Transformations - Transformations and Projections in Computer Graphics - page 44

u

u

P

C

θ

C

θ

P*

r *

P*

φ

O

r

P

Q

(a) (b)

Figure 1.26: A General Rotation.

Using Equations (A.3) and (A.5) (page 222), we can rewrite this as r ∗ =( uu T ) r +
cos θ r cos θ ( uu T ) r +sin θ Ur ,where

−

⎛ ⎞

0 −u z u y

⎝ ⎠ .

U = u z 0 − u x

− u y u x 0

The result can now be summarized as r ∗ = Mr ,where

M = uu T +cos θ ( I uu T )+sin θ U

− (1.32)

⎡ ⎤

u x +cos θ (1 − u x ) u x u y (1 − cos θ ) − u z sin θu x u z (1 − cos θ )+ u y sin θ

⎣ ⎦

cos θ )+ u z sin θu y +cos θ (1 u y )

= u x u y (1 − − u y u z (1 − cos θ ) − u x sin θ .

u x u z (1 − cos θ ) − u y sin θu y u z (1 − cos θ )+ u x sin θu z +cos θ (1 − u z )

Direction cosines .If v =( v x ,v y ,v z ) is a three-dimensional vector, its direction
cosines are defined as

v x
| v |

v y
| v |

v z
| v |

N 1 = , N 2 = , N 3 = .

These are the cosines of the angles between the direction of v and the three coordinate
axes. It is easy to verify that N 1 + N 2 + N 3 =1. If u =( u x ,u y ,u z )isaunitvector,

then =1and u x , u y ,and u z are the direction cosines of u .

It can be shown that a rotation through an angle

| u |

θ is performed by the transpose
M T . Consider the two successive and opposite rotations r ∗ = Mr and r = M T r ∗ .On
the one hand, they can be expressed as the product r = M T r ∗ = M T Mr .Onthe
other hand, they rotate in opposite directions, so they return all points to their original
positions; therefore r must be equal to r .Weendupwith r = M T Mr or MM T

−

= I ,