Graphics Reference
In-Depth Information
Under this new force model, the force on a ball with velocity v = 0 m/s is
m/s 2 , while the force on a ball that is already falling with velocity
9.81 kg
·
m/s 2 . This new force model still
does not depend on time or position. But we could imagine a more sophisticated
model that includes air currents, whose effects depend on the location in space
time at which the current is experienced.
Now, consider the ball moving in three spatial dimensions. How do the types
of the functions change? Positions and velocities are now 3-vectors:
v =
2 m/s has lower magnitude:
8.81 kg
·
R 3
x : R
(35.27)
f : R 1
R 3
R 3
R 3 :( t , y , v )
×
×
f ( t , y , v )
(35.28)
: R 7
R 3 :( t , y , v )
f ( t , y , v )
(35.29)
The force function still takes three arguments. Because it is always applied to
position and velocity, people sometimes write f ( t , x , x ) , treating the x and x func-
tions as if they were variables. 2 Recall the pitfalls of that notation identified earlier
in the chapter. So, although you may see that as a convenience in animation notes
and research papers, we will continue to be careful to distinguish the function and
its value here.
35.6.3 Piecewise-Constant Approximation
Assume for the moment that the force function is constant with respect to time, and
that acceleration is therefore also constant. In a physics textbook on Newtonian
mechanics, the final position of a body experiencing constant acceleration would
be expressed as
x 1 = x 0 + v 0 t + 1
2 a 0 t 2 ,
(35.30)
x ( t )= x ( 0 )+ x ( 0 ) t + 1
2 x ( 0 ) t 2 , and
(35.31)
= x ( 0 )+ x ( 0 ) t + 1
2
f ( 0, x ( 0 ) , x ( 0 ))
m
t 2 ,
(35.32)
where the second version follows our notation. The right side is quadratic in t
and all other factors are constant, so this describes a parabola. That is the arc of a
thrown ball, which experiences essentially constant acceleration due to gravity, so
this matches our intuition for the position of an object as a function of time. These
equations arise from integrating Equation 35.21 under the constant acceleration
assumption:
2. We could make this notation meaningful by redefining force as a higher-order function,
f : R × ( R R 3
) R 3 , but contorting ourselves to make the notation
consistent is not useful because in practice we will apply force function to points and
velocities from the previous frame, not functions over all time.
) × ( R R 3
 
 
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