Physically Based Animation - Computer Animation: Algorithms and Techniques - page 219

Colliding bodies
As objects move relative to one another, there are two issues that must be addressed: (1) detecting the
occurrence of collision and (2) computing the appropriate response to those collisions. The former is
strictly a kinematic issue in that it has to do with the positions and orientations of objects and how they
change over time. The latter is usually a dynamic issue in that forces that are a result of the collision are
computed and used to produce new motions for the objects involved.
Collision detection considers the movement of one object relative to another. In its most basic form,
testing for a collision amounts to determining whether there is intersection in the static position of two
objects at a specific instance in time. In a more sophisticated form, the movement of one object relative
to the other object during a finite time interval is tested for overlap. These computations can become
quite involved when dealing with complex geometries.
Collision response is a consideration in physically based simulation. The geometric extent of the
object is not of concern but rather the distribution of its mass. Localized forces at specific points on
the object impart linear and rotational forces onto the other objects involved.
In dealing with the time of collision, there are two options. The first is to proceed as best as one can
from this point in time by calculating an appropriate reaction to the current situation by the particle
involved in the collision (the penalty method). This option allows penetration of the particle before
the collision reaction takes place. Of course, if the particle is moving rapidly, this penetration might
be visually significant. If multiple collisions occur in a time interval, they are treated as occurring
simultaneously even though handling them in their correct sequential order may have produced differ-
ent results. While more inaccurate than the second option, this is simpler to implement and often gives
acceptable results.
The second option is to back up time t i to the first instant that a collision occurred and determine the
appropriate response at the time of collision. If multiple collisions occur in a time interval, then time is
backed up to the point at which the first collision took place. In complex environments in which col-
lisions happen at a high rate, this constant backing up of time and recomputing the motion of objects
can become quite time-consuming.
There are three common options for collision response: a strictly kinematic response, the penalty
method, and the calculation of an impulse force. The kinematic response is quick and easy. It pro-
duces good visual results for particles and spherically shaped objects. The penalty method introduces
a temporary, nonphysically based force in order to restore nonpenetration. It is typically used when
response to the collision occurs at the time step when penetration is detected (as opposed to backing
up time). The advantage of this technique is that it is easy to compute and the force is easily in-
corporated into the computational mechanism that simulates rigid body movement. Calculating
the impulse force is a more precise way of introducing a force into the system and is typically used
when time is backed up to the point of first contact. Detecting collisions and reacting to them are
discussed next.
Particle-plane collision and kinematic response
One of the simplest illustrative situations to consider for collision detection and response is that of a
particle traveling at some velocity toward a stationary plane at an arbitrary angle (see Figure 7.14 ).
The task is to detect when the particle collides with the plane and have it bounce off the plane.
Because a simple plane is involved, its planar equation can be used ( Eq. 7.51 ). E ( p )istheplanar