Transformation - Mathematics for Computer Graphics - page 82

and

p =( x p i + y p j + z p k )cos α +( a i + b j + c k )( ax p + by p + cz p ) K

bx p ) k ]sin α
p =[ x p cos α + a ( ax p + by p + cz p ) K +( bz p −

+[( bz p − cy p ) i +( cx p − az p ) j +( ay p −

cy p )sin α ] i

+[ y p cos α + b ( ax p + by p + cz p ) K +( cx p − az p )sin α ] j

+[ z p cos α + c ( ax p + by p + cz p ) K +( ay p − bx p )sin α ] k
p = x p a 2 K +cos α + y p ( abK

c sin α )+ z p ( acK + b sin α ) i
+ x p ( abK + c sin α )+ y p b 2 K +cos α + z p ( bcK − a sin α ) j
+ x p ( acK − b sin α )+ y p ( bcK + a sin α )+ z p c 2 K +cos α k

−

and the transformation becomes
⎡ ⎤ ⎡ ⎤ ⎡ ⎤

x p
y p
z p

a 2 K +cos α bK

c sin αacK + b sin α
abK + c sin α 2 K +cos α cK

− x p

y p
z p

⎣ ⎦ = ⎣ ⎦ · ⎣ ⎦

− a sin α

acK − b sin α cK + a sin α 2 K +cos α

where K =1 − cos α .

Now let's approach the problem using transforms and trigonometric identities.
Figure 7.25 shows a point P ( x , y , z ) to be rotated through an angle α to
P ( x ,y ,z ) about an axis defined by v = a i + b j + c k where =1.

The transforms to achieve this operation can be expressed as follows
⎡

v

⎤ ⎡ ⎤

x
y
z

x
y
z

⎣ ⎦ = T 5 × ⎣ ⎦

T 4 × T 3 × T 2 × T 1 ×

which aligns the axis of rotation with the x -axis, performs the rotation of Pα
about the x -axis, and returns the axis of rotation to its original position.

Y

P P'

a

v

b

q

a

f

Z

c

X

Fig. 7.25. The geometry associated with rotating a point about an arbitrary axis.