Transformation - Mathematics for Computer Graphics - page 81

To find −− NQ

−− NQ = NQ

NQ
NP NP r =

r =cos α · r

but

.

p = n + r = n ( n p )+ r

therefore

. p )

r = p − n ( n

and

−− NQ =[ p − n ( n

. p )] cos α

To find −−→ QP

Let

n × p = w

where

w = 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 = w sin α =( n × p )sin α

then

. p )+[ p − n ( n . p )] cos α +( n × p )sin α

p = n ( n

and

. p )(1 − cos α )+( n × p )sin α

p = p cos α + n ( n

Let

K =1 − cos α

then

p = p cos α + n ( n

.

p ) K +( n × p )sin α