Physically Based Animation - Computer Animation: Algorithms and Techniques - page 209

It should be noted that the difference between material commonly taught in standard physics texts and
that used in computer animation is in how the equations of motion are used. In standard physics, the
emphasis is usually in analyzing the equations of motion for times at which significant events happen,
such as an object hitting the ground. In computer animation, the concern is with modeling the motion
of objects at discrete time steps [ 12 ] as well as the significant events and their aftermath. While
the fundamental principles and basic equations are the same, the discrete time sampling creates numerical
issues that must be dealt with carefully. See Appendix B.7 for equations of motion from basic physics.

7.4.1 Bodies in free fall
To understand the basics in modeling physically based motion, the motion of a point in space will be
considered first. The position of the point at discrete time steps is desired, where the interval between
these time steps is some uniform Dt . To update the position of the point over time, its position, velocity,
and acceleration are used, which are modeled as functions of time, ( t ), respectively.

If there are no forces applied to a point, then a point's acceleration is zero and its velocity remains
constant (possibly non-zero). In the absence of acceleration, the point's position,

x

( t ),

v

( t ),

a

( t ), is updated by its
velocity, v ( t ), as in Equation 7.17 . A point's velocity, v ( t ), is updated by its acceleration, a ( t ). Accel-
eration arises from forces applied to an object over time. To simplify the computation, a point's accel-
eration is usually assumed to be constant over the time period Dt (see Eq. 7.18 ) . A point's position is
updated by the average velocity during the time period Dt . Under the constant-acceleration assumption,
the average velocity during a time period is the average of its beginning velocity and ending velocity, as
in Equation 7.19 . By substituting Equation 7.18 into Equation 7.19 , one defines the updated position in
terms of the starting position, velocity, and acceleration ( Eq. 7.20 ) .

x

x t þ Dt ð Þ x ðtÞþ v ðtÞDt (7.17)

v t þ Dt ð Þ v ðtÞþ a ðtÞDt (7.18)

v ðtÞþ v t þ Dt ð Þ

x t þ Dt ð Þ x ðtÞþ Dt (7.19)

2

v ðtÞþ v ðÞþ a ðÞDt
2

x t þ Dt ð Þ x ðtÞþ Dt

(7.20)

1
2 a ðÞDt

2

x t þ Dt ð Þ x ðtÞþ v ðÞDt þ

A simple example
Using a standard two-dimensional physics example, consider a point with an initial position of (0, 0),
with an initial velocity of (100, 100) feet per second, and under the force of gravity resulting in a uni-
form acceleration of (0, 232) feet per second. Assume a delta time interval of 1/30 of a second (corre-
sponding roughly to the frame interval in the National Television Standards Committee video). In this
example, the acceleration is uniformly applied throughout the sequence, and the velocity is modified at
each time step by the downward acceleration. For each time interval, the average of the beginning and
ending velocities is used to update the position of the point. This process is repeated for each step in
time (see Eq. 7.21 and Figures 7.7 and 7.8 ).