# Fairness of surface and _CurvatureAnalysis in Rhino?

I would like to ask you question to confirm if I understand “fairness of surface” correctly.

## Question

Are my following three perceptions correct?

1. When evaluating the “smoothness” of a surface, there are 2 indicators: continuity and fairness.
2. When the curvature changes gradually, the surface is fair. When the curvature changes suddenly, it is unfair.
3. In Rhino, fairness is checked with the _ CurvatureAnalysis command. A gradual change in color means fair. A sudden change in color means unfair.
The images below are examples of when fairness is better than continuity for evaluating smoothness.
The 3dm file is continuity vs fairness.3dm (1.5 MB) .

### (2)Curvature Analysis

The surface on the right of the image is relatively unfair because of the rapid color change.

## References

I think it will be useful for people who have the same questions as me, so I will attach a link to the information source that I looked up when I created the question.

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Continuity is well defined and can be directly evaluated explicitly with mathematical formulas.

Fairness is subjective and depends on how the surface behaves compared to what the observer expects. There is no simple, abstract mathematical definition of fairness.

I disagree. A surface on a car may have a very gradual waviness and would be considered not to be fair. But the curvature may be very small and change gradually.

An airfoil shape will typically have rapidly changing curvature near the type but be considered fair.

The Zebra command and the EMap command with fluorescent_tube.bmp can be very powerful tools to evaluate the smoothness and fairness of a surface. I use those commands as my primary tools rather than CurvatureAnalysis. I look at the surface while moving and rotating it.

A difficulty with using CurvatureAnalysis is the colors depends on the scaling used. Depending on the scaling a major problem can be missed or a satisfactory surface can appear to have significant problems.

That depends on the design intent and the context of the surface.

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Are both of these surfaces fair or unfail? Are they acceptable for an automobile outer panel or a high end boat deck?
Surfaces20200319.3dm (239.1 KB)

Didn’t the term derive from the fairing surfaces on airplanes and boats?
IE the surfaces built between wing and body on a plane to improve airflow over the fuselage. Not fair or unfair, just fair or not…

possibly produced with different tools, or maybe two different programs?

Both made with Rhino and started out as identical single span degree 3 surfaces. One was modified in the flatter area. These surfaces are intended to demonstrate that “fairness” in not a simple matter of looking for rapid change of curvature. One corner has intentional tight curvature but I consider the unmodified surface to be fair. The modified area on the one surface does not have tight curvature or rapid rate of change of curvature but for me it would be unacceptable for a show surface.

Try using Zebra with Thinnest stripes to find the area in the blue surface which is not fair. Also use EMap with fluorescent_tube.bmp. Rotate the surface with both and watch what happens in the flatter area. Compare the blue surface to the grey surface. The surfaces are 254.71 x 193.04 x 178.30 inches and the maximum deviation between the surfaces is less than 0.1 inch.

Thank you very much for the detailed explanation and specific examples.

I understood that the important things is the quality of the surface that meet the intent and purpose of the design because Fairness is a subjective thing.

I checked the surfaces of your 3dm.
Yes, the area where the control points are dense (circled in pink) appears to be a little wavy wrinkles in zebra.
Also, I think it is noteworthy that the gray surface has a span of 1 while the blue surface has a span of 5 for U and 6 for V.

Thank you very much.
Due to my inexperience, I didn’t know the term “failing surfaces”.
Now that you’ve taught me these words, I’m able to search and insight them more appropriately.
I would like to thank you once again.

The part that was missing from my understanding seemed to be “what is a high quality surface”.
It turns out that it is good to refer to Alias’s Golden Rule and Theory Builders.

Updated file with a green surface added which is 5 spans x 6 spans with the same arrangement of control points as the blue surface, and as fair and as smooth as the gray surface.
Surfaces20200321.3dm (325.4 KB)

Multiple spans do not mean the surface will not be fair. They do provide opportunity to create unfair surfaces.

The blue surface was created by adding knots to a mirrored copy of the blue surface, and then moving control points to purposely create the unfairness.

Thank you very much. I looked at the file you gave me.
I was surprised to see that the green surface is a very smooth while it is multi span(U:5, V:6).

I have one question.
If you don’t mind me asking, can you tell me how to make a surface like the green surface that is multi span and has good internal continuity?

My understating is that

• Multi span surface has better local controllability, but has worse internal continuity.
• Single span has worse local controllability, but it has better internal continuity.
• It is difficult to achieve both local controllability and internal continuity.

Therefore, if my understanding is correct, in general, it should be difficult to create something like the green surface that you have presented, which is multi span and has good internal continuity.
If you don’t mind, could you tell me how you created such surface?

Very carefully. The green surface was intended as a demonstration that multi-span does not automatically mean unfair surfaces, nothing more. It was made by inserting knots into a single span surface.

Correct if comparing a higher degree single span surface and a lower degree multispan surface when both have the same number of control points. But what level of continuity is needed?

A different way to think about this is for a given level on curvature continuity required what are the alternatives between degree of the surfaces, single span vs multispan and the resulting number of control points. That though is a topic for a different thread.

Very carefully.

Your carefulness and skill is amazing…!

That though is a topic for a different thread.

Yes. I’ll continue learning and may open a new thread for that.

Thank you very much again.

Thanks but it is a mistake to assume the multi-span surface in the previous post is an indicator of my skill or patience… The method I used to make it multi-span,the InsertKnot command, increases the number of control points without changing the overall shape.

Thank you, I understand.
I’m sorry for confusing you due to my lack of English and Rhino experience.