Normalized Plane
(Unit Normal Vector)
| Non-normalized Plane
|
|---|
Ax + By + Cz + D = 0
N = Ai + Bj + Ck
|
Distance d from a point P to the plane
|
d = | N · P + D |
|
d2 = |
( N · P + D )
( N · P + D ) |
|
N · N |
| |
d = |
| N · P + D | |
|
|N| |
|
Closest point P' on the plane to a point P
|
P' = P - ( N · P + D )
N
|
P' = P - |
( N · P + D ) N |
|
N · N |
|
Intersection of the line L(t) = B + t M
with the plane
|
|
|
Projection V' of a vector V onto the plane.
(Subtract from the vector its projection onto the normal.)
|
V' = V - (N · V) N
|
|
The line L(t) = B + t M of intersection
between planes N1, D1 and
N2, D2
|
det = ( N1 · N2 ) ( N1 · N2 ) -
( N1 · N1 ) ( N2 · N2 )
Note: N1 || N2 => det = 0
s1 = |
( N2 · N2 ) D1 - ( N1 · N2 ) D2 |
|
det |
| |
s2 = |
( N1 · N1 ) D2 - ( N1 · N2 ) D1 |
|
det |
M = N1 × N2
B = s1N1 + s2N2
|
|
|
|
Line
|
|---|
L(t) = B + tM
|
Closest point on the line from a point P
|
|
|
Closest point on line L1(t)
to line L2(u)
|
N = M2 × ( M2
× M1 )
D = - ( N · B2 )
|