The Plane - Vector Analysis for Computer Graphics - page 114

Let the plane equation be

ax + by + cz = d

Therefore, the surface normal n is given by

n = a i + b j + c k

Let Qxyz be the nearest point on the plane to P. Therefore,

q = x i + y j + z k

and

n · q = d (4.12)

Let

− QP

= r = n

Therefore,

q = p − n (4.13)

To find q , we need to find .
Using n , take the dot product of Eq. (4.13):

n · q = n · p − n · n (4.14)

Substitute Eq. (4.12) in Eq. (4.14):

d = n · p − n · n

and

n · p − d n · p − d

=

n =

2

n ·

n

If

n = 1

then

= n · p − d

Q is given by

q = p − n (4.15)