Graphics Reference
In-Depth Information
x
(
t
)
t
˙
x
(
t
)
Figure 35.28: Visualization of
D
for the state space of one 1D particle under some arbitrary
forces.
Consider the tangents to solution curves for
t
versus
X
, like those in
Figure 35.27. They define some vector field over the time-state space such that
if the universe we are simulating has solution
X
, the value of this vector field is
X
(
t
)
at
(
t
,
X
(
t
))
.Formally,let
D
be the vector-valued function such that
D
(
t
,
Y
)=
X
(
t
)
if
Y
=
X
(
t
)
,
∀
physically possible
X
.
(35.63)
In the context of physical simulation under Equation 35.62, this means that
D
(
t
,
Y
)=
Y
[
n
+
1..2
n
]
f
(
t
,
Y
)
m
,
(35.64)
+
j
(
t
,
Y
)
Δ
t
although in the following derivation we will make no assumptions about
D
,
X
,
or
X
, so as to keep the discussion valid for all ordinary differential equations.
From here on, these will simply be arbitrary functions, which may not even be
vector-valued, and our goal is to trace a flow curve through the tangent field.
We chose the letter “D” because the tangent field function is reminiscent of
the derivative
X
. It is not actually the derivative:
D
is a field over time-state space
(as shown in Figure 35.28) and
X
is the tangent function to one particular curve
X
through that space. That is, every solution
X
is a
flow curve
of the field
D
.
If we were following a particular solution
X
and somehow stepped off that
curve, the
D
field would guide us along some other flow curve that quite likely
diverged from our original solution. Figure 35.29 depicts two instances of this
situation for a simple tangent field. Although it is undesirable, this will prove to
be unavoidable during our numerical evaluation, and the best that we can do is to
try to minimize divergence and encourage transitions that are at least plausible.
35.6.6.2 Following the Tangent Field
Consider a series of
state vectors
{
Y
1
,
Y
2
,
...}
such that
Y
i
≈
X
(
t
i
)
, where the
sampling period
t
i
+
1
−
t
i
=Δ
t
is constant. These are the regularly spaced samples
of the state function
X
that we want the simulation system to provide.
