There are a lot of technical details and assumptions wrapped up in this topic. I’ll start from the beginning in case somebody unfamiliar with the setting reads this thread.
Why do we care if a curve is subd-friendly?
If you are using a Rhino command like SubDLoft and you want the subd surface to exactly pass through the the curves, then you need to use subd-friendly curves that have the same number of control points.
How do you make a subd-friendly curve?
Run the Rhino Curve command and set the option SubDFriendly=Yes.
Now, some terminology and details.
A “closed” curve starts and ends at the same point, like a circle.
An “open” curve starts and ends at different points, like a line.
Every kind of subd-friendly curve is a non-rational uniform cubic.
Non-rational means the NURBS control point weights have to be 1.
Uniform means the NURBS knots increase the the same amount, typically 1, as you go from one knot to the next. (Most users never see or care about the knots.
Cubic means the NURBS curve has degree=3.
You can use the Rhino What command to see what properties a NURBS curve has.
Here is a degree 4, rational, non-uniform example.
Here is a SubD-friendly example.
The context for talking about the 1/3-2/3 rule is NURBS curves that are:
- open
- cubic
- non-rational
- uniform
A NURBS curve that is open, cubic, non-rational, and uniform is subd-friendly if and only if the 2nd derivative at the start and end point is zero. This condition is sometimes called “natural.” In this sense the word “natural” has been assigned a technical mathematical meaning that is a strong restriction. This is often counter intuitive to the way the word “natural” is typically used in any ordinary conversation about curve appearance or aesthetics.
Many Rhino users, if not the vast majority, are not comfortable talking about things like the 2nd derivative of a curve. It turns out there is another way to state this condition.
If you have an open, cubic, non-rational, uniform NURBS curve AND the 1st, 2nd, and 3rd control points are colinear AND the distance from the 2nd to the 3rd is twice the distance from the 1st to the second, then the 2nd derivative is zero at the start. This is still a mouthful, BUT, it can be described visually as the 1/3 - 2/3 rule.
Everything I just said about the 1st, 2nd and 3rd control points and the start of the curve also applies to the last three control points and the end of the curve.
How is this useful?
If you need SubD-friendly curves and you’re starting out with ordinary curves, looking at the first 3 control points and seeing how close they are to satisfying the 1/3 - 2/3 rule will help you understand how far away your curve is from being subd-friendly.