<p>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:</p>

<p>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.</p>

<p>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.</p>

<p>How can you solve this problem ?</p>

CC BY-SA 3.0
2016-07-02T17:40:24.233Z
This is a [duplicate question](http://computergraphics.stackexchange.com/questions/1669/why-for-perfect-reflections-a-surface-must-have-g2-continuity) 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
joojaa
CC BY-SA 3.0
2016-07-02T18:30:09.230Z
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.
trichoplax is on Codidact now
CC BY-SA 3.0
2016-07-02T18:56:37.067Z
@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.
joojaa
CC BY-SA 3.0
2016-07-02T19:22:00.157Z
@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.
trichoplax is on Codidact now
CC BY-SA 3.0
2016-07-03T10:03:59.617Z
They aren't 3 definitions. It is an unique definition; all 3 conditions must hold.
Valerio
CC BY-SA 3.0
2016-07-04T08:18:30.957Z
@Valerio yes and if you read the linked post you would know why. Basically eyes structure will see up to second derivate. For obvious reasons this means inflection will be seen and uneven variation will be seen by a human because of the nature of the system.
joojaa
<curve>
Curves fairness
Valerio
# Curves fairness

(Asked by: username_0 on 2016-07-02)

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 ?

**Tags:** curve

> **Comments:**
> **username_1**: This is a [duplicate question](http://computergraphics.stackexchange.com/questions/1669/why-for-perfect-reflections-a-surface-must-have-g2-continuity) 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  
> **username_2**: 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.  
> **username_1**: @username_2 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.  
> **username_2**: @username_1 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.  
> **username_0**: They aren't 3 definitions. It is an unique definition; all 3 conditions must hold.  
> **username_1**: @username_0 yes and if you read the linked post you would know why. Basically eyes structure will see up to second derivate. For obvious reasons this means inflection will be seen and uneven variation will be seen by a human because of the nature of the system.  

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