| This class implements the representation of an interpolated Spline curve.
The curve described by such an object interpolates an arbitrary number of
fixed points called nodes. The distance between two nodes should
currently be constant. This is about to change in a later version but it can
last a while as it's not really needed. Nevertheless, if you need the
feature, just write me
a note and I'll write it asap.
The interpolated Spline curve can't be described by an polynomial analytic
equation, the degree of which would be as high as the number of nodes, which
would cause extreme oscillations of the curve on the edges.
The solution is to split the curve accross a lot of little intervals :
an interval starts at one node and ends at the next one. Then, the
interpolation is done on each interval, according to the following conditions :
- the interpolated curve is degree 3 : it's a cubic curve ;
- the interpolated curve contains the two points delimiting the interval.
This condition obviously implies the curve is continuous ;
- the interpolated curve has a smooth slope : the curvature has to be the
same on the left and the right sides of each node ;
- the curvature of the global curve is 0 at both edges.
Every part of the global curve is represented by a cubic (degree-3)
polynomial, the coefficients of which have to be computed in order to meet
the above conditions.
This leads to a n-unknow n-equation system to resolve. One can resolve an
equation system by several manners ; this class uses the Jacobi iterative
method, particularly well adapted to this situation, as the diagonal of the
system matrix is strong compared to the other elements. This implies the
algorithm always converges ! This is not the case of the Gauss-Seidel
algorithm, which is quite faster (it uses intermediate results of each
iteration to speed up the convergence) but it doesn't converge in all the
cases or it converges to a wrong value. This is not acceptable and that's why
the Jacobi method is safer. Anyway, the gain of speed is about a factor of 3
but, for a 100x100 system, it means 10 ms instead of 30 ms, which is a pretty
good reason not to explore the question any further :)
Here is a little piece of code showing how to use this class :
// ... float[] nodes = {3F, 2F, 4F, 1F, 2.5F, 5F, 3F}; Spline3 curve =
new Spline3(nodes); // ... public void paint(Graphics g) { int[] plot =
curve.getPlots(); for (int i = 1; i < n; i++) { g.drawLine(i - 1, plot[i -
1], i, plot[i]); } } // ...
Have fun with it !
Any comments, feedback, bug reports or suggestions will be appreciated.
author:
Jean-Pierre Norguet
version:
$Revison$ updated $Date: 2007-01-07 17:09:09 +0000 (Sun, 07 Jan 2007) $ |
