Interpolation/Extrapolation/Splines

[Michael Steinberg]

Oh wait.
The article sais that that would have been the definition of a spline-function of a control-point problem. So that seems to go conform with your explanation.
Since I've got Interpolation/Extrapolation as main topic, I will only need the splines which will go through the control-points. So bezier splines fall out. Is there anywhere a list of all existant splines going through the control points?
Well, the I still need to know in which part of mathematics these problems fall (interpolation with normal polynoms and splines), so I can look for a book! If anybody has a hint, I would be very grateful!
The internet tutorials only focus on how to render curves (mostly bezier), but I need the theory. Have got 5 weeks left. Argh.

[martin_marinov]

Ok - the definition which your teacher gave you is not very exact : I think he meant a cubic interpolation spline which is a partial case of a spline curve.
But in fact this is one of the most used ))
This part of mathematics is called numerical analysis(methods) or computational science.
But probably you need to have some knowledge in linear algebra, calculus and diff. geometry to understand a mathematical book on this topic.
The simple Bezier Curve interpolates only its first and last point - in fact the bezier curve is a polynom, not a "real" spline. But the polynom is a partial case of the spline... ). The polynoms which interpolate a given set of point are the Lagrange polynom, and more generaly the Hermite polynom.
I think you teacher wants the classic interpolation cubic spline curve, which is calculated by solving a linear equation system mapping the interpolated control points to the another set, which is used with aproximation algorithm to generate the interpolation curve. However the process is a bit complicated, so can't explain it here.(At least it deserves writing some math formulas, not only plain text ) So try to find a good book on the topic...

Regards
Martin

[Sven Clauw]

hi,
i actually follow a (introduction) course on curves at the university (in belgium). We covered 3 types of curves: polynomal curves (using Lagrange polynomals), Bézier curves using Bernstein polynomals) and B-Spline curves (using normalized B-spline polynomals).

Polynomal curves go through the controlpoints, but show heavy oscillations when the degree is high.

Beziercurves are smooth, but require composition of beziercurves to model complex surfaces or curves and when a controlpoint is changed, the whole curve is affected.

B-Splines are better as they interpolate the controlpoints best (using quadratic B-splies) and smoothly (using cubic B-Splines) and can easily be used to model complex surfaces. Also, when a controlpoint is moved, only a small portion of the curve is affected.
B-Splines are used for NURBS : Non-Uniform Rational B-Splines.
If you want more information i can look it up in my course-notes.

Sven

[martin_marinov]

So the book I had here is "Curves And Surfaces for Computer Aided Geometric Design" by Gerald Farin
Academic Press. Inc
G. Farin is one of the most famous researchers in the area. His home page is : http://eros.eas.asu.edu/~farin/personal/gf.html
You may find there something about.
The book contains a fully explanantion on the topic and source code too. So I suggest you to go for it.

Regards
Martin

[Michael Steinberg]

Oh ****, martin. That contains really a LOT. Is there any chance, that most types of splines and curves covered in this do not go through the control points?

I appreciate any help here. You're a big aid for me!
Michael

[martin_marinov]

It has a chapter specially for the spline curves you need - they are called cubic interpolation spline curves. It contains source code also.
If you wish we can discuss it further on ICQ or e-mail
ICQ#:21770883
medo@sirma.bg

[XBCT]

Hi!
Sorry for answering that late....

>What topic do you have XBCT?<

I don´t have a topic so far and I have to make a proposition myself for the topic of my "Facharbeit"....
After reading your post I thougt it would be cool to do something about "Beziers" ´cause I already implemented them and therefore it wouldn´t be too much work.

>Hey, you wrote your own nick incorrectly!<

Ahrrrrg....I *DON`T* wrote my nick incorrectly! The problem is that I wrote it wrong as I created my account. I´m called XBTC everywhere else....

Greets, XBTC!

[This message has been edited by XBCT (edited 02-21-2001).]