Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 479

radius R and mass M , the inertia tensor is given by Equation B.145 . If the inertia tensor is known at one
position by I , then the inertia tensor I 0 for parallel axes at a new position ( X , Y , Z ) relative to the original
position is calculated by Equation B.146 . If the inertia tensor for one set of axes is known by I , then the
inertia tensor for a rotated frame is calculated by Equation B.147 ,where R is the rotation matrix describ-
ing the rotated frame relative to the original frame.

2
4

3
5

I xx I xy I xz

I yx I yy I yz

I ¼ (B.141)

I zx I zy I zz

I xx ¼ R ðy

2 2

þ z Þdm

I yy ¼ R ðx

2

þ z

2

Þdm

I zz ¼ R ðx

2 2

þ y Þdm

(B.142)

I xy ¼ R xydm
I xz ¼ R xzdm
I yz ¼ R yzdm

2
4

3
5

I xx 00
0 I yy 0
00 I zz

I ¼ (B.143)

2
4

3
5

2 2

Mðb þ c Þ 0 0

1
12

2 2

I ¼ 0 Mða þ c Þ 0 (B.144)

2 2

0 0 Mða þ b Þ

2
4

3
5

100
010
001

2

2 MR

I ¼ (B.145)

5

2 3

2 2

I xx þ MðY þ Z ÞI xy MXY I xz MXZ

4 5

2 2

I translated ¼ I xy MXY I yy þ MðX þ Z ÞI yz MXYZ (B.146)

2 2

I xz MXZ I yz MZYX I zz þ MX þ Y

I rotated ¼ RI object R 1 (B.147)

B.8 Numerical integration techniques
Numerical integration is useful for finding the arc length of the curve, updating arbitrary function
values using derivative information, and specifically updating the position of an object over time.
A useful technique for arc length computation is Gaussian quadrature. With regard to general function
value updating, Runge-Kutta, explicit Euler integration, and implicit Euler integration are discussed.
Huen, Verlet, and Leapfrog integration are covered specifically in the context of position update using a
known acceleration. As with many numerical techniques, Press et al. [ 16 ] is an extremely valuable
reference.