Principle curvature: min and max curvature are always perpendicular?

[Relative0]

I am trying to understand Gaussian curvature. This led me into looking at principle curvature. Now If one takes a look at the picture of the "Saddle Surface" on Wikipedia here: http://en.wikipedia.org/wiki/Principal_curvature

I see that at the point p on the saddle where curvature goes both positive and negative in different directions that there is a normal vector at point p. There are two planes of "principle curvature" which intersect at the normal vector at p. But furthermore that each normal plane (containing the normal p) intersects the surface and that at the point p a tangent plane is made which intersects any particular normal plane through p. That also for each normal plane going through p, the intersection of the normal plane with the surface gives rise to a curve on the normal plane - each curve different depending on the normal plane chosen. Now According to Wikipedia:

"The directions of the normal plane where the curvature takes its maximum and minimum values are always perpendicular, if k1 does not equal k2"

This part I don't understand.. I see that this result is due to Euler, but don't see why (or where to find an explanation on as to why this is true.

Any thoughts?

Thanks,

Brian

 

[quasar987]

Hi,

It is easier to see what's going on if you define the principal curvatures as the eigenvalues of the second fundamental form II. Since II is symmetric, the spectral theorem guarantees that the eigenvalues k1,k2 of II are real and that there exists an orthonormal basis t1, t2 with respect to which II is diagonal (with diagonal entries its 2 eigenvalues). Define the principal curvatures as k1, k2.

To see now that they correspond to the values that minimize/maximize the normal curvature, just pick a curve c(t) (unit speed), and write c' in terms of t1,t2 and the angle µ between t1 and c': c'=cos(µ)t1+sin(µ)t2. Then use that the normal curvature of c is (c')^T II c' (by definition of II). Write this matrix equation wrt to the basis t1,t2. It yields the following expression for the normal curvature of c: k_n(c)=k1cos²(µ)+k2sin²(µ). Use calculus to minimize/maximize this function. You will find that the min/max values of k_n(c) are when µ=0 and pi/2, which correspond to k_n=k1 and k2.

 

[Relative0]

Thanks Quasar! I am a little confused and am hoping you might clarify

write c' in terms of t1,t2 and the angle µ between t1 and c': c'=cos(µ)t1+sin(µ)t2. Then use that the normal curvature of c is (c')^T II c' (by definition of II)

I am guessing you mean maybe c and c' as t1 and t2 are orthonormal? Also that the normal curvature of c is (c')^T II c'. I don't see what that could be.

Perhaps you might be able to clarify?

Thanks much though for your insight!

Brian

 

[quasar987]

I am guessing you mean maybe c and c' as t1 and t2 are orthonormal?

No, I do mean c' and t1, why? What seems weird about that to you? (Note by the way that c and c' does not make sense as c is a curve, not a vector, so you can't really compare the two).

All I'm saying is that since t1,t2 is a basis of the tangent plane of the surface (vector space), you can write c' as a linear combination of them. Moreover, you can use plane trigonometry to express the coefficients in terms of the angle µ between t1 and c'.

Also that the normal curvature of c is (c')^T II c'. I don't see what that could be.

This is a matrix equation, with (c')^T being the transpose of c'. This is a compact form of the expression

$L\dot{u}^2+2M\dot{u}\dot{v}+N\dot{v}^2$

with which you are perhaps more familiar? In this expression, it is implicit that we have chosen local coordinates (u,v) for the surface that that wrt those coordinates, the curve c(t) is given by (u(t),v(t)). Dots mean derivatives wrt t. That the second fundamental form gives the normal curvature via this expression is practically by definition of the second fundamental form.

 

[blue_raver22]

I can provide some insight into the concept of principle curvature and its relationship to Gaussian curvature. Principle curvature is a geometric property of a surface that describes the curvature at a specific point in two perpendicular directions, called the maximum and minimum curvature directions. These directions are always perpendicular to each other, as stated in the Wikipedia article you referenced.

This result is a consequence of the definition of principle curvature and can be explained using the concept of normal curvature. Normal curvature is the curvature of a surface in a specific direction, measured along the normal vector to the surface at a given point. In the case of principle curvature, the maximum and minimum curvature directions are the directions in which the normal curvature is either the largest or smallest, respectively.

Now, let's consider the case of a saddle surface, as shown in the Wikipedia image. At the point p on the surface, the normal vector is perpendicular to the surface, and there are two planes of principle curvature that intersect at this point. These planes are also perpendicular to the normal vector. This means that the normal curvature in these two directions is zero, as the normal vector is perpendicular to both planes.

In other words, the maximum and minimum curvature directions are perpendicular to the normal vector, and therefore, they are also perpendicular to each other. This is true for any point on a surface, not just at the point p on a saddle surface.

I hope this explanation helps to clarify the concept of principle curvature and its relationship to Gaussian curvature. If you would like further information, I suggest consulting a textbook or speaking with a mathematician who specializes in differential geometry.