The statement you're referring to when you say
"So when you talk about adding cv's without adding knots"
may have been inaccurate. I'm assuming it had to do with
elevating bezier curve degree(? G2 constraints Alias vs. Pro/E).
Any statement to the effect of "adds CVs vs. knots" should have
said "internal" or "mid span" knots. More below ...
> degree, cv's, and knots were all dependent on each other.
They are, but it can be confusing.
> "control points are a list of
> at least degree+1 points"
It is important to focus on "at least" and know that describes a bezier curve,
i.e. four CVs for a degree 3, five CVs degree six, etc. It must be remembered
that a bezier has no 'internal knots', e.g. all of the knots are are at either
end of the curve (see 'multiplicity' below).
(It may be worth noting ...
. the "order" of the curve is degree + 1 as well and
. the CV 'count' for a bezier is 0 -> degree, i.e. degree 3 is 0,1,2,3.
If that confuses more than clarifies; forget it.)
> "knots are a list of degree+N-1 numbers,
> where N is the number of control points"
What might be missing to make the whole comprehensible is the term 'full
muliplicity' or 'fully multiple' knots. The typical NURBS curve, to include
beziers and b-splines, as used in typical(?) CAD has fully multiple knots at
each end.*
So for a degree 3 bezier the expression would be; [3 + 4 - 1] or 6, three on
each end.
[*
I don't know how 'accurate' that statement is and it should be noted
that not all splines have fully multiple knots at each end.
(This is a very informative treatment of basic concepts and definitions
if you don't have a copy)
NURB Curves: A Guide for the Uninitiated, P.J. Schneider
[url]http://devworld.apple.com/dev/techsupport/develop/issue25/sc hneider.html[/url]
Look at fig. 9 & 13 for quick ref, note the ends of the curve relative CVs.
I think I've also seen something similar in CADDS NURBS manual, but would
have to search.
Some older systems(?) placed degree + 1 knots on each end of a curve. IGES
uses that convention for compatibility. Newer systems add on write or
disregard on read the extra knot, should you ever come across that confusing
little bit of trivia.
]
If we create a b-spline (by 'picking' points between start and end) or if a
bezier is converted to b-spline (to satisfy imposed Constraint conditions)
a CV is added in conjunction with each knot. So ...
_ For a degree three b-spline defined by picking three points
[degree + N - 1] = 3 + 5 - 1 = 7 = three each end and one in the middle.
_ For a degree 4 b-spline defined by picking three points
[degree + n - 1] = 4 + 6 - 1 = 9 = four each end and one in the middle.
And so forth.
_ _ _ _
If that doesn't seem to make sense, holler back. I may have, again, confused
something in the telling.
