# Curves fairness

Curves fairness is a strange problem in Computer Aided Geometric Design and every author gives his personal definition. One of the simplest is the following:

A curve is said to be fair if (1) it is G2 continuous, (2) it has no undesirable inflection points, and (3) its curvature varies in an even way.

Yes, but why is a curve with these properties fair ? It is only a definition and I can define what I want. I need a theorem to state something in mathematics. Where are theorems about fairness ? The problem is that in mathematics I can speak about sets,functions and other mathematical objects, but not about fairness.

How can you solve this problem ?

• This is a duplicate question Basically its because you want the reflective properties of the surface to continue the same way over the gap. Otherwise humans will interpret a sudden change as a crease Commented Jul 2, 2016 at 17:40
• Although this question mentions two additional properties in addition to the previous question, it is not clear what is being asked. If a term has multiple conflicting definitions, then each one is probably for a specific purpose. Without knowing your intention it is difficult to judge what you need to know. Commented Jul 2, 2016 at 18:30
• @trichoplax actually all the 3 definitions seem to say the same thing. Curvature must not suddenly change. 1 specifies how deep your continuity must be 2 that it may not abruptly change 3 that the change must not be too fast. All of those have been discussed n the duplicate, i would agree. But yes the question is not so clear. Also note G2 curvature does not apply to curves as such for silhouettes it could be less than G2 is its not too abrupt change. However CAD deals with surfaces a curve must have same properties as a surface it models or it will destroy the surfaces continuity requirement. Commented Jul 2, 2016 at 18:56
• @joojaa I don't have the knowledge to judge how close to being a duplicate this is - thanks for the extra information. If this is closed as a duplicate then people searching for the terms in this question will still be redirected to the previous question, so we just need to judge whether anything new would be present in answers to this new question. Commented Jul 2, 2016 at 19:22
• They aren't 3 definitions. It is an unique definition; all 3 conditions must hold. Commented Jul 3, 2016 at 10:03