mean and gauss curvature

[Poirot]

How do you tell from a picture of a surface what the signs of the mean and gauss curvature are?

 

[Fantini]

I don't think that is possible. Gaussian and mean curvatures can vary at each point, therefore how would you be able to describe them by simply looking at the picture?

 

[Poirot]

For the mean curvature, I have been given a surface with the uni normal shown. For the gauss curvature, just a surface so I'm guessing it's constant on theses surfaces.

 

[Fantini]

It's not enough information. Without being able to calculate the first and second fundamental forms coefficients you can't say anything else other than "guesses". Even if it's an educated guess, it's of no use unless you can quantify it.

 

[Poirot]

Remember it's only the sign i'm interested in. Anyone?

 

[Fantini]

Even if it's solely the sign, with the curvatures varying pointwise how can you pinpoint it at each one?

 

[Poirot]

Even if it's solely the sign, with the curvatures varying pointwise how can you pinpoint it at each one?

Because the sign is constant.

 

[Fantini]

What if the curvatures vanish?

 

[Opalg]

How do you tell from a picture of a surface what the signs of the mean and gauss curvature are?

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As a very rough intuitive guide, suppose that you have a picture of an oriented surface, so that it makes sense to talk about the regions inside and outside the surface, as with the objects pictured above. You then have to estimate the principal curvatures. These are the maximal and minimal values of the cross-sectional curvatures at a given point (where the curvature is considered positive if it is directed towards the inside of the surface, and negative if it is directed towards the outside. For every point in each of the three objects in the picture, the maximal curvature occurs for a horizontal cross-section and the minimal curvature occurs for a vertical cross-section, as indicated by the lines in the picture.

The Gaussian curvature is the product of the two principal curvatures, so it is positive if these have the same sign, negative if they have opposite signs, and zero if either of them is zero. From that, it should be obvious that the Gaussian curvature is negative (at each point) for the object on the left in the picture, zero for the cylinder in the centre of the picture, and positive for the ball on the right.

The mean curvature is the arithmetic mean of the two principal curvatures. So it is positive if they are both positive. If they have opposite signs then you have to judge which of them is larger in absolute value.