8.3: Principal Curvatures as Extrema of Normal Curvature


The formula for the normal curvature depends on the coefficients of the first and second fundamental forms at a point and the direction (u'(0),v'(0) of a vector Xu(u0,v0)u'(0) + Xv(u0,v0)v'(0) in the tangent space.

κN(0) = (L11u'(0)2 + 2L12u'(0)v'(0) + L22v'(0)2) / (g11u'(0)2 + 2g12u'(0)v'(0) + g22v'(0)2)

In the first section of this lab, we claimed that the principal curvatures at a point on a surface could be defined as the maximum and minimum values of the normal curvature. Using our expression for κN(t) in terms of the coefficients of the first and second fundamental forms, we can now prove it. In the case where u'(0) 0, we set λ = v'(0)/u'(0). We may then write:

κN(λ) = (L11 + 2L12λ + L22λ2) / (g11 + 2g12λ + g22λ2)

Since λ specifies a direction, we are interested in the maximum and minimum values of the normal curvature as a function of λ. So, we differentiate with respect to λ and set the result equal to 0. We obtain

N'(λ) = [(g11 + 2g12λ + g22λ2)(2L12 + 2L22λ) - (L11 + 2L12λ + L22λ2)(2g12 + 2g22λ)]/[g11 + 2g12λ + g22λ2]2 = 0,

which leads to

(g11 + g12λ)(L12 + L22λ) = (L11 + L12λ)(g12 + g22λ) ,    
or
(L12 + L22λ)/(g12 + g22λ) = (L11 + L21λ)/(g11 + g21λ) = c

If we set this common value temporarily equal to c and we find that

[L11 + L12λ] = c[g11 + g12λ], and
[L12 + L22λ] = c[ g12 + g22λ]        

Plugging these values back into the formula for N, we see that c = N.

We can then solve for λ in terms of c to get

L11 - cg11 = (L12 - cg12)λ, and
L12 - cg12 = (L22 - cg22)λ        

From this we obtain

(L11 - cg11)(L22 - cg22) - (L12 - cg12)2 = 0,      so              

L11L22 - L122 + (-g11L22 + 2g12L12 - g22L11)c + (g11g22 - g122)c2 = 0.

Dividing by g gives Κ - 2Hc + c2 = 0, which means that c (and N) satisfy the characteristic equation of the L matrix, as predicted.


Demonstration

In this demonstration we show a surface and its domain. The two vector fields corresponding to the two principal directions for points (u,v) in the domain are drawn in the domain window. Using the Show Vec Field 1 and Show Vec Field 2 checkboxes in the control panel, the vector fields can be toggled on and off. Also, in the domain window, we select a point (u(0),v(0)) and draw two lines in the directions of the red and green arrows. The corresponding point X(u(0),v(0)) and the red and green curves appear on the surface.

The demonstration provides visualizations of two properties of the principal directions. We just proved that the extrema of normal curvature occur along the principal directions. In the surface window, we show κNN for both the red and green curves. As we change the directions of the red and green curves, we can, therefore, observe how the normal curvature is affected. When the directions of the red and green curves are set so that the normal curvatures are at their extrema values, the red and green vectors in the domain should line up with the two vector fields.

A second property of the principal directions is that they are perpendicular except at umbilic points, where every direction is principal. In a third window, we show the tangent plane at the point X(u(0),v(0)) as well as the red and green tangent vectors. The projection of the red and green curves into the tangent space is also shown for reference. Note that the tangent vectors are perpendicular whenever the curves are lined up with the principal directions.

Exercises

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