Three-dimensional transformations can be defined synthetically as follows: point X on a "subjective" 3-space must be transformed to a point T also on the subjective space. The transformations uses these elements: a pair of "observation points" P and Q , and an "objective" 3-space. The subjective and objective spaces and the two points all lie in four-dimensional space, and the two 3-spaces can intersect at some plane.
Draw line l 1 through points X and P . This line intersects the objective space at point R . Draw line l 2 through points R and Q . Line l2 intersects the projective plane at point T . Then T is the transform of X .
Let
X
:
(
x
,
y
,
z
,
0
)
,
{\displaystyle X:(x,y,z,0),}
T
:
(
T
x
,
T
y
,
T
z
,
0
)
,
{\displaystyle T:(T_{x},T_{y},T_{z},0),}
P
:
(
P
x
,
P
y
,
P
z
,
P
t
)
,
{\displaystyle P:(P_{x},P_{y},P_{z},P_{t}),}
Q
:
(
Q
x
,
Q
y
,
Q
z
,
Q
t
)
.
{\displaystyle Q:(Q_{x},Q_{y},Q_{z},Q_{t}).}
Let there be an "objective" 3-space described by
t
=
f
(
x
,
y
,
z
)
=
m
x
+
n
y
+
k
z
+
b
{\displaystyle t=f(x,y,z)=mx+ny+kz+b}
Draw line l 1 through points P and X . This line intersects the objective plane at R . This intersection can be described parametrically as follows:
(
1
−
λ
1
)
X
+
λ
1
P
=
(
R
x
,
R
y
,
R
z
,
m
R
x
+
n
R
y
+
k
R
z
+
b
)
.
{\displaystyle (1-\lambda _{1})X+\lambda _{1}P=(R_{x},R_{y},R_{z},mR_{x}+nR_{y}+kR_{z}+b).}
This implies the following four equations:
R
x
=
x
+
λ
1
(
P
x
−
x
)
{\displaystyle R_{x}=x+\lambda _{1}(P_{x}-x)}
R
y
=
y
+
λ
1
(
P
y
−
y
)
{\displaystyle R_{y}=y+\lambda _{1}(P_{y}-y)}
R
z
=
z
+
λ
1
(
P
z
−
z
)
{\displaystyle R_{z}=z+\lambda _{1}(P_{z}-z)}
R
t
=
λ
1
P
t
=
m
R
x
+
n
R
y
+
k
R
z
+
b
{\displaystyle R_{t}=\lambda _{1}P_{t}=mR_{x}+nR_{y}+kR_{z}+b}
Substitute the first three equations into the last one:
(
m
x
+
n
y
+
k
z
)
+
λ
1
(
m
P
x
+
n
P
y
+
k
P
z
−
m
x
−
n
y
−
k
z
−
P
t
)
+
b
=
0
{\displaystyle (mx+ny+kz)+\lambda _{1}(mP_{x}+nP_{y}+kP_{z}-mx-ny-kz-P_{t})+b=0}
Solve for λ1 ,
λ
1
=
−
(
b
+
m
x
+
n
y
+
k
z
)
m
(
P
x
−
x
)
+
n
(
P
y
−
y
)
+
k
(
P
z
−
z
)
−
P
t
=
λ
1
N
λ
1
D
.
{\displaystyle \lambda _{1}={-(b+mx+ny+kz) \over m(P_{x}-x)+n(P_{y}-y)+k(P_{z}-z)-P_{t}}={\lambda _{1N} \over \lambda _{1D}}.}
Draw line l 2 through points R and Q . This line intersects the subjective 3-space at T . This intersection can be represented parametrically as follows:
(
1
−
λ
2
)
R
+
λ
2
Q
=
(
T
x
,
T
y
,
T
z
,
0
)
{\displaystyle (1-\lambda _{2})R+\lambda _{2}Q=(T_{x},T_{y},T_{z},0)}
This implies the following four equations:
T
x
=
R
x
+
λ
2
(
Q
x
−
R
x
)
,
{\displaystyle T_{x}=R_{x}+\lambda _{2}(Q_{x}-R_{x}),}
T
y
=
R
y
+
λ
2
(
Q
y
−
R
y
)
,
{\displaystyle T_{y}=R_{y}+\lambda _{2}(Q_{y}-R_{y}),}
T
z
=
R
z
+
λ
2
(
Q
z
−
R
z
)
,
{\displaystyle T_{z}=R_{z}+\lambda _{2}(Q_{z}-R_{z}),}
R
t
+
λ
2
(
Q
t
−
R
t
)
=
0.
{\displaystyle R_{t}+\lambda _{2}(Q_{t}-R_{t})=0.}
The last equation can be solved for λ2 ,
λ
2
=
R
t
R
t
−
Q
t
{\displaystyle \lambda _{2}={R_{t} \over R_{t}-Q_{t}}}
which can then be substituted into the other three equations:
T
x
=
R
x
+
R
t
Q
x
−
R
x
R
t
−
Q
t
=
R
t
Q
x
−
R
x
Q
t
R
t
−
Q
t
,
{\displaystyle T_{x}=R_{x}+R_{t}{Q_{x}-R_{x} \over R_{t}-Q_{t}}={R_{t}Q_{x}-R_{x}Q_{t} \over R_{t}-Q_{t}},}
T
y
=
R
y
+
R
t
Q
y
−
R
y
R
t
−
Q
t
=
R
t
Q
y
−
R
y
Q
t
R
t
−
Q
t
,
{\displaystyle T_{y}=R_{y}+R_{t}{Q_{y}-R_{y} \over R_{t}-Q_{t}}={R_{t}Q_{y}-R_{y}Q_{t} \over R_{t}-Q_{t}},}
T
z
=
R
z
+
R
t
Q
z
−
R
z
R
t
−
Q
t
=
R
t
Q
z
−
R
z
Q
t
R
t
−
Q
t
.
{\displaystyle T_{z}=R_{z}+R_{t}{Q_{z}-R_{z} \over R_{t}-Q_{t}}={R_{t}Q_{z}-R_{z}Q_{t} \over R_{t}-Q_{t}}.}
Substitute the values for Rx , Ry , Rz , and Rt obtained from the first intersection into the above equations for Tx , Ty , and Tz ,
T
x
=
λ
1
P
t
Q
x
−
[
x
+
λ
1
(
P
x
−
x
)
]
Q
t
λ
1
P
t
−
Q
t
=
λ
1
[
P
t
Q
x
−
Q
t
(
P
x
−
x
)
]
−
x
Q
t
λ
1
P
t
−
Q
t
,
{\displaystyle T_{x}={\lambda _{1}P_{t}Q_{x}-[x+\lambda _{1}(P_{x}-x)]Q_{t} \over \lambda _{1}P_{t}-Q_{t}}={\lambda _{1}[P_{t}Q_{x}-Q_{t}(P_{x}-x)]-xQ_{t} \over \lambda _{1}P_{t}-Q_{t}},}
T
y
=
λ
1
P
t
Q
y
−
[
y
+
λ
1
(
P
y
−
y
)
]
Q
t
λ
1
P
t
−
Q
t
=
λ
1
[
P
t
Q
y
−
Q
t
(
P
y
−
y
)
]
−
y
Q
t
λ
1
P
t
−
Q
t
,
{\displaystyle T_{y}={\lambda _{1}P_{t}Q_{y}-[y+\lambda _{1}(P_{y}-y)]Q_{t} \over \lambda _{1}P_{t}-Q_{t}}={\lambda _{1}[P_{t}Q_{y}-Q_{t}(P_{y}-y)]-yQ_{t} \over \lambda _{1}P_{t}-Q_{t}},}
T
z
=
λ
1
P
t
Q
z
−
[
z
+
λ
1
(
P
z
−
z
)
]
Q
t
λ
1
P
t
−
Q
t
=
λ
1
[
P
t
Q
z
−
Q
t
(
P
z
−
z
)
]
−
z
Q
t
λ
1
P
t
−
Q
t
.
{\displaystyle T_{z}={\lambda _{1}P_{t}Q_{z}-[z+\lambda _{1}(P_{z}-z)]Q_{t} \over \lambda _{1}P_{t}-Q_{t}}={\lambda _{1}[P_{t}Q_{z}-Q_{t}(P_{z}-z)]-zQ_{t} \over \lambda _{1}P_{t}-Q_{t}}.}
Multiply both numerators and denominators of the above three equations by the denominator of lambda1 : λ1D ,
T
x
=
λ
1
N
[
P
t
Q
x
−
Q
t
(
P
x
−
x
)
]
−
x
Q
t
λ
1
D
P
t
λ
1
N
−
Q
t
λ
1
D
,
{\displaystyle T_{x}={\lambda _{1N}[P_{t}Q_{x}-Q_{t}(P_{x}-x)]-xQ_{t}\lambda _{1D} \over P_{t}\lambda _{1N}-Q_{t}\lambda _{1D}},}
T
y
=
λ
1
N
[
P
t
Q
y
−
Q
t
(
P
y
−
y
)
]
−
y
Q
t
λ
1
D
P
t
λ
1
N
−
Q
t
λ
1
D
,
{\displaystyle T_{y}={\lambda _{1N}[P_{t}Q_{y}-Q_{t}(P_{y}-y)]-yQ_{t}\lambda _{1D} \over P_{t}\lambda _{1N}-Q_{t}\lambda _{1D}},}
T
z
=
λ
1
N
[
P
t
Q
z
−
Q
t
(
P
z
−
z
)
]
−
z
Q
t
λ
1
D
P
t
λ
1
N
−
Q
t
λ
1
D
,
{\displaystyle T_{z}={\lambda _{1N}[P_{t}Q_{z}-Q_{t}(P_{z}-z)]-zQ_{t}\lambda _{1D} \over P_{t}\lambda _{1N}-Q_{t}\lambda _{1D}},}
Plug in the values of the numerator and denominator of lambda1 :
λ
1
N
=
b
+
m
x
+
n
y
+
k
z
{\displaystyle \lambda _{1N}=b+mx+ny+kz}
λ
1
D
=
P
t
+
m
(
x
−
P
x
)
+
n
(
y
−
P
y
)
+
k
(
z
−
P
z
)
{\displaystyle \lambda _{1D}=P_{t}+m(x-P_{x})+n(y-P_{y})+k(z-P_{z})}
to obtain
T
x
=
T
x
N
T
x
D
=
(
b
+
m
x
+
n
y
+
k
z
)
[
P
t
Q
x
−
Q
t
(
P
x
−
x
)
]
−
x
Q
t
[
P
t
+
m
(
x
−
P
x
)
+
n
(
y
−
P
y
)
+
k
(
z
−
P
z
)
]
P
t
(
b
+
m
x
+
n
y
+
k
z
)
−
Q
t
[
P
t
+
m
(
x
−
P
x
)
+
n
(
y
−
P
y
)
+
k
(
z
−
P
z
)
]
.
{\displaystyle T_{x}={T_{xN} \over T_{xD}}={(b+mx+ny+kz)[P_{t}Q_{x}-Q_{t}(P_{x}-x)]-xQ_{t}[P_{t}+m(x-P_{x})+n(y-P_{y})+k(z-P_{z})] \over P_{t}(b+mx+ny+kz)-Q_{t}[P_{t}+m(x-P_{x})+n(y-P_{y})+k(z-P_{z})]}.}
T
y
=
T
y
N
T
x
D
{\displaystyle T_{y}={T_{yN} \over T_{xD}}}
,
T
y
N
=
(
b
+
m
x
+
n
y
+
k
z
)
[
P
t
Q
y
−
Q
t
(
P
y
−
y
)
]
−
y
Q
t
[
P
t
+
m
(
x
−
P
x
)
+
n
(
y
−
P
y
)
+
k
(
z
−
P
z
)
]
,
{\displaystyle T_{yN}=(b+mx+ny+kz)[P_{t}Q_{y}-Q_{t}(P_{y}-y)]-yQ_{t}[P_{t}+m(x-P_{x})+n(y-P_{y})+k(z-P_{z})],}
T
z
=
T
z
N
T
x
D
{\displaystyle T_{z}={T_{zN} \over T_{xD}}}
.
The numerator TxN can be expanded. It will be found that second-degree terms of x , y , and z will cancel each other out. Then collecting terms with common x , y , and z yields
T
x
N
=
x
(
m
P
t
Q
x
+
n
P
y
Q
t
+
k
P
z
Q
t
+
Q
t
(
b
−
P
t
)
)
+
y
n
(
P
t
Q
x
−
P
x
Q
t
)
+
z
k
(
P
t
Q
x
−
P
x
Q
t
)
+
b
(
P
t
Q
x
−
P
x
Q
t
)
{\displaystyle T_{xN}=x(mP_{t}Q_{x}+nP_{y}Q_{t}+kP_{z}Q_{t}+Q_{t}(b-P_{t}))+yn(P_{t}Q_{x}-P_{x}Q_{t})+zk(P_{t}Q_{x}-P_{x}Q_{t})+b(P_{t}Q_{x}-P_{x}Q_{t})}
Likewise, the denominator becomes
T
x
D
=
(
m
x
+
n
y
+
k
z
)
(
P
t
−
Q
t
)
+
(
m
P
x
+
n
P
y
+
k
P
z
)
Q
t
+
P
t
(
b
−
Q
t
)
.
{\displaystyle T_{xD}=(mx+ny+kz)(P_{t}-Q_{t})+(mP_{x}+nP_{y}+kP_{z})Q_{t}+P_{t}(b-Q_{t}).}
The numerator TyN , when expanded and then simplified, becomes
T
y
N
=
x
m
(
P
t
Q
y
−
P
y
Q
t
)
+
y
(
m
P
x
Q
t
+
n
P
t
Q
y
+
k
P
z
Q
t
+
Q
t
(
b
−
P
t
)
)
+
z
k
(
P
t
Q
y
−
P
y
Q
t
)
+
b
(
P
t
Q
y
−
P
y
Q
t
)
.
{\displaystyle T_{yN}=xm(P_{t}Q_{y}-P_{y}Q_{t})+y(mP_{x}Q_{t}+nP_{t}Q_{y}+kP_{z}Q_{t}+Q_{t}(b-P_{t}))+zk(P_{t}Q_{y}-P_{y}Q_{t})+b(P_{t}Q_{y}-P_{y}Q_{t}).}
Likewise, the numerator TzN becomes
T
z
N
=
x
m
(
P
t
Q
z
−
P
z
Q
t
)
+
y
n
(
P
t
Q
z
−
P
z
Q
t
)
+
z
(
m
P
x
Q
t
+
n
P
y
Q
t
+
k
P
t
Q
z
+
Q
t
(
b
−
P
t
)
)
+
b
(
P
t
Q
z
−
P
z
Q
t
)
.
{\displaystyle T_{zN}=xm(P_{t}Q_{z}-P_{z}Q_{t})+yn(P_{t}Q_{z}-P_{z}Q_{t})+z(mP_{x}Q_{t}+nP_{y}Q_{t}+kP_{t}Q_{z}+Q_{t}(b-P_{t}))+b(P_{t}Q_{z}-P_{z}Q_{t}).}
Let
α
=
m
P
t
Q
x
+
n
P
y
Q
t
+
k
P
z
Q
t
+
Q
t
(
b
−
P
t
)
,
{\displaystyle \alpha =mP_{t}Q_{x}+nP_{y}Q_{t}+kP_{z}Q_{t}+Q_{t}(b-P_{t}),}
β
=
n
(
P
t
Q
x
−
P
x
Q
t
)
,
{\displaystyle \beta =n(P_{t}Q_{x}-P_{x}Q_{t}),}
γ
=
k
(
P
t
Q
x
−
P
x
Q
t
)
,
{\displaystyle \gamma =k(P_{t}Q_{x}-P_{x}Q_{t}),}
δ
=
b
(
P
t
Q
x
−
P
x
Q
t
)
,
{\displaystyle \delta =b(P_{t}Q_{x}-P_{x}Q_{t}),}
ϵ
=
m
(
P
t
−
Q
t
)
,
{\displaystyle \epsilon =m(P_{t}-Q_{t}),}
ζ
=
n
(
P
t
−
Q
t
)
,
{\displaystyle \zeta =n(P_{t}-Q_{t}),}
η
=
k
(
P
t
−
Q
t
)
,
{\displaystyle \eta =k(P_{t}-Q_{t}),}
θ
=
(
m
P
x
+
n
P
y
+
k
P
z
)
Q
t
+
P
t
(
b
−
Q
t
)
,
{\displaystyle \theta =(mP_{x}+nP_{y}+kP_{z})Q_{t}+P_{t}(b-Q_{t}),}
ι
=
m
(
P
t
Q
y
−
P
y
Q
t
)
,
{\displaystyle \iota =m(P_{t}Q_{y}-P_{y}Q_{t}),}
κ
=
m
P
x
Q
t
+
n
P
t
Q
y
+
k
P
z
Q
t
+
Q
t
(
b
−
P
t
)
,
{\displaystyle \kappa =mP_{x}Q_{t}+nP_{t}Q_{y}+kP_{z}Q_{t}+Q_{t}(b-P_{t}),}
λ
=
k
(
P
t
Q
y
−
P
y
Q
t
)
,
{\displaystyle \lambda =k(P_{t}Q_{y}-P_{y}Q_{t}),}
μ
=
b
(
P
t
Q
y
−
P
y
Q
t
)
,
{\displaystyle \mu =b(P_{t}Q_{y}-P_{y}Q_{t}),}
ν
=
m
(
P
t
Q
z
−
P
z
Q
t
)
,
{\displaystyle \nu =m(P_{t}Q_{z}-P_{z}Q_{t}),}
ξ
=
n
(
P
t
Q
z
−
P
z
Q
t
)
,
{\displaystyle \xi =n(P_{t}Q_{z}-P_{z}Q_{t}),}
o
=
m
P
x
Q
t
+
n
P
y
Q
t
+
k
P
t
Q
z
+
Q
t
(
b
−
P
t
)
,
{\displaystyle o=mP_{x}Q_{t}+nP_{y}Q_{t}+kP_{t}Q_{z}+Q_{t}(b-P_{t}),}
ρ
=
b
(
P
t
Q
z
−
P
z
Q
t
)
.
{\displaystyle \rho =b(P_{t}Q_{z}-P_{z}Q_{t}).}
Then the transformation in 3-space can be expressed as follows,
T
x
=
α
x
+
β
y
+
γ
z
+
δ
ϵ
x
+
ζ
y
+
η
z
+
θ
,
{\displaystyle T_{x}={\alpha x+\beta y+\gamma z+\delta \over \epsilon x+\zeta y+\eta z+\theta },}
T
y
=
ι
x
+
κ
y
+
λ
z
+
μ
ϵ
x
+
ζ
y
+
η
z
+
θ
,
{\displaystyle T_{y}={\iota x+\kappa y+\lambda z+\mu \over \epsilon x+\zeta y+\eta z+\theta },}
T
z
=
ν
x
+
ξ
y
+
o
z
+
ρ
ϵ
x
+
ζ
y
+
η
z
+
θ
.
{\displaystyle T_{z}={\nu x+\xi y+oz+\rho \over \epsilon x+\zeta y+\eta z+\theta }.}
The sixteen coefficients of this transformation can be arranged in a coefficient matrix
M
T
=
[
α
β
γ
δ
ι
κ
λ
μ
ν
ξ
o
ρ
ϵ
ζ
η
θ
]
.
{\displaystyle M_{T}={\begin{bmatrix}\alpha &\beta &\gamma &\delta \\\iota &\kappa &\lambda &\mu \\\nu &\xi &o&\rho \\\epsilon &\zeta &\eta &\theta \end{bmatrix}}.}
Whenever this matrix is invertible, its coefficients will describe a quadrilinear fractional transformation.
Transformation T in 3-space can also be represented in terms of homogeneous coordinates as
T
:
[
x
:
y
:
z
:
1
]
→
[
α
x
+
β
y
+
γ
z
+
δ
:
ι
x
+
κ
y
+
λ
z
+
μ
:
ν
x
+
ξ
y
+
o
z
+
ρ
:
ϵ
x
+
ζ
y
+
η
z
+
θ
]
.
{\displaystyle T:[x:y:z:1]\rightarrow [\alpha x+\beta y+\gamma z+\delta :\iota x+\kappa y+\lambda z+\mu :\nu x+\xi y+oz+\rho :\epsilon x+\zeta y+\eta z+\theta ].}
This means that the coefficient matrix of T can operate directly on 4-component vectors of homogeneous coordinates. Transformation of a point can be effected simply by multiplying the coefficient matrix with the position vector of the point in homogeneous coordinates. Therefore, if T transforms a point on the plane at infinity , the result will be
T
:
[
x
:
y
:
z
:
0
]
→
[
α
x
+
β
y
+
γ
z
:
ι
x
+
κ
y
+
λ
z
:
ν
x
+
ξ
y
+
o
z
:
ϵ
x
+
ζ
y
+
η
z
]
.
{\displaystyle T:[x:y:z:0]\rightarrow [\alpha x+\beta y+\gamma z:\iota x+\kappa y+\lambda z:\nu x+\xi y+oz:\epsilon x+\zeta y+\eta z].}
If ε, ζ, and η are not all equal to zero, then T will transform the plane at infinity into a locus of points which lie mostly in affine space. If ε, ζ, and η are all zero, then T will be a special kind of projective transformation called an affine transformation , which transforms affine points into affine points and ideal points (i.e. points at infinity) into ideal points.
The group of affine transformations has a subgroup of affine rotations whose matrices have the form
M
A
R
=
[
α
β
γ
0
ι
κ
λ
0
ν
ξ
o
0
0
0
0
1
]
{\displaystyle M_{AR}={\begin{bmatrix}\alpha &\beta &\gamma &0\\\iota &\kappa &\lambda &0\\\nu &\xi &o&0\\0&0&0&1\end{bmatrix}}}
such that the submatrix
[
α
β
γ
ι
κ
λ
ν
ξ
o
]
{\displaystyle {\begin{bmatrix}\alpha &\beta &\gamma \\\iota &\kappa &\lambda \\\nu &\xi &o\end{bmatrix}}}
is orthogonal .
Given a pair of quadrilinear fractional transformations T 1 and T 2 , whose coefficient matrices are
M
T
1
{\displaystyle M_{T_{1}}}
and
M
T
2
{\displaystyle M_{T_{2}}}
, then the composition of these pair of transformations is another quadrilinear transformation T 3 whose coefficient matrix
M
T
3
{\displaystyle M_{T_{3}}}
is equal to the product of the first and second coefficient matrices,
(
T
3
=
T
2
∘
T
1
)
↔
(
M
T
3
=
M
T
2
M
T
1
)
.
{\displaystyle (T_{3}=T_{2}\circ T_{1})\leftrightarrow (M_{T_{3}}=M_{T_{2}}M_{T_{1}}).}
The identity quadrilinear fractional transformation TI is the transformation whose coefficient matrix is the identity matrix .
Given a spatial projectivity T1 whose coefficient matrix is
M
T
1
{\displaystyle M_{T_{1}}}
, the inverse of this projectivity is another projectivity T −1 whose coefficient matrix
M
T
−
1
{\displaystyle M_{T_{-1}}}
is the inverse of T 1 ′s coefficient matrix,
(
T
−
1
∘
T
1
=
T
I
)
↔
(
M
T
−
1
M
T
1
=
I
)
{\displaystyle (T_{-1}\circ T_{1}=T_{I})\leftrightarrow (M_{T_{-1}}M_{T_{1}}=I)}
.
Composition of quadrilinear transformations is associative, therefore the set of all quadrilinear transformations, together with the operation of composition, form a group .
This group of quadrilinear transformations contains subgroups of trilinear transformations. For example, the subgroup of all quadrilinear transformations whose coefficient matrices have the form
[
α
β
0
δ
ι
κ
0
μ
0
0
0
0
ϵ
ζ
0
θ
]
{\displaystyle {\begin{bmatrix}\alpha &\beta &0&\delta \\\iota &\kappa &0&\mu \\0&0&0&0\\\epsilon &\zeta &0&\theta \end{bmatrix}}}
is isomorphic to the group of all trilinear transformations whose coefficient matrices are
[
α
β
δ
ι
κ
μ
ϵ
ζ
θ
]
.
{\displaystyle {\begin{bmatrix}\alpha &\beta &\delta \\\iota &\kappa &\mu \\\epsilon &\zeta &\theta \end{bmatrix}}.}
This subgroup of quadrilinear transformations all have the form
T
:
(
x
,
y
,
z
)
→
(
α
x
+
β
y
+
δ
ϵ
x
+
ζ
y
+
θ
,
ι
x
+
κ
y
+
μ
ϵ
x
+
ζ
y
+
θ
,
0
)
.
{\displaystyle T:(x,y,z)\rightarrow \left({\alpha x+\beta y+\delta \over \epsilon x+\zeta y+\theta },{\iota x+\kappa y+\mu \over \epsilon x+\zeta y+\theta },0\right).}
This means that this subgroup of transformations will act on the plane z = 0 just like a group of trilinear transformations.
Projective transformations in 3-space transform planes into planes. This can be demonstrated more easily using homogeneous coordinates.
Let
z
=
m
x
+
n
y
+
b
{\displaystyle z=mx+ny+b}
be the equation of a plane. This is equivalent to
m
x
+
n
y
−
z
+
b
=
0.
(
21
)
{\displaystyle mx+ny-z+b=0.\qquad \qquad (21)}
Equation (21) can be expressed as a matrix product:
[
m
n
−
1
b
]
[
x
.
.
y
.
.
z
.
.
1
]
=
0.
{\displaystyle [m\ n\ -1\ b]{\begin{bmatrix}x\\.\ .\\y\\.\ .\\z\\.\ .\\1\end{bmatrix}}=0.}
A permutation matrix can be interposed between the two vectors, in order to make the plane vector have homogeneous coordinates:
[
m
:
n
:
b
:
1
]
[
1
0
0
0
0
1
0
0
0
0
0
1
0
0
−
1
0
]
[
x
.
.
y
.
.
z
.
.
1
]
=
0.
(
22
)
{\displaystyle [m:n:b:1]{\begin{bmatrix}1&0&0&0\\\ &\ &\ &\ \\0&1&0&0\\\ &\ &\ &\ \\0&0&0&1\\\ &\ &\ &\ \\0&0&-1&0\end{bmatrix}}{\begin{bmatrix}x\\.\ .\\y\\.\ .\\z\\.\ .\\1\end{bmatrix}}=0.\qquad \qquad (22)}
A quadrilinear transformation should convert this to
[
T
m
:
T
n
:
T
b
:
1
]
[
1
0
0
0
0
1
0
0
0
0
0
1
0
0
−
1
0
]
[
T
x
.
.
T
y
.
.
T
z
.
.
1
]
=
0
(
23
)
{\displaystyle [T_{m}:T_{n}:T_{b}:1]{\begin{bmatrix}1&0&0&0\\\ &\ &\ &\ \\0&1&0&0\\\ &\ &\ &\ \\0&0&0&1\\\ &\ &\ &\ \\0&0&-1&0\end{bmatrix}}{\begin{bmatrix}T_{x}\\.\ .\\T_{y}\\.\ .\\T_{z}\\.\ .\\1\end{bmatrix}}=0\qquad \qquad (23)}
where
[
T
x
.
.
T
y
.
.
T
z
.
.
1
]
=
[
α
β
γ
δ
ι
κ
λ
μ
ν
ξ
o
ρ
ϵ
ζ
η
θ
]
[
x
.
.
y
.
.
z
.
.
1
]
.
(
24
)
{\displaystyle {\begin{bmatrix}T_{x}\\.\ .\\T_{y}\\.\ .\\T_{z}\\.\ .\\1\end{bmatrix}}={\begin{bmatrix}\alpha &\beta &\gamma &\delta \\\ &\ &\ &\ \\\iota &\kappa &\lambda &\mu \\\ &\ &\ &\ \\\nu &\xi &o&\rho \\\ &\ &\ &\ \\\epsilon &\zeta &\eta &\theta \end{bmatrix}}{\begin{bmatrix}x\\.\ .\\y\\.\ .\\z\\.\ .\\1\end{bmatrix}}.\qquad \qquad (24)}
Equation (22) is equivalent to
[
m
:
n
:
b
:
1
]
[
1
0
0
0
0
1
0
0
0
0
0
1
0
0
−
1
0
]
[
α
¯
ι
¯
ν
¯
ϵ
¯
β
¯
κ
¯
ξ
¯
ζ
¯
γ
¯
λ
¯
o
¯
η
¯
δ
¯
μ
¯
ρ
¯
θ
¯
]
[
α
β
γ
δ
ι
κ
λ
μ
ν
ξ
o
ρ
ϵ
ζ
η
θ
]
[
x
.
.
y
.
.
z
.
.
1
]
=
0
(
25
)
{\displaystyle [m:n:b:1]{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}}{\begin{bmatrix}{\bar {\alpha }}&{\bar {\iota }}&{\bar {\nu }}&{\bar {\epsilon }}\\{\bar {\beta }}&{\bar {\kappa }}&{\bar {\xi }}&{\bar {\zeta }}\\{\bar {\gamma }}&{\bar {\lambda }}&{\bar {o}}&{\bar {\eta }}\\{\bar {\delta }}&{\bar {\mu }}&{\bar {\rho }}&{\bar {\theta }}\end{bmatrix}}{\begin{bmatrix}\alpha &\beta &\gamma &\delta \\\iota &\kappa &\lambda &\mu \\\nu &\xi &o&\rho \\\epsilon &\zeta &\eta &\theta \end{bmatrix}}{\begin{bmatrix}x\\.\ .\\y\\.\ .\\z\\.\ .\\1\end{bmatrix}}=0\qquad \qquad (25)}
where
α
¯
=
|
κ
λ
μ
ξ
o
ρ
ζ
η
θ
|
;
β
¯
=
|
λ
μ
ι
o
ρ
ν
η
θ
ϵ
|
,
{\displaystyle {\bar {\alpha }}=\left|{\begin{matrix}\kappa &\lambda &\mu \\\xi &o&\rho \\\zeta &\eta &\theta \end{matrix}}\right|;\qquad {\bar {\beta }}=\left|{\begin{matrix}\lambda &\mu &\iota \\o&\rho &\nu \\\eta &\theta &\epsilon \end{matrix}}\right|,}
etc.
Applying equation (24) to equation (25) yields
[
m
:
n
:
b
:
1
]
[
1
0
0
0
0
1
0
0
0
0
0
1
0
0
−
1
0
]
[
α
¯
ι
¯
ν
¯
ϵ
¯
β
¯
κ
¯
ξ
¯
ζ
¯
γ
¯
λ
¯
o
¯
η
¯
δ
¯
μ
¯
ρ
¯
θ
¯
]
[
T
x
.
.
T
y
.
.
T
z
.
.
1
]
=
0.
(
26
)
{\displaystyle [m:n:b:1]{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}}{\begin{bmatrix}{\bar {\alpha }}&{\bar {\iota }}&{\bar {\nu }}&{\bar {\epsilon }}\\{\bar {\beta }}&{\bar {\kappa }}&{\bar {\xi }}&{\bar {\zeta }}\\{\bar {\gamma }}&{\bar {\lambda }}&{\bar {o}}&{\bar {\eta }}\\{\bar {\delta }}&{\bar {\mu }}&{\bar {\rho }}&{\bar {\theta }}\end{bmatrix}}{\begin{bmatrix}T_{x}\\.\ .\\T_{y}\\.\ .\\T_{z}\\.\ .\\1\end{bmatrix}}=0.\qquad \qquad (26)}
Combining equations (26) and (23) produces
[
α
¯
β
¯
γ
¯
δ
¯
ι
¯
κ
¯
λ
¯
μ
¯
ν
¯
ξ
¯
o
¯
ρ
¯
ϵ
¯
ζ
¯
η
¯
θ
¯
]
[
1
0
0
0
0
1
0
0
0
0
0
−
1
0
0
1
0
]
[
m
.
.
n
.
.
b
.
.
1
]
=
[
1
0
0
0
0
1
0
0
0
0
0
−
1
0
0
1
0
]
[
T
m
.
.
T
n
.
.
T
b
.
.
1
]
.
{\displaystyle {\begin{bmatrix}{\bar {\alpha }}&{\bar {\beta }}&{\bar {\gamma }}&{\bar {\delta }}\\{\bar {\iota }}&{\bar {\kappa }}&{\bar {\lambda }}&{\bar {\mu }}\\{\bar {\nu }}&{\bar {\xi }}&{\bar {o}}&{\bar {\rho }}\\{\bar {\epsilon }}&{\bar {\zeta }}&{\bar {\eta }}&{\bar {\theta }}\end{bmatrix}}{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}}{\begin{bmatrix}m\\.\ .\\n\\.\ .\\b\\.\ .\\1\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}}{\begin{bmatrix}T_{m}\\.\ .\\T_{n}\\.\ .\\T_{b}\\.\ .\\1\end{bmatrix}}.}
Solve for
[
T
m
:
T
n
:
T
b
:
1
]
T
{\displaystyle [T_{m}:T_{n}:T_{b}:1]^{T}}
,
[
T
m
.
.
T
n
.
.
T
b
.
.
1
]
=
[
α
¯
β
¯
δ
¯
−
γ
¯
ι
¯
κ
¯
μ
¯
−
λ
¯
ϵ
¯
ζ
¯
θ
¯
−
η
¯
−
ν
¯
−
ξ
¯
−
ρ
¯
o
¯
]
[
m
.
.
n
.
.
b
.
.
1
]
.
(
27
)
{\displaystyle {\begin{bmatrix}T_{m}\\.\ .\\T_{n}\\.\ .\\T_{b}\\.\ .\\1\end{bmatrix}}={\begin{bmatrix}{\bar {\alpha }}&{\bar {\beta }}&{\bar {\delta }}&-{\bar {\gamma }}\\{\bar {\iota }}&{\bar {\kappa }}&{\bar {\mu }}&-{\bar {\lambda }}\\{\bar {\epsilon }}&{\bar {\zeta }}&{\bar {\theta }}&-{\bar {\eta }}\\-{\bar {\nu }}&-{\bar {\xi }}&-{\bar {\rho }}&{\bar {o}}\end{bmatrix}}{\begin{bmatrix}m\\.\ .\\n\\.\ .\\b\\.\ .\\1\end{bmatrix}}.\qquad \qquad (27)}
Equation (27) describes how 3-space transformations convert a plane (m , n , b ) into another plane (Tm , Tn , Tb ) where
T
m
=
α
¯
m
+
β
¯
n
+
δ
¯
b
−
γ
¯
−
ν
¯
m
−
ξ
¯
n
−
ρ
¯
b
+
o
¯
,
{\displaystyle T_{m}={{\bar {\alpha }}m+{\bar {\beta }}n+{\bar {\delta }}b-{\bar {\gamma }} \over -{\bar {\nu }}m-{\bar {\xi }}n-{\bar {\rho }}b+{\bar {o}}},}
T
n
=
ι
¯
m
+
κ
¯
n
+
μ
¯
b
−
λ
¯
−
ν
¯
m
−
ξ
¯
n
−
ρ
¯
b
+
o
¯
,
{\displaystyle T_{n}={{\bar {\iota }}m+{\bar {\kappa }}n+{\bar {\mu }}b-{\bar {\lambda }} \over -{\bar {\nu }}m-{\bar {\xi }}n-{\bar {\rho }}b+{\bar {o}}},}
T
b
=
ϵ
¯
m
+
ζ
¯
n
+
θ
¯
b
−
η
¯
−
ν
¯
m
−
ξ
¯
n
−
ρ
¯
b
+
o
¯
.
{\displaystyle T_{b}={{\bar {\epsilon }}m+{\bar {\zeta }}n+{\bar {\theta }}b-{\bar {\eta }} \over -{\bar {\nu }}m-{\bar {\xi }}n-{\bar {\rho }}b+{\bar {o}}}.}