Game Engine Architecture

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12.4.1.2. Separability of Linear and Angular Dynamics

An unconstrained rigid body is one that can translate freely along all three
Cartesian axes and that can rotate freely about these three axes as well. We say
that such a body has six degrees of freedom (DOF).
It is perhaps somewhat surprising that the motion of an unconstrained
rigid body can be separated into two independent components:

z Linear dynamics. This is a description of the motion of the body when

we ignore all rotational eff ects. (We can use linear dynamics alone to

describe the motion of an idealized point mass—i.e., a mass that is infi ni-

tesimally small and cannot rotate.)

z Angular dynamics. This is a description of the rotational motion of the body.

As you can well imagine, this ability to separate the linear and angular com-
ponents of a rigid body’s motion is extremely helpful when analyzing or sim-
ulating its behavior. It means that we can calculate a body’s linear motion
without regard to rotation—as if it were an idealized point mass—and then
layer its angular motion on top in order to arrive at a complete description of
the body’s motion.

12.4.1.3. Center of Mass

For the purposes of linear dynamics, an unconstrained rigid body acts as
though all of its mass were concentrated at a single point known as the center
of mass (abbreviated CM, or sometimes COM). The center of mass is essen-
tially the balancing point of the body for all possible orientations. In other
words, the mass of a rigid body is distributed evenly around its center of mass
in all directions.
For a body with uniform density, the center of mass lies at the centroid of
the body. That is, if we were to divide the body up into N very small pieces,
add up the positions of all these pieces as a vector sum, and then divide by the
number of pieces, we’d end up with a prett y good approximation to the loca-
tion of the center of mass. If the body’s density is not uniform, the position of
each litt le piece would need to be weighted by that piece’s mass, meaning that
in general the center of mass is really a weighted average of the pieces’ positions.
So we have

,

ii ii

CM ii

i

i

mm

m m

∀∀

∀

==

∑∑

∑

rr

r

where the symbol r represents a radius vector or position vector —i.e., a vector
extending from the world space origin to the point in question. (These sums

12.4. Rigid Body Dynamics