Physically Based Animation - Computer Animation: Algorithms and Techniques - page 227

A ( t )

v B ( t )

p A

x A ( t )

p B

B ( t )

x B ( t )

v A ( t )

FIGURE 7.23
Configuration of colliding objects.

p A ðtÞ¼ v A ðtÞþo A ðtÞ r A ðtÞ
_

(7.59)

p B ðtÞ¼ v B ðtÞþo B ðtÞ r B ðtÞ

ð v A ; w

B

The linear and angular velocities of the objects before the collision Þ are updated by the

ð v A ; w þ

B

impulse to form the linear and angular velocities after the collision . The linear velocities are

updated by adding in the effect of the impulse scaled by one over the mass ( Eq. 7.60 ). The angular
velocities are updated by computing the effect of the impulse on the angular velocity of the objects
( Eq. 7.61 ).

Þ

j N
M A

v A ðtÞ¼ v A ðtÞþ

(7.60)

j N
M B

v B ðtÞ¼ v B ðtÞþ

o A ðtÞ¼o A ðtÞþI A ðr A ðtÞjNÞ
o B ðtÞ¼o B ðtÞþI B ðr B ðtÞjNÞ

(7.61)

To solve for the impulse, form the difference between the velocities of the contact points after
collision in the direction of the normal to the surface of contact ( Eq. 7.62 ) . The version of
Equation 7.59 for velocities after collision is substituted into Equation 7.62 . Equations 7.60 and
7.61 are then substituted into that to produce Equation 7.63 . Finally, substituting into Equation 7.56
and solving for j produces Equation 7.64 .
v A ðtÞ¼ðð p A ðtÞ p B ðtÞÞ

N Þ N (7.62)

v A ðtÞþo A ðtÞ v rel ðtÞ¼ v B ðtÞþo B

N ð r A ðtÞð r B ðtÞÞÞ (7.63)

ðð 1 þ kÞv rel

j ¼ (7.64)

M A þ

1

M B þ N ðI 1

1

A ðtÞð r A ðtÞ N Þ r A ðtÞI 1 B ðtÞð r B ðtÞ N Þ r B ðtÞÞ

Contact between two objects is defined by the point on each object involved in the contact and the
normal to the surface of contact. A point of contact is tested to see if an actual collision is taking place.
When collision is tested for, the velocity of the contact point on each object is calculated. A collision is
detected if the component of the relative velocity of the two points in the direction of the normal indi-
cates the contact points are approaching each other.