Graphics Reference
In-Depth Information
Although
X
will be a computationally convenient representation, there is also
some physical motivation for it. Both position and velocity (as a proxy for momen-
tum) are included in the state vector because they are properties of objects that
appear in mechanical models of the real world. The laws of physics tell us how to
compute forces, which are proportional to acceleration. The inputs to force func-
tions are always position and velocity. Forces never turn out to be proportional to
acceleration, so we don't explicitly store acceleration in the state. The exception
is the “normal” force model, for which we have already described the problems
arising from including acceleration as an input.
Note that
X
(
t
)
describes the second and third parameters of the
f
and impulse
functions; we can redefine them to take only two parameters and thus write
X
(
t
)=
x
(
t
)
x
(
t
)
=
X
(
t
)[
n
+
1. .2
n
]
f
(
t
,
X
(
t
))
m
.
(35.62)
+
d
j
(
t
,
X
(
t
))
dt
When considering multiple particles with varying mass, we could replace
f
/
m
M
−
1
for some diagonal matrix of masses
M
. Note that differential Equa-
tion 35.62 is the system version of the single-particle relation previously intro-
duced in Equation 35.22.
Chapter 29 framed light transport in the rendering equation, which was an inte-
gral equation. As was the case there, considering general-purpose numeric meth-
ods will lead us to broader computer science than needed for the specific dynamics
problem at hand. This has two advantages. We can build on previous work that is
not graphics-specific, including not just mathematics but also numerical ODE-
solving software libraries. We can also take the algorithms that we develop and
apply them to other problems beyond dynamics, both in computer graphics and in
other fields.
with
f
·
35.6.6.1 Time-State Space
Let's look at a concrete example to gain some intuition for the 2
n
+
1-dimensional
time-state space defined by
t
,
X
(
t
)
.
Figure 35.27 shows the path (described by its
y
-coordinate) of a cannonball
that is modeled as a particle in a 1D universe. The upper-left plot shows the famil-
iar time-space path plot of
x
(
t
)
versus
t
, which is a parabola. The upper-right plot
shows the time-velocity path plot of
x
(
t
)
versus
t
, which is linear because the ball
experiences constant negative acceleration. This path crosses
x
(
t
)=
0 at the apex
of the cannonball's flight. The lower-left image combines these to show the time-
state path plot of
X
(
t
)
versus
t
. Keep in mind that
X
describes the entire universe,
not just one object. In this example, they happen to be the same because we're
considering a universe with a single particle. Were there more particles, the plot
would have many more dimensions.
The thick line in the lower-left subfigure shows one particular
X
of the many
that could exist. The thin lines are its projection onto the
(
t
,
x
)
- and
(
t
,
x
)
-planes as
“shadows” for reference. Those shadows are exactly the upper-left and upper-right
figures.
From the perspective of
t
=
0, the dark
(
t
,
X
(
t
))
-curve describes the entire
future fate of the universe. From the perspective of some point at the high end of
the timeline, this is the history of the universe. But that curve is only one possi-
ble reality. Were there different initial conditions,
X
would have been a different
function and traced a different curve.
The lower-right subfigures depicts three alternative curves that
could have
been
solutions to
X
, given different initial conditions. Here only the shadows on
