Physically Based Animation - Computer Animation: Algorithms and Techniques - page 213

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FIGURE 7.11
Euler method and midpoint method. (A) Using step sizes of 2 with the Euler method. (B) Using step sizes of 2 with
the midpoint method.

Equations of motion for a rigid body
To develop the equations of motion for a rigid body, several concepts from physics are presented first
[ 12 ]. The rotational equivalent of linear force, or torque , needs to be considered when a force is applied
to an object not directly in line with its center of mass . To uniquely solve for the resulting motions of
interacting objects, linear momentum and angular momentum have to be conserved. And, finally, to
calculate the angular momentum, the distribution of an object's mass in space must be characterized
by its inertia tensor . These concepts are discussed in the following sections and are followed by the
equations of motion.

Orientation and rotational movement
Similar to linear attributes of position, velocity, and acceleration, three-dimensional objects have rota-
tional attributes of orientation, angular velocity, and angular acceleration as functions of time. If an
individual point in space is modeled, such as in a particle system, then its rotational information
can be ignored. Otherwise, the physical extent of the mass of an object needs to be taken into consid-
eration in realistic physical simulations.
For current purposes, consider an object's orientation to be represented by a rotation matrix, R ( t ).
Angular velocity is the rate at which the object is rotating irrespective of its linear velocity. It is repre-
sented by a vector, o ( t ). The direction of the vector indicates the orientation of the axis about which
the object is rotating; the magnitude of the angular velocity vector gives the speed of the rotation in
revolutions per unit of time. For a given number of rotations in a given time period, the angular veloc-
ity of an object is the same whether the object is rotating about its own axis or rotating about an axis
some distance away. Consider a point that is rotating about an axis that passes through the point and
that the rate of rotation is two revolutions per minute. Now consider a second point that is rotating