Hermite splines
The Bezier spline assumed that
you know the point at the beginning and end of each spline segment,
as well as two intermediate influencer points.
If instead you want to start with positions
P0
and
P1
at both the beginning and end of the spline segment,
as well as velocity
R0
and
R1
at both the beginning and end of the spline segment,
then you have a Hermite spline.
The problem to solve remains the same:
Convert the four numbers you have
along each coordinate into the four cubic polynomial coefficients
(a,b,c,d) that will allow you to compute at3+bt2+ct+d.
To do this we need four cubic primitive functions
that allow us selective control over different parts of the curve:
| function
| value
at t=0
| value
at t=1
| velocity
at t=0
| velocity
at t=1
|
| 2t3-3t2+1
| 1 | 0 | 0 | 0
|
| -2t3+3t2
| 0 | 1 | 0 | 0
|
| t3-2t2+t
| 0 | 0 | 1 | 0
|
| t3-t2
| 0 | 0 | 0 | 1
|
We can use the above functions to describe the value along
the spline for any given t:
(2t3-3t2+1) P0 +
(-2t3+3t2) P1 +
(t3-2t2+t) R0 +
(t3-t2) R1
To get the four cubic polynomial coefficients (a,b,c,d)
for x,y or z
we can rewrite the above as a matrix-vector multiplication:
ax
bx
cx
dx
|
| ←
|
|
| 2
| -2
| 1
| 1
|
| -3
| 3
| -2
| -1
|
| 0
| 0
| 1
| 0
|
| 1
| 0
| 0
| 0
|
|
| •
|
|
Px at t=0
Px at t=1
Rx at t=0
Rx at t=1
|
Now to find each coordinate value x,y or z
at a given t,
along that spline segment
we can just evaluate a cubic polynomial:
x =
axt3 +
bxt2 +
cxt + dx
y =
ayt3 +
byt2 +
cyt + dy
z =
azt3 +
bzt2 +
czt + dz
Catmull-Rom splines
If you want a spline that simply goes through
a given set of points, you can use the Catmull-Rom spline.
Given a set of points
P0, ...
Pn-1,
we can compute the smooth Catmull-Rom spline segment
between
Pi and Pi+1 as follows:
ax
bx
cx
dx
|
| ←
|
|
| -1/2
| 1
| -1/2
| 0
|
| 3/2
| -5/2
| 0
| 1
|
| -3/2
| 2
| 1/2
| 0
|
| 1/2
| -1/2
| 0
| 0
|
|
| •
|
|
Pi-1
Pi
Pi+1
Pi+2
|
For the first or last segment,
Pi-1
or
Pi+2,
respectively,
can just be set to the first
point
P0
or the last point
Pn-1.
To create a looping spline:
- For the first segment, when i==0: replace Pi-1 by Pn-1
- For the last segment, when i==n-2: replace Pi+2 by P0
Then, as usual,
to find each coordinate value x,y or z
at a given t
along that spline segment
we can just evaluate a cubic polynomial:
x =
axt3 +
bxt2 +
cxt + dx
y =
ayt3 +
byt2 +
cyt + dy
z =
azt3 +
bzt2 +
czt + dz
At the end of class we watched some video clips containing CGI effects