipulations. Thus the discriminant is always greater than or equal to zero and (3.43) has real roots. The discriminant becomes zero if and only if and or if and only if there is a constant such that

$\displaystyle L = kE,\;\;\; M=kF,\;\;\; N=kG\;.$

(3.45)

Such a point is called an umbilic and the normal curvature is the same in all directions. Therefore (3.43) has either two distinct real roots, or a double root. If we set

$\displaystyle K = \frac{LN - M^2}{EG - F^2}\;,$			(3.46)
$\displaystyle H = \frac{EN + GL -2FM}{2(EG - F^2)}\;,$			(3.47)

the quadratic equation for $\kappa_n$ (3.43) simplifies to:

$\displaystyle \kappa_{n}^2 -2H\kappa_n + K =0\;.$

(3.48)

The quantities and are called Gaussian (Gauss) curvature and mean curvature, respectively. Upon solving (3.48) for the extreme values of curvature, we have

$\displaystyle \kappa_{max} = H + \sqrt{H^2 - K}\;,$			(3.49)
$\displaystyle \kappa_{min} = H - \sqrt{H^2 - K}\;,$			(3.50)

where $\kappa_{max}$ is the maximum principal curvature and $\kappa_{min}$ is the minimum principal curvature. The directions in the tangent plane for which $\kappa_n$ takes maximum and minimum values are called principal directions. The corresponding directions in the -plane can be determined by using (3.40), which leads to

$\displaystyle \lambda = - \frac{M-\kappa_nF}{N- \kappa_nG}\;,$

(3.51)

$\displaystyle \lambda = - \frac{L-\kappa_nE}{M- \kappa_nF}\;,$

(3.52)

where $\kappa_n$ is replaced by either $\kappa_{max}$ or $\kappa_{min}$ .

When the discriminant is zero or , $\kappa_n$ is a double root with value equal to and the corresponding point of the surface is an umbilical point. At an umbilical point a surface is locally a part of sphere with radius of curvature $\frac{1}{\vert H\vert}$ . In the special case where both and vanish, the point is a flat or planar point.

Alternatively we can derive the principal directions by solving a quadratic equation in $\lambda$

$\displaystyle (FN-GM)\lambda^2 + (EN-GL)\lambda + (EM-FL)=0\;,$

(3.53)

which is deduced from (3.37). The discriminant of this equation is easily shown to be the same as that of (3.43), and hence it is greater than or equal to zero. At an umbilical point the discriminant vanishes and (3.45) hold, thus we have , and . Therefore, the coefficients of the quadratic equation become all zero and thus the principal directions are not defined. When a point on the surface is a non-umbilical point, there are always two principal directions determined by the quadratic equations. Let $\lambda_{max}$ and $\lambda_{min}$ be the directions of maximum and minimum principal curvature in the -plane. Then, $\lambda_{max}$ and $\lambda_{min}$ satisfy the quadratic equation (3.53):

$\displaystyle (FN-GM)\lambda_{max}^2 + (EN-GL)\lambda_{max} + (EM-FL)$	$\displaystyle =$	$\displaystyle 0\;,$	(3.54)
$\displaystyle (FN-GM)\lambda_{min}^2 + (EN-GL)\lambda_{min} + (EM-FL)$	$\displaystyle =$	$\displaystyle 0\;.$	(3.55)

From these equations we can deduce

$\displaystyle \lambda_{max}+\lambda_{min} = - \frac{EN-GL}{FN-GM}\;,$			(3.56)
$\displaystyle \lambda_{max}\lambda_{min} = \frac{EM-FL}{FN-GM}\;,$			(3.57)

thus,

$\displaystyle E + F(\lambda_{max}+\lambda_{min}) + G \lambda_{max}\lambda_{min} = \frac{1}{FN-GM}[E(FN-GM) - F(EN-GL) + G(EM-FL)] = 0\;.$

(3.58)

Consequently, it is evident from (3.18) that the two tangent vectors in the principal directions are orthogonal.

A curve on a surface whose tangent at each point is in a principal direction at that point is called a line of curvature. Since at each (non-umbilical) point there are two principal directions that are orthogonal, the lines of curvatures form an orthogonal net of lines. Figure 3.10 shows an example of the lines of curvature on a saddle-shaped surface where all points are hyperbolic. The solid lines correspond to the maximum principal curvature direction, while the dashed lines correspond to the minimum principal curvature direction (convention (a) is used). Since there is no umbilical point on the surface, we do not encounter any singularity on the net of lines of curvature. The lines of curvature in the presence of umbilical points are discussed in Chap. 9.

**Figure 3.10:** Lines of curvature
$\begin{figure}\centerline{ \psfig{file=fig/loc_saddle.ps,height=3.5in}}\end{figure}$

This orthogonal net of lines can be used as a parametrization of a surface. In such cases, we have (see (3.19)), and (3.41) reduce to

$\displaystyle (L-\kappa_nE)du + Mdv=0,\;\;\; Mdu + (N - \kappa_nG)dv =0\;.$

(3.59)

If these equations are satisfied by and by , this implies , and the two principal curvatures are $\kappa_1 = \frac{L}{E}$ and $\kappa_2 = \frac{N}{G}$ , in the absence of umbilical points. Therefore the necessary condition for the parametric lines to be lines of curvature is

$\displaystyle F=M=0\;.$

(3.60)

The converse is also true and the condition is also sufficient.

Example 3.4.1. As a curve in the -plane , revolves about the -axis, it generates a surface of revolution . The curves in different rotated positions are called the meridians of , while the circles generated by each point on are called the parallels of . If we denote the rotation angle in the -plane as $\theta$ , the surface of revolution can be parametrized as

$\displaystyle {\bf r}=(f(t)\cos\theta, f(t)\sin\theta, g(t))^T\;.$

Thus,

$\displaystyle {\bf r}_t=(\dot{f}(t)\cos\theta, \dot{f}(t)\sin\theta, \dot{g}(t))^T, \;\;\;\;\;\;{\bf r}_{\theta}=(-f(t)\sin\theta, f(t)\cos\theta, 0)^T,$

and hence

$\displaystyle E = \dot{f}^2(t) + \dot{g}^2(t),\;\;\;\;\;\;F=0,\;\;\;\;\;\;G=f^2(t)\;.$

Since , (3.19) shows that the meridians and parallels are orthogonal. Furthermore we have

$\displaystyle L = \frac{-\ddot{f}\dot{g}+\dot{f}\ddot{g}}{\sqrt{\dot{f}^2(t) + ... ...;\;\;\;M=0,\;\;\;\;\;\;N=\frac{f\dot{g}}{\sqrt{\dot{f}^2(t) + \dot{g}^2(t)}}\;,$

which lead us to the conclusion that the meridians and parallels of a surface of revolution are the lines of curvature.

Next: 3.5 Gaussian and mean Up: 3. Differential Geometry of Previous: 3.3 Second fundamental form Contents Index

December 2009