thoughts on curvature combs,splines control

ok thanks Jeff,
my memory of differential calculus etc is somewhat diminished after 25
yrs.... I think I need to freshen up some : ) .. maybe I will dig out some
old textbooks.
neil

 

Control points on the curve/surface, generally. IIRC the (polygon)
hull vertices/mesh are computed from those and the constraints, if
any, on the curve/surface.

In general, for any curve that must exactly fit N points in one
segment the degree of the curve needs be no greater than N-1.

Thus a line fits two points, a circle (or other conic) three. Lines
have degree 1, as in Y = M*X +B, conics degree 2 as in Y = A*X**2 +B
or (a circle) (X-A)**2 +(Y-B)**2 = R**2 (in the 2D or planar case).

Note that all (AFAIK) CAD or CAD/CAM systems are limited to
polynomial approximations of limited maximum degree. This causes
problems when the desired curve or surface has things like trig or
exponential functions as it's desired basis.

Try to model & then extrapolate a sine curve, as an example.

This is not to say that special API programs cannot deal with such
in many ways. Just that they cannot be well modeled, in general.

Bear in mind that we are talking *exact* fits and analytic curves
(to computational limits) here.

Well, all NURBS software deals with rational polynomials.

Probably pretty close I'd expect. Or three points and two end
tangents. Each system may well supply their users different
tools & terms to get to about the same places.
(Part of what makes jb so amusing ...... new buzzword from a
blurb for something that was old 20 years ago and he wets his
pants & gets confused <G>. IF he had actually ever used any of it he'd
have known it was the same thing with a new buzzword, ad or his
total lack of comprehension.)

Often systems have unique commands to provide the system

Also, consider a cubic polynomial: f1(X) = A*X**3 + B* X**2 + C*X +D
What if A=0?

Consider a second cubic polynomial:
f2(X) = E*X**3 + F* X**2 + G*X +H

A rational cubic would be f1(X)/f2(X). This makes things a bit more
interesting, right?

For a parabola, as an example, we get something like A= E=F=H=0
and H=1 (and to be simple C=0) resulting in B*X**2+D = f1(X)/f2(X)
which we might call Y.

"Smooth" is often hard to define as it means different things to
different people and in different applications.

"Internal knots"? Do you mean hull vertex points that cross the
parent curve?

BTW, Have you ever placesd a spline along the hull vertex points,
then across that spline's hull's vertex points, etc?

There can be a huge number of possible "blends". That poses a
problem IMHO as few systems or people will do such in the same
way.

I'm a bit confused ..... any curve has a tangent at any point on it,
including the ends.
That would be G1/C1.
The *rate of change* of that tangent would be G2/C2 and be the
second derivative of the parent curve (the tangent is the first).

If two curves have a G2/C2 relationship at their common end
point that just means that their second derivatives are the same
*at that point*.

That can work too as 5>3. 3 is the minimum degree that can *have*
a non-constant (=zero) second derivative.

Higher degrees have their uses I expect.
Cubics are fast to compute and work well for most applications.

ComputerVision & UG both support up to degree 17 or 18 IIRC. I
don't know about Pro-E.

As you remain at low degree and add points that the curve must fit
you have to add segments to the curve (or the system does and you
don't see it.) Remember, the degree is 1 less than the number of
points for an exact polynomial (single "segment") fit, in general.

Here's a good, though a bit dated (I suppose) book:
"Computational Geometry for Design and Manufacture"
by I.D. Faux, M.J.Pratt

http://www.amazon.com/exec/obidos/s...ne/qid=1106818409/sr=12-1/102-0340510-2011328

Often some good things here too, if you search a bit:
http://www.mel.nist.gov/
http://www.iges5x.org/ (look for the appendix data IIRC)
http://www.nist.gov/iges/ (look for the appendix data IIRC)

<G>

 

There are some programs that don't have Insert / Conic. Hard to believe,
huh?
---

Defining "smooth" is an easy one for me; no unintended or unwanted
excursions. I'm easy.
---

Generally, polynomial expressions don't interest me. Aside from my lack of
comprehension I just don't see how to apply them using program interface /
inputs. Would be interested in hearing how you apply them. Well, I
lied.... if I could assimilate the calculus I might not have to beg
questions like....

http://www.mcadcentral.com/proe/forum/forum_posts.asp?TID=26084&PN=3

If you have a solution it would be cool. Basically, I want to solve for
"a" to define a catenary in the form; y = a*cosh(x/a)-a, for a given
bounding rectangle, 0,0 is bottom center. From what I've read I think it
might be unsolvable, but even my comprehension of explanations is too
sketchy for me to be sure.
---

Points / control vertices / hull points / knots .... What we have here is
a failure to communicate (to borrow one of my favorite lines).

http://www.rhino3d.com/nurbs.htm covers all the definitions. It also
encapsulates about all the theory I have under my belt.
---

Higher degree entities do have uses. I glimpse elements of a few
(smoothing, manual point editing). "They take longer to compute"; I'd say
it's virtually irrelevant for most applications (ok, what do I know of
"most"?). Isn't a more common and oft'times worse processing bog treating
third degree curves as though they were polylines; increasing the point
density to increase resolution? Or, maybe you are talking about solving
increasing degrees of continuity? Definitely true.

 

Just knowing that about all you can have ARE polynomials can be a
huge help ....

For those with an Engineering bent I'd suggest a text by Thomas.
This is too new and may well be a different Thomas than the one
I recall from HS but check with a library ..... and it has far too
many pages IIRC. Think the 60s ....

http://www.amazon.com/exec/obidos/t...f=sr_1_1/102-0340510-2011328?v=glance&s=books

Actually, I don't recall catenary curves to be anything like that in
form.

I *MAY* come back & look at it again later .... time permitting ...
you want "a" for given X & Y, right?

Another good book:
http://www.amazon.com/exec/obidos/s...pes%2522&store-name=books/102-0340510-2011328

(Be advised that I've heard that there were code errors in the C
code version which was translated from the FORTRAN.)
A search of the Web will find several groups with these books

Quick !!! Hide this from jb!!

Lines are fast to compute. Computers got faster too <G>.
Still, adding compute time & cycles slows things down. People
that insist on overly-complicating things (like CAD models) often

You too. Later.

 

What happens to this entity when it's translated to another
system that supports only cubics & lower?

Is it approxinated in some way by one system or just not
read by the system that got it, thus leaving a gap?

Might explain a few problems if so.

I wonder how jb is coming along with his latest demo fixing
things that he may be unable to save <G>.

 

Oops .... that IS sort of a catenary function. My recall was bad.
Ages have passed.
It's generally y=a*cosh(x/a)+b though.
y = a*cosh(x/a)-a = a*(cosh(x/a)-1)

LET C=x/a
Then y = (x/C)*(cosh(C)-1) = (x/C)*(1 + C**2/2! + C**4/4! + C**6/6!
+ .... -1)
= (x/C)*(C**2/2! + C**4/4! + C**6/6! + ....) ..

None of which helps <G>.

 

BTW, As only polynomial curves can be created curves
that are not polynomial in their basis cannot.

This includes anything with a trig function in
it.
Which is why some look at others a bit oddly when
they think that they can model (many) gears or cams
and then machine them to the CAD design.

There's probably a way to write an applications program
to measure the error between the desired curve & the
CAD model or to create a fair-fitting CAD curve (probably
with a great many control points) that's within an error
tolerance of the desired curve.

There's also probably an entire branch of mathematics <G>.

 

explanations of splines as they would apply to SW

All this curve discussing with Cliff <G> prompted me to spend some time
digging into application...
http://www.mcadforums.com/forums/viewtopic.php?t=2448
.... not SW specific, but I believe the basic methods should be applicable.
The shapes were all defined using conic "arcs" and two (end) point splines
with tangent constraints for loft / blend guide curves between the
paraboloid (is that a word?) nose cap and the cylinderical section.
Altering the shape and trim line on the nose cap (changes the "take off"
tangent direction) and the tangent "weight" of the curves is all that's
done to adjust the shapes. The only three point splines (unconstrained; no
tangent conditions) used were for the windscreen and the trim line for the
bottom windscreen blend. Need to rework the canopy / windscreen
intersection; there's definite mach rumble potential there. 8~) HTH a
bit.

 

"beware the "healer""

Try telling that to jb.
There's usually no way for these things to actually *know*
what is *right* when things are wrong. Too many options ...
and the original data might be wrong too <G>.