Piecewise Polynomial Interpolation

(c)  1999-2013 Stephen H. Noskowicz All Rights Reserved

  Revised & Updated Jan 17 2013

The FULL text is now available at:
 http://home.comcast.net/~k9dci/site/?/page/Piecewise_Polynomial_Interpolation/

PREFACE - What got me started on this...

  One of my hobbies is a small laser light show system I designed, built and 
wrote the software for in the early '80s. I do demos, small shows locally and 
I have students display their simple images on one of my projectors in a 
microprocessor course I taught.  Sometime later, the light show professional 
who originally interested me in laser graphics saw one of my animations.  
I did something very simple which turned out looking quite good - a bouncing star.  
He asked me if I used splines.  I queried, "What's a spline?", to which he 
replied, "Never, mind."  So I dropped it, but kept the word in the back of my mind.

  A few years later, another friend gave me a copy of the SIGGRAPH '84 
video tape showing the Kochanek-Bartels curve drawing method, also called TCB.  
This is a cousin to a very common curve method you may have heard of called Bezier.  
I thought it could dramatically reduce the work in laser show animations.  I thought 
it was extremely interesting, but well beyond my understanding.
  Key-frame animation is a technique where the master animator only draws a few 
important (key) frames of an animation and a junior animator, or in this case 
the computer, smoothly fills-in the frames in-between (“tweening”). Very 
similar techniques are now widely used in many aspects of animated films. 

  Several years later, I started reading the original Internet Usenet "group", 
comp.graphics.algorithms, searched around on the net and got copies of the 
leading Graphics texts (Foley/van Dam et al.   - Rogers/Adams - Bartels/Beatty/Barsky).   
Texts were either purely math, or graduate level graphics books that helped a 
little, but they all used varying terminology.
  I had hoped, but failed to find a book explaining the basics and thought that 
one must certainly exist with a collection of the various common interpolation 
methods, but found nothing of the kind.

  Early in 1996, I decided to look at what I had collected, and started to slowly 
figure things out.  Conversations, started on the Internet over questions 
relating to interpolation, prompted me to begin recording my thoughts for future 
reference and this is the result.  The very book I wanted originally.

  This work overtook my laser show work and to this date, because my focus has 
changed, I have not used these techniques on laser show animations.

  Parts of my explanation of the Kochanek-Bartels method have been referenced 
on Wikipedia:
 http://en.wikipedia.org/wiki/Kochanek%E2%80%93Bartels_spline 

Regards, Steve, Noskowicz