I got a bit confused, as I found some sources stating that the curvature unit is radians / linear unit (m,ft,mm…).
Perhaps this is strictly related with curve curvature?
While surface curvature is always expressed in 1 / linear unit (m,ft,mm)?
“Radians” are somewhat odd when it comes to units. It can be defined as a ratio which is independent systems of units of length, time, mass, etc and the associate absolute standards (platinum bar in a vault, number of oscillations of a particular atom, etc).
If curvature is determined using derivatives then the result has units of 1/length units and radians are not involved. Likewise if curvature is determined by fitting a arc then it also has units of 1/length units and radians are not involved. However if curvature is determined using angle change and length then the units are radians/length units. The magnitude of the curvature does not depend on whether radians are included or not.
A surface may have a mean curvature of 0.0005 1/mm or 0.0005 radians/mm. They are the same. However if meters are used instead of millimeters then the mean curvature would be 0.5 1/m or 0.5 radians/m.
Is there a particular reason why including or not including radians makes a difference in your work? You can consider curvature to have units of radians/length units or 1/length units. The number is exactly the same.
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:
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.
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.
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:
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 directionsor 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?
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
There are actually 3 cases which must be distinguished.
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.