# Gaussian curvature and developability

Hey guys,

it is clear, that in order to get a devolable surface, the Gaussian curvature has to be 0. However when optimizing geometry with different tools (D.Loft, Kangaroo, DevLoft command etc.), I never get geometry that has exactly 0 Gaussian curvature.
Is there a certain value from which on the geometry is develobable? I guess it depends on the material (e.g. glass has a different behavior than sheet metal), bending temerature, speed of deformation etc… but how can I judge, if the surface is usable?

Thanks a lot!

p.s.: Bonus question: I read here, that the curvature vallue in the Rhino curvature analysis is 1/length. However mathematically, Gaussian curvature is defined as product of two main curvatures, which would mean that it has to be 1/length²… Am I missing something here?

This depends on the material - the numbers are in model units - generally a material will be developable within a certain range - it gets complicated with directional materials etc so the key is to know what you’ve got and how much less than strictly developable it can be.

-Pascal

Ok, let´s say for example I am working with sheet metal. Are there any standard charts with values?

“Curvature”, not “Guassian curvature” has units of 1/length as described in the linked thread.

“Guassian curvature” has units of 1/length^2. It is the product of two curvatures, not a simple curvature.

Gaussian curvature is K1 * K2 where K1 and K2 are the principal curvatures and equal to 1/radii of curvatures in the principal directions. So Gaussain curvature has units of 1/length^2. If millimeters are the length unit then Gaussian curvature has units of 1/mm^2. If meters are used then Gaussian curvature has units of 1/m^2. So the numeric value for the Gaussian curvature of a surface will be 1,000,000 larger if meters are used rather than millimeters.

Copied from an earlier thread: Verifying developable surfaces

Guassian curvature is a very common metric suggested for assessing if a surface is developable, but it has a significant drawbacks for assessing if a surface is close enough to exactly developable.

An exactly developable surface has exactly zero Guassian curvature. But what is the surface is not “exactly developable” as sometimes happens in design.

How small is small enough for Guassian curvature? That is a non-trivial question, and one that I have rarely seen an answer to.

Is 0.1 small enough? How about 0.001 - that seems like a small number? Or should it be even smaller, perhaps 0.000001?

Gaussain curvature has units of 1/length^2. If millimeters are the length unit then Gaussian curvature has units of 1/mm^2. If meters are used then Gaussian curvature has units of 1/m^2. So the numeric value for the Gaussian curvature of a surface will be 1,000,000 larger if meters are used rather than millimeters. A Guassian curvature of 0.1 if meters are the length units use is the same as 0.009 if feet are used, the same as 0.00065 if inches are used, and the same as and 0.000001 if millimeters are used.

A second drawback to using Guassian curvature to assess if surface is close enough to exactly developable is it is difficult to relate Guassian curvature to other measures such as the amount of twist.

Not that I’ve seen. Allowable amount of Guassian curvature would depend on:
Material
Thickness
Size of panel
Aspect ratio of panel
Fabrication method
Shape of the surface and minimum curvature
Allowable deviation of fabricated item from designed shape
Allowable stresses after fabrication

I use the Curvature command to check for developability. The command shows the principal curvature circles. A developable surface will have one circle with zero curvature, ie a straight line. Deviation from an exact developable surface can be seen by the deviation from a straight line.