Game Development Reference
In-Depth Information
Spline interpolation
Spline interpolation is a form of interpolation where the interpolant is a special
type of piecewise polynomial called a spline. Spline interpolation is preferred over
polynomial interpolation because the interpolation error can be minimized. Also,
spline interpolation is better for a higher degree of polynomials because it avoids
Runge's phenomenon
(
http://en.wikipedia.org/wiki/Runge's_phenomenon
)
found in polynomial interpolation. It is used when the function changes over the
value of
x
. For example, if
x>0
, then
f(x)=x
2
, and if
x<0
, then
f(x)=1/x
2
. Piecewise
polynomial refers to a different function for ranges of
x
.
Spline interpolation and linear algebra are complex topics. Refer to the link
http://www.vis.uni-stuttgart.de/~kraus/LiveGraphics3D/cagd/
. This web
page contains good Java applets (
Chapter 8
,
B-spline Curves
on the web page) to
understand spline interpolation.
We are not very interested in spline interpolation but very interested in its specific
type called B-spline interpolation. This is the most used form of interpolation in
tweening. Basically, when we want to generate a curved path, we can use this kind
of interpolation.
B-spline or Basis spline is defined by its order, a set of weighted control points and
a knot vector. The control points determine the shape of the curve. Typically, each
point of the curve is computed by taking a weighted sum of a number of control
points. However, the curve usually does not go through the control point.
P
6
t
7
Q7
P
4
Q6
t
8
P
7
P
5
t
6
P
1
Q5
t
3
t
5
P
0
Q4
P
3
t
4
control point
Q3
knot
P
2
