Now I understand your answer!

I can not give you anything else than a single like to your post, even though I owe you a lot for this given knowledge and an answer to my question.

Thank you David!

Hello @davidcockey

Can I ask just one more question?

The rhino help link you gave me:

http://docs.mcneel.com/rhino/5/help/en-us/commands/curvatureanalysis.htm

provides explanations about Gaussian curvature values, where positive Gaussian curvature value means the surface is bowl-like. And negative value means the surface is saddle-like. Zero value means the surface is flat in at least one direction:

But there is no similar explanation for Mean curvature values.

Can it be said that:

- positive Mean curvature value means the surface is saddle-like
- negative Mean curvature value means the surface is bowl-like
- zero Mean curvature value means the surface is flat in both directions (not at least one)?

You can download nice document prepared by Rajaa Issa. Personally I like this document as quick reference.

http://files.mcneel.com/grasshopper/1.0/docs/en/TheEssentialMathematics_ThirdEdition_rev3.zip

and go to page 62 in pdf or 57 in document.

Thank you Nosorozec

Have you perhaps thought of page 63 in pdf or 58 in document, instead?

Because on page 62 in pdf or 57 in document, this is the only mention of Mean curvature:

The principal curvatures are used to compute the Gaussian and mean curvatures of the surface.

Wich does not answer my upper question.

Here is a screen shot of the page 63 in pdf or 58 in document, instead:

This page only mentions this when it comes to Mean curvature values:

Any point with zero mean curvature has negative or zero Gaussian curvature.

But what about positive Mean curvature values, and negative Mean curvature values?

The “mean curvature” displayed by the CurvatureAnalysis command is always non-negative. The command line report for the command decribes it as “unsigned”. It appears to be the absolute value of the mathematical mean curvature. If it is zero then the surface at that point is either flat or a “minimal surface” which is “saddle like”.

The “mean curvature” reported at a point in the command line by the Curvature command is signed. If it is zero then the surface at that point is either flat or a “minimal surface” which is “saddle like” at the point analyzed. If it is positive then the surface will be concave when viewed from the side the surface normal is pointing to or “saddle like” at the point analyzed at the point analyzed, If it is negative then it will either be convex when viewed from the side the surface normal is pointing to or “saddle like” at the point analyzed.

Edited with correction

.

@davidcockey thank you for the detailed reply!

It looks like the whole confusion is caused due to _CurvatureAnalysis command, which for some reason threats negative mean curvature values as positive - like you noticed, it shows the absolute mean curvature value.

I posted this question to another Rhino user on different forum who is skillful in programming. He used some Rhino Common command which evaluates the Mean curvature at particular point. This Rhino Common command:

http://developer.rhino3d.com/api/RhinoCommonWin/html/P_Rhino_Geometry_SurfaceCurvature_Mean.htm

He then evaluated the Mean curvature at a couple of points and made an analysis mesh from it with a legend. That is the number 3) photo:

Number 1) photo is a surface for which he performed the Mean analysis. When _Curvature command is applied to the two red points of that surface, they show the same results as the 3) photo (the same results for “Mean curvature”).

Photo 2) has been made by using Rhino’s _CurvatureAnalysis command.

As you already mentioned, for some reason: negative mean curvature values are always threated as positive, which is what caused the confusion.

So can we conclude:

If I use the _Curvature command, or that SurfaceCurvature.Mean Rhino Common programming command, and if the following values are returned:

- negative Mean curvature values: then that part of the surface is bowl-like - concave (viewed from the side of the surface normal)
- positive Mean curvature values: then that part of the surface is saddle-like - convex (viewed from the side of the surface normal)
- zero Mean curvature value: then that part of the surface is flat in both directions

Is this incorrect?

Here is the .3dm file with all three mentioned images:

Thank you.

Incorrect.

*negative Mean curvature values: then that part of the surface is bowl-like - convex (viewed from the side of the surface normal)* **or saddle like with one principal curvature positive and one principle curvature negative.**

*positive Mean curvature values: then that part of the surface is saddle-like - concave (viewed from the side of the surface normal)* **or saddle like with one principal curvature positive and one principle curvature negative.**

*zero Mean curvature value: then that part of the surface is flat in both directions* **or a minimum surface with the principle curvatures equal in magnitude but opposite in sign**

Is there a reason you want to use mean curvature rather than Guassian curvature to analyze whether the shape is concave, convex or saddle-like?

Edited: concave and convex corrected.

Thank you for the reply @davidcockey.

Because when a user is using either the _CurvatureAnalysis or _Curvature or Rhino Common command, the Gaussian curvature always shows the saddle-like or bowl-like parts of the surface to have similar curvature values.

However, If when a user uses the _Curvature function and mentioned Rhino Common programming commands, Mean curvature will show the difference between the saddle-like or bowl-like parts of the surface both visually (different colors) and by returning positive and negative curvature values, as can be seen on previously attached .3dm file and last two photos.

A bowl-like area of a surface will have a positive Gaussian curvature. A saddle-like area of a surface will have a negative Gaussian curvature. Depending on the scaling used for the display the Gaussian curvature may appear to be “similar”. This illustration shows the areas with negative Gaussian curvature in blue which are saddle-like. The areas in red are bowl-like, and the areas in green are close to being curved in direction or flat.

The sign of mean curvature does not reliably distinguish between bowl-like and saddle-like areas of a surface. In the illustrations in the post above with the surface color coded with mean curvature most of the areas with both positive and negative mean curvature are “bowl-like”.

A negative mean curvature only means that at least one of the principal curvatures is negative. The other principal curvature could also be negative if the surface is bowl-like and convex when viewed from the side of the surface normal. Or the surface could be saddle like with the other principal curvature could be positive but smaller in magnitude than the negative mean curvature, so that the average is negative,

A positive mean curvature only means that at least one of the principal curvatures is positive. The other principal curvature could also be positive if the surface is bowl-like and concave when viewed from the side of the surface normal. Or the surface could be saddle like with the other principal curvature could be negative but smaller in magnitude than the positive mean curvature, so that the average is positive,

A mean curvature is defined as H=(P1+P2)/2.

So according to this formula the negative mean curvature exists when (P1+P2)<0, positive mean curvature when (P1+P2)>0 and zero mean curvature when (P1+P2)=0. In last case it means the principal directions P1 and P2 have the same magnitude but opposite sign P1=-P2 and such surface is called minimal surface.

The negative mean curvature tells us the one principal direction is more negative then the second one (except case when both P1 and P2 are negative).

http://help.autodesk.com/cloudhelp/2016/ENU/Alias-Tutorials-Legacy/images/GUID-C27ECC73-50DE-41F6-8052-B5FAEB6CD789.png

The Gaussian curvature is defined as K=P1*P2 and is useful

- for determining the inflection points
- for decision if the surface is developable (K=0)

excerpt from:

http://brickisland.net/cs177/?p=144

https://knowledge.autodesk.com/support/alias-products/getting-started/caas/CloudHelp/cloudhelp/2016/ENU/Alias-Tutorials-Legacy/files/GUID-E44AB3A3-18DF-42F4-B4D4-870EFB42534A-htm.html

http://homepages.inf.ed.ac.uk/rbf/BOOKS/FSTO/node28.html

Thank you for the complete replies both @davidcockey and @Nosorozec !!!

Which curvature type should be used to for visually making a difference between the convex and concave shapes of the surface?

Not Gaussian, not mean?

There are actually 3 cases which must be distinguished.

- k>0 convex
- k>0 concave
- k<=0

where k is Gaussian curvature. So maybe display signed mean curvature as a false color but mask the entire region of k<=0 with grey, or draw the k=0 contour on the display so that the user can distinguish the positively curved areas.

@GregArden I appreciate your reply.

But if convex and concave shapes are both when k, Gaussian curvature > 0, then how can the difference between these two be identified?

Signed mean curvature can differentiate these cases if k>0.

What is a signed mean curvature?

Rhino _CurvatureAnalysis command supports only: Gaussian, Mean, Max radius, Mean radius.

Mean curvature is usually defined as the average of the minimum and maximum curvature. Curvature can be either positive or negative. Therefore mean curvature as usually defined can be either positive or negative.

CurvatureAnalysis shows the absolute value of curvature which is always positive. In this discussion “signed mean curvature” means as usually defined which can be positive or negative rather than the absolute value.

Thank you for the explanation @davidcockey !

But I still do not understand which curvature type should be used for visually making a difference between the convex and concave shapes of the surface?

If Guassian curvature is negative then the surface is anticlastic or saddle-like at that location and convex/concave do not apply.

If Guassian curvature is positive then the surface is synclastic or dome/bowl like at that location and signed mean curvature can be used for differentiating between convex and concave. However the CurvatureAnalysis command currently only shows the magnitude of the mean curvature but not the sign so it does not help. The Curvature command can be used to examine which way a surface is curved at a point.

Thank you for the clarification David!

Sorry for taking me too long to see this reply.