Physically Based Animation - Computer Animation: Algorithms and Techniques - page 217

animation, the computation is usually summed over mass points that make up the object ( Eq. 7.39 ).
Notice that angular momentum is not directly dependent on the linear components of the object's
motion.

L ðÞ¼ P ð ð q i ðÞ x ðÞÞm i ð q i ðÞ _ v ðÞ Þ Þ

¼ P RðÞ ð q m i o ðÞ ð ð q ðÞ x ðÞ Þ Þ Þ (7.39)

¼ P m i RðÞ ð ð q o ðÞRðÞ ð q Þ Þ Þ

In a manner similar to the relation between linear force and the change in linear momentum, torque
equals the change in angular momentum ( Eq. 7.40 ). If no torque is acting on an object, then angular
momentum is constant. However, the angular velocity of an object does not necessarily remain con-
stant even in the case of no torque. The angular velocity can change if the distribution of mass of an
object changes, such as when an ice skater spinning on his skates pulls his arms in toward his body to
spin faster. This action brings the mass of the skater closer to the center of mass. Angular momentum
is a function of angular velocity, mass, and the distance the mass is from the center of mass. To main-
tain a constant angular momentum, the angular velocity must increase if the distance of the mass
decreases.

L ðÞ¼ t ðÞ

(7.40)

Inertia tensor
Angular momentum is related to angular velocity in much the same way that linear momentum is
related to linear velocity, ( t ) (see Eq. 7.41 ). However, in the case of angular momentum,

a matrix is needed to describe the distribution of mass of the object in space, the inertia tensor ,
I ( t ). The inertia tensor is a symmetric 3

P ( t ) ¼ Mv

3 matrix. The initial inertia tensor defined for the untrans-
formed object is denoted as I object ( Eq. 7.42 ) . Terms of the matrix are calculated by integrating over the
object (e.g., Eq. 7.43 ) . In Equation 7.43 , the density of an object point, q ¼

( q x , q y , q z ), is r ( q ). For the
discrete case, Equation 7.44 is used. In a center-of-mass-centered object space, the inertia tensor for a
transformed object depends on the orientation, R ( t ), of the object but not on its position in space. Thus,
it is dependent on time. It can be transformed according to the transformation of the object by
Equation 7.45 . Its inverse, which we will need later, is transformed in the same way.

LðÞ¼IðÞo ðÞ (7.41)

2
4

3
5

I xx I xy I xz

I object ¼ I xy I yy I yz (7.42)

I xz I yz I zz

I xx ¼ RRR ðrðqÞðq y

2 2

þ q z ÞÞdxdydz (7.43)

I xx ¼ P m i ðy i I xy ¼ P m i x i y i

2

þ z i

2

Þ

I yy ¼ P m i ðx i I xz ¼ P m i x i z i

2

þ z i

2

Þ (7.44)

I zz ¼ P m i ðx i I yz ¼ P m i x i z i

2 2

þ y i Þ

T

IðtÞ¼RðtÞI object RðtÞ (7.45)