Interpolation/Extrapolation/Splines

[Michael Steinberg]

Hi! I'm sorry for stressing you with this, but I had some problems when looking for infos for my work in mathematics.
The title is "Interpolation/Extrapolation", subtitle "Spline-curves". Now, when I looked through the internet, one mentioned Bezier-Splines, another one Bezier-Curves etc. Now, is a Bezier curve a special case of splines, or are there Bezier curves AND bezier curves?

Can anyone recommend me a tutorial where I can read about the theory of Spline and Bezier-Splines not about the actual implementation?

I know that this is a bit unrelated, but any answers are highly appreciated! Thanks!
Michael

[Michael Steinberg]

After looking another few minutes through the online material, it gets more confusing. One guy shows an example where the spline curve goes through the given control points, where the other guy shows an example where the curve does not go through the control points.
I get the feeling that the word "spline" does not mean anything different than "any kind of curve".

[Elixer]

Well, the Catmall-Rom (sp?) spline is the one that goes through all control points. Wait, actually, I think there are some others that do it to, like hermite and koch something or another ). Bezier splines don't go through all the control points though. There are like 7 or 8 different versions of Bezier, all with different names. It really depends if what you need with them, and what properties you need. (For example Bezier curves have a convex hull defined by control points, Catmull-Rom doesnt)

As for reference, I remember this stuff from Computer Graphics principles and practice (otherwise known as the bible for gfx guys )
Maybe you can look up stuff by Foley, vandam, and I forgot the other guys. Sorry.

[Michael Steinberg]

Well, I'd need any interpolative splines, and I would suppose that are the ones that go through the points. So far I saw hermite splines( p(t) = [x(t),y(t)] ), explicite splines ( y=f(X) ). Implicite splines seem to be very unpractical. I'll look for your authors! Thanks Elixer!
If there's a book which covers all interresting types, I would buy that as well! Any ideas? Thanks!

[XBCT]

Hi!
Michael: Do you do this work for a "Facharbeit"?
I´m just asking ´cause I thougt about doing Bezier stuff for my "Facharbeit" ´cause I already implemented them in my Q3 clone so it wouldn´t be too much work....

Greets, XBTC!

[Michael Steinberg]

Yes, XBCT, it's for my "Facharbeit". After all I came to the conclusion, that bezier spline don't belong to my topic, since they don't interpolate. They would have to go through the "Stützpunkte" to interpolate.
What topic do you have XBCT?

[Michael Steinberg]

Hey, you wrote your own nick incorrectly!

[martin_marinov]

Hi,
The common definition for spline has nothing to do with interpolation/aproximation. A spline is a continius function defined over sequence of sucessive knots[t0,t1,..tn], which is piecewize polynomial and has a fixed defgree of continuity equal to the degree of the polynomials.
The simple Bezier curve is a polynomial curve, and thus is a only a very limited spline - curve; The composite Bezier curve is a another example of a spline curve, although it is not a optimal one in term of continuity.
It's a well known fact, that the spline functions give one of the best methods to doing interpolation/aproximation of numerical (point) data. The so called NURBSplines are the most commonly used basis functions for calculating spline curves, although there some different approaches mainly in subdivision area.
If you wish to know more about spline function, you better read one book in numerical methods or/and computational science. I haven't seen any good guide in the web for that . If you wish to see some algorithms for calcuating/displaying NURBS curves, I can send you some from my university "Facharbeits" .
Regards (Gute Wuenschen
Martin

[Michael Steinberg]

K, what I currently think is, that all splines which go through the control points, are interpolative. Any kind of bezier splines doen't belong to this class. I will most probably buy a book this week, but I'm not too sure about which one. Which books will cover Interpolation, and will the ones always cover splines? Which part of mathematics is it?

Thanks!
Michael

[Michael Steinberg]

Oh, the resource our teacher provided has a word on splines.
It sais a spline is defined with these criterias:

1. It interpolates the controlpoints
2. It is identical to a polynom of maximum 3rd degree between two control points
3. It min. two times differencible (well, to the hell with english...)

That article also has a word that interpolation would be equal to finding a function which will go through the control-points. So, since bezier splines don't go through the controlpoints (most times), the can't be splines. What is it all about. Are there many different definitions out in the world? That is driving me to hell.