Physically Based Animation - Computer Animation: Algorithms and Techniques - page 229

toward the supporting object, and if the supporting object is considered immovable, such as the ground
or a table, the normal force is immediately and completely canceled by a force equal and opposite in
direction that is supplied by the supporting object. If the supporting object is movable, then the normal
force is applied transitively to the supporting object and added to the forces being applied to the sup-
porting object. If the normal force is directed away from the supporting object, then it is simply used to
move the object up and away from the supporting object.
The component of the force applied to the object that is parallel to the surface is responsible for
sliding (or rolling) the object along the surface of the supporting object. If the object is initially sta-
tionary with respect to the supporting object, then there is typically some threshold force, due to static
friction , that must be exceeded before the object will start to move. This static friction force, F s ,isa
function of the normal force, F N ( Eq. 7.65 ) . The constant m s is the coefficient of static friction and is a
function of the two surfaces in contact.

F s ¼ m s F N (7.65)

Once the object is moving along the surface of the supporting object, there is a kinetic friction that
works opposite to the direction of travel. This friction creates a force, opposite to the direction of travel,
which is a linear function of the normal force,

F k , on the object ( Eq. 7.66 ). The constant m k is the coef-
ficient of kinetic friction and is a function of the two surfaces in contact.

F k ¼ m k F N (7.66)

Resting contact
Computing the forces involved in resting contact is one of the more difficult dynamics problems for com-
puter animation. The exposition here follows that found in Baraff's work [ 1 ] . The solution requires access
to quadratic programming, the implementation of which is beyond the scope of this topic. An example
situation in which several objects are resting on one another is shown in Figure 7.26 .
For each contact point, there is a force normal to the surface of contact, just as in colliding contact.
The objective is to find the magnitude of that force for a given configuration of objects. These forces (1)

n 2 ( t 0 )

p 3

p 2

n 3 ( t 0 )

n 5 ( t 0 )

n 1 ( t 0 )

n 4 ( t 0 )

p 1 p 4 p 5

FIGURE 7.26
Multiple-object resting contacts.