Physically Based Animation - Computer Animation: Algorithms and Techniques - page 199

7.1 Basic physics—a review
The physics most useful to animators is that found in high school text books. It is the physics based
on Newton's laws of motion. The basics will be reviewed here; more in-depth discussion of specific
animation techniques follows. More physics can be found in Appendix B.7 as well as any standard
physics text (e.g., [ 6 ]).
The most basic (and most useful) equation relates force to the acceleration of a mass, Equation 7.1 .
f ¼ m a

(7.1)

When setting up a physically based animation, the animator will specify all of the possible forces
acting in that environment. In calculating a frame of the animation, each object will be considered
and all the forces acting on that object will be determined. Once that is done, the object's linear accel-
eration (ignoring rotational dynamics for now) can be calculated based on Equation 7.1 using
Equation 7.2 .

f
m

a ¼ (7.2)

From the object's current velocity, v , and newly determined acceleration, a new velocity, v 0 , can
be determined as in Equation 7.3 .

v 0 ¼ v þ a D t

(7.3)
The object's new position, p 0 , can be updated from its old position, p , by the average of its old veloc-
ity and its new velocity. 1

1
2 ð v þ v 0 Þ D t

p 0 ¼ p þ (7.4)

Notice that these are vector-valued equations; force, acceleration, and velocity are all vectors (indi-
cated in bold in these equations).
There are a variety of forces that might be considered in an animation. Typically, forces are applied
to an object because of its velocity and/or position in space.
The gravitational force, f, between two objects at a given distance, d , can be calculated if their
masses are known ( G is the gravitation constant of 6.67 3 10 11 m 3 s 2 kg 1 ). This magnitude of
the force,

f , is applied along the direction between the two objects ( Eq. 7.5 ).

k¼G m 1 m 2
d

f ¼k f (7.5)

2

If the earth is one of the objects, then Equations 7.1 and 7.5 can be simplified by estimating the
distance to be the radius of the earth ( r e ), using the mass of the earth ( m e ), and canceling out the mass
of the other object, m o . This produces the magnitude of the gravitation acceleration, a e , and is directed
downward (see Eq. 7.6 ).

m o ¼ G m e

f

8 meter
sec

a e ¼

e ¼

9 : (7.6)

r

2 2

1 Better alternatives for updating these values will be discussed later in the chapter, but these equations should be familiar to
the reader.