Discussion of Homework on Subdivision Curves
You showed an impressive effort and interesting results in this assignment,
and I bet you all learned a whole lot in trying to invent your own
subdivision schemes.
Some of you implemented -- or reinvented -- known schemes, with predictable
good results.
The quadratic and cubic B-splines got "rediscovered" by a few,
and the cubic interpolatory scheme was adopted by several.
I am particularly impressed by those of you who dared to cook up their
own combination of coefficients,
often with less robust results -- but probably got deeper insights
into the nature of subdivision in trying.
Some even added an extra slider to study the influence of a parameter
on their scheme, -- this I consider A+ work !
When you invent a new scheme, and have to judge whether "it works" and
how successful it might be,
it is important to be critical and to subject it to really stressful
tests,
i.e., sharp corners with uneven lengths of the attached control polygon
sides.
Any "misbehavior" of your scheme should alert you to check your basic
concept, as well as your implementation.
Thes are some of the checks that you should apply, and warning signs
which may indicate that something is wrong:
-
The number of points should increase by a factor >1.0 in each generation,
typically by a factor 2.0.
-
The curve should remain connected.
-
The curve should become smoother and smoother.
-
At interpolating points, you should get at least G1 continuity.
-
The curve should not develop additional inflection points after the first
subdivision step.
-
The (control-)points along the curve should become more and more evenly
spaced.
In general, when you design any methods or systems, explicitly write down
your expectations,
and then check your results against those expectations.