Rotation Transforms in Space - Rotation Transforms for Computer Graphics - page 139

and

p

p ) cos α.

−− NQ

= − n ( n ·

To find −− QP :
Let

n ˆ × p = w

where

| w |=|ˆ n |·| p | sin θ =| p | sin θ

but

| r |=| p | sin θ

therefore,

| |=| |

w r .

Now

QP
NP =

QP
|

QP
|

= = sin α

r | w |

therefore,

−− QP =

= ˆ ×

w sin α n p sin α

then

p ) + p p ) cos α + ˆ

p = ˆ n ( ˆ n · − ˆ n ( ˆ n · n × p sin α

and

p = p cos α + ˆ n ( ˆ n · p )( 1 − cos α) + ˆ n × p sin α.

This is known as the Rodrigues rotation formula, as it was developed by the French
mathematician, Olinde Rodrigues (1795-1851), who had also invented the ideas be-
hind quaternions before Hamilton. This has been documented by Simon Altmann in
the Mathematics Magazine under the title “ Hamilton, Rodrigues and the quaternion
scandal ”[7].
If we let

K = 1 − cos α

then

p = p cos α + ˆ n ( n ˆ · p )K + ˆ n × p sin α

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

+ (bz p − bx p ) k sin α cy p ) i + (cx p − az p ) j + (ay p −

= x p cos α

cy p ) sin α i

+

a(ax p + by p +

+

(bz p −

cz p )K

+ y p cos α az p ) sin α j

+ b(ax p + by p + cz p )K + (cx p −