Physically Based Animation - Computer Animation: Algorithms and Techniques - page 226

Tangential component of velocity ( v tangent )

point p at t i 1

Velocity ( v )

point p at t i

Normal to surface

Normal component of velocity ( v n )

Components of a particle's velocity colliding with a plane

v n

Resulting velocity ( k v n ) v tangent

( k . v n )

Computing velocity resulting from a collision ( k is the coefficient of restitution)

FIGURE 7.22
Impact response of a point with a plane.

velocity after the collision in the direction normal to the surface of contact, N ( Eq. 7.56 ) . Impulse is a

vector quantity in the direction of the normal to the surface of contact, , the equa-

tions of how the velocities must change based on the equations of motion before and after the collision
are used. These equations use the magnitude of the impulse, j , and they are used to solve for its value.
v rel ðtÞ¼k v rel ðtÞ

J ¼ jN. To compute J

(7.56)

Assume that the collision of two objects, A and B , has been detected at time t . Each object has a
position for its center of mass ( x A ( t ), x B ( t )), linear velocity ( v A ( t ), v B ( t )), and angular velocity ( v A ( t ),
v B ( t )). The points of collision ( p A , p B ) have been identified on each object (see Figure 7.23 ).
At the point of intersection, the normal to the surface of contact,

N

, is determined depending on
whether it is a vertex-face contact or an edge-edge contact. The relative positions of the contact points
with respect to the center of masses are

r B , respectively ( Eq. 7.57 ) . The relative velocity of the
contact points of the two objects in the direction of the normal to the surface is computed by
Equation 7.58 . The velocities of the points of contact are computed as in Equation 7.59 .

r A and

r A ðtÞ¼ p A ðtÞ x A ðtÞ

(7.57)

r B ðtÞ¼ p B ðtÞ x B ðtÞ

v rel ðtÞ¼ðð_ p A ðtÞ_ p B ðtÞÞ N Þ N (7.58)