# Projective Geometry/Classic/Projective Transformations/Transformations of projective 3-space

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 l1 through points X and P. This line intersects the objective space at point R. Draw line l2 through points R and Q. Line l2 intersects the projective plane at point T. Then T is the transform of X.

## Analysis

Let

${\displaystyle X:(x,y,z,0),}$
${\displaystyle T:(T_{x},T_{y},T_{z},0),}$
${\displaystyle P:(P_{x},P_{y},P_{z},P_{t}),}$
${\displaystyle Q:(Q_{x},Q_{y},Q_{z},Q_{t}).}$

Let there be an "objective" 3-space described by

${\displaystyle t=f(x,y,z)=mx+ny+kz+b}$

Draw line l1 through points P and X. This line intersects the objective plane at R. This intersection can be described parametrically as follows:

${\displaystyle (1-\lambda _{1})X+\lambda _{1}P=(R_{x},R_{y},R_{z},mR_{x}+nR_{y}+kR_{z}+b).}$

This implies the following four equations:

${\displaystyle R_{x}=x+\lambda _{1}(P_{x}-x)}$
${\displaystyle R_{y}=y+\lambda _{1}(P_{y}-y)}$
${\displaystyle R_{z}=z+\lambda _{1}(P_{z}-z)}$
${\displaystyle R_{t}=\lambda _{1}P_{t}=mR_{x}+nR_{y}+kR_{z}+b}$

Substitute the first three equations into the last one:

${\displaystyle (mx+ny+kz)+\lambda _{1}(mP_{x}+nP_{y}+kP_{z}-mx-ny-kz-P_{t})+b=0}$

Solve for λ1,

${\displaystyle \lambda _{1}={-(b+mx+ny+kz) \over m(P_{x}-x)+n(P_{y}-y)+k(P_{z}-z)-P_{t}}={\lambda _{1N} \over \lambda _{1D}}.}$

Draw line l2 through points R and Q. This line intersects the subjective 3-space at T. This intersection can be represented parametrically as follows:

${\displaystyle (1-\lambda _{2})R+\lambda _{2}Q=(T_{x},T_{y},T_{z},0)}$

This implies the following four equations:

${\displaystyle T_{x}=R_{x}+\lambda _{2}(Q_{x}-R_{x}),}$
${\displaystyle T_{y}=R_{y}+\lambda _{2}(Q_{y}-R_{y}),}$
${\displaystyle T_{z}=R_{z}+\lambda _{2}(Q_{z}-R_{z}),}$
${\displaystyle R_{t}+\lambda _{2}(Q_{t}-R_{t})=0.}$

The last equation can be solved for λ2,

${\displaystyle \lambda _{2}={R_{t} \over R_{t}-Q_{t}}}$

which can then be substituted into the other three equations:

${\displaystyle T_{x}=R_{x}+R_{t}{Q_{x}-R_{x} \over R_{t}-Q_{t}}={R_{t}Q_{x}-R_{x}Q_{t} \over R_{t}-Q_{t}},}$
${\displaystyle T_{y}=R_{y}+R_{t}{Q_{y}-R_{y} \over R_{t}-Q_{t}}={R_{t}Q_{y}-R_{y}Q_{t} \over R_{t}-Q_{t}},}$
${\displaystyle T_{z}=R_{z}+R_{t}{Q_{z}-R_{z} \over R_{t}-Q_{t}}={R_{t}Q_{z}-R_{z}Q_{t} \over R_{t}-Q_{t}}.}$

Substitute the values for Rx, Ry, Rz, and Rt obtained from the first intersection into the above equations for Tx, Ty, and Tz,

${\displaystyle T_{x}={\lambda _{1}P_{t}Q_{x}-[x+\lambda _{1}(P_{x}-x)]Q_{t} \over \lambda _{1}P_{t}-Q_{t}}={\lambda _{1}[P_{t}Q_{x}-Q_{t}(P_{x}-x)]-xQ_{t} \over \lambda _{1}P_{t}-Q_{t}},}$
${\displaystyle T_{y}={\lambda _{1}P_{t}Q_{y}-[y+\lambda _{1}(P_{y}-y)]Q_{t} \over \lambda _{1}P_{t}-Q_{t}}={\lambda _{1}[P_{t}Q_{y}-Q_{t}(P_{y}-y)]-yQ_{t} \over \lambda _{1}P_{t}-Q_{t}},}$
${\displaystyle T_{z}={\lambda _{1}P_{t}Q_{z}-[z+\lambda _{1}(P_{z}-z)]Q_{t} \over \lambda _{1}P_{t}-Q_{t}}={\lambda _{1}[P_{t}Q_{z}-Q_{t}(P_{z}-z)]-zQ_{t} \over \lambda _{1}P_{t}-Q_{t}}.}$

Multiply both numerators and denominators of the above three equations by the denominator of lambda1: λ1D,

${\displaystyle T_{x}={\lambda _{1N}[P_{t}Q_{x}-Q_{t}(P_{x}-x)]-xQ_{t}\lambda _{1D} \over P_{t}\lambda _{1N}-Q_{t}\lambda _{1D}},}$
${\displaystyle T_{y}={\lambda _{1N}[P_{t}Q_{y}-Q_{t}(P_{y}-y)]-yQ_{t}\lambda _{1D} \over P_{t}\lambda _{1N}-Q_{t}\lambda _{1D}},}$
${\displaystyle T_{z}={\lambda _{1N}[P_{t}Q_{z}-Q_{t}(P_{z}-z)]-zQ_{t}\lambda _{1D} \over P_{t}\lambda _{1N}-Q_{t}\lambda _{1D}},}$

Plug in the values of the numerator and denominator of lambda1:

${\displaystyle \lambda _{1N}=b+mx+ny+kz}$
${\displaystyle \lambda _{1D}=P_{t}+m(x-P_{x})+n(y-P_{y})+k(z-P_{z})}$

to obtain

${\displaystyle T_{x}={T_{xN} \over T_{xD}}={(b+mx+ny+kz)[P_{t}Q_{x}-Q_{t}(P_{x}-x)]-xQ_{t}[P_{t}+m(x-P_{x})+n(y-P_{y})+k(z-P_{z})] \over P_{t}(b+mx+ny+kz)-Q_{t}[P_{t}+m(x-P_{x})+n(y-P_{y})+k(z-P_{z})]}.}$
${\displaystyle T_{y}={T_{yN} \over T_{xD}}}$,
${\displaystyle T_{yN}=(b+mx+ny+kz)[P_{t}Q_{y}-Q_{t}(P_{y}-y)]-yQ_{t}[P_{t}+m(x-P_{x})+n(y-P_{y})+k(z-P_{z})],}$
${\displaystyle T_{z}={T_{zN} \over T_{xD}}}$.

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

${\displaystyle T_{xN}=x(mP_{t}Q_{x}+nP_{y}Q_{t}+kP_{z}Q_{t}+Q_{t}(b-P_{t}))+yn(P_{t}Q_{x}-P_{x}Q_{t})+zk(P_{t}Q_{x}-P_{x}Q_{t})+b(P_{t}Q_{x}-P_{x}Q_{t})}$

Likewise, the denominator becomes

${\displaystyle T_{xD}=(mx+ny+kz)(P_{t}-Q_{t})+(mP_{x}+nP_{y}+kP_{z})Q_{t}+P_{t}(b-Q_{t}).}$

The numerator TyN, when expanded and then simplified, becomes

${\displaystyle T_{yN}=xm(P_{t}Q_{y}-P_{y}Q_{t})+y(mP_{x}Q_{t}+nP_{t}Q_{y}+kP_{z}Q_{t}+Q_{t}(b-P_{t}))+zk(P_{t}Q_{y}-P_{y}Q_{t})+b(P_{t}Q_{y}-P_{y}Q_{t}).}$

Likewise, the numerator TzN becomes

${\displaystyle T_{zN}=xm(P_{t}Q_{z}-P_{z}Q_{t})+yn(P_{t}Q_{z}-P_{z}Q_{t})+z(mP_{x}Q_{t}+nP_{y}Q_{t}+kP_{t}Q_{z}+Q_{t}(b-P_{t}))+b(P_{t}Q_{z}-P_{z}Q_{t}).}$

Let

${\displaystyle \alpha =mP_{t}Q_{x}+nP_{y}Q_{t}+kP_{z}Q_{t}+Q_{t}(b-P_{t}),}$
${\displaystyle \beta =n(P_{t}Q_{x}-P_{x}Q_{t}),}$
${\displaystyle \gamma =k(P_{t}Q_{x}-P_{x}Q_{t}),}$
${\displaystyle \delta =b(P_{t}Q_{x}-P_{x}Q_{t}),}$
${\displaystyle \epsilon =m(P_{t}-Q_{t}),}$
${\displaystyle \zeta =n(P_{t}-Q_{t}),}$
${\displaystyle \eta =k(P_{t}-Q_{t}),}$
${\displaystyle \theta =(mP_{x}+nP_{y}+kP_{z})Q_{t}+P_{t}(b-Q_{t}),}$
${\displaystyle \iota =m(P_{t}Q_{y}-P_{y}Q_{t}),}$
${\displaystyle \kappa =mP_{x}Q_{t}+nP_{t}Q_{y}+kP_{z}Q_{t}+Q_{t}(b-P_{t}),}$
${\displaystyle \lambda =k(P_{t}Q_{y}-P_{y}Q_{t}),}$
${\displaystyle \mu =b(P_{t}Q_{y}-P_{y}Q_{t}),}$
${\displaystyle \nu =m(P_{t}Q_{z}-P_{z}Q_{t}),}$
${\displaystyle \xi =n(P_{t}Q_{z}-P_{z}Q_{t}),}$
${\displaystyle o=mP_{x}Q_{t}+nP_{y}Q_{t}+kP_{t}Q_{z}+Q_{t}(b-P_{t}),}$
${\displaystyle \rho =b(P_{t}Q_{z}-P_{z}Q_{t}).}$

Then the transformation in 3-space can be expressed as follows,

${\displaystyle T_{x}={\alpha x+\beta y+\gamma z+\delta \over \epsilon x+\zeta y+\eta z+\theta },}$
${\displaystyle T_{y}={\iota x+\kappa y+\lambda z+\mu \over \epsilon x+\zeta y+\eta z+\theta },}$
${\displaystyle T_{z}={\nu x+\xi y+oz+\rho \over \epsilon x+\zeta y+\eta z+\theta }.}$

The sixteen coefficients of this transformation can be arranged in a coefficient matrix

${\displaystyle M_{T}={\begin{bmatrix}\alpha &\beta &\gamma &\delta \\\iota &\kappa &\lambda &\mu \\\nu &\xi &o&\rho \\\epsilon &\zeta &\eta &\theta \end{bmatrix}}.}$

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

${\displaystyle T:[x:y:z:1]\rightarrow [\alpha x+\beta y+\gamma z+\delta :\iota x+\kappa y+\lambda z+\mu :\nu x+\xi y+oz+\rho :\epsilon x+\zeta y+\eta z+\theta ].}$

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

${\displaystyle T:[x:y:z:0]\rightarrow [\alpha x+\beta y+\gamma z:\iota x+\kappa y+\lambda z:\nu x+\xi y+oz:\epsilon x+\zeta y+\eta z].}$

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

${\displaystyle M_{AR}={\begin{bmatrix}\alpha &\beta &\gamma &0\\\iota &\kappa &\lambda &0\\\nu &\xi &o&0\\0&0&0&1\end{bmatrix}}}$

such that the submatrix

${\displaystyle {\begin{bmatrix}\alpha &\beta &\gamma \\\iota &\kappa &\lambda \\\nu &\xi &o\end{bmatrix}}}$

is orthogonal.

## Properties of quadrilinear fractional transformations

Given a pair of quadrilinear fractional transformations T1 and T2, whose coefficient matrices are ${\displaystyle M_{T_{1}}}$ and ${\displaystyle M_{T_{2}}}$, then the composition of these pair of transformations is another quadrilinear transformation T3 whose coefficient matrix ${\displaystyle M_{T_{3}}}$ is equal to the product of the first and second coefficient matrices,

${\displaystyle (T_{3}=T_{2}\circ T_{1})\leftrightarrow (M_{T_{3}}=M_{T_{2}}M_{T_{1}}).}$

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 ${\displaystyle M_{T_{1}}}$, the inverse of this projectivity is another projectivity T−1 whose coefficient matrix ${\displaystyle M_{T_{-1}}}$ is the inverse of T1′s coefficient matrix,

${\displaystyle (T_{-1}\circ T_{1}=T_{I})\leftrightarrow (M_{T_{-1}}M_{T_{1}}=I)}$.

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

${\displaystyle {\begin{bmatrix}\alpha &\beta &0&\delta \\\iota &\kappa &0&\mu \\0&0&0&0\\\epsilon &\zeta &0&\theta \end{bmatrix}}}$

is isomorphic to the group of all trilinear transformations whose coefficient matrices are

${\displaystyle {\begin{bmatrix}\alpha &\beta &\delta \\\iota &\kappa &\mu \\\epsilon &\zeta &\theta \end{bmatrix}}.}$

This subgroup of quadrilinear transformations all have the form

${\displaystyle T:(x,y,z)\rightarrow \left({\alpha x+\beta y+\delta \over \epsilon x+\zeta y+\theta },{\iota x+\kappa y+\mu \over \epsilon x+\zeta y+\theta },0\right).}$

This means that this subgroup of transformations will act on the plane z = 0 just like a group of trilinear transformations.

## Spatial transformations of planes

Projective transformations in 3-space transform planes into planes. This can be demonstrated more easily using homogeneous coordinates.

Let

${\displaystyle z=mx+ny+b}$

be the equation of a plane. This is equivalent to

${\displaystyle mx+ny-z+b=0.\qquad \qquad (21)}$

Equation (21) can be expressed as a matrix product:

${\displaystyle [m\ n\ -1\ b]{\begin{bmatrix}x\\.\ .\\y\\.\ .\\z\\.\ .\\1\end{bmatrix}}=0.}$

A permutation matrix can be interposed between the two vectors, in order to make the plane vector have homogeneous coordinates:

${\displaystyle [m:n:b:1]{\begin{bmatrix}1&0&0&0\\\ &\ &\ &\ \\0&1&0&0\\\ &\ &\ &\ \\0&0&0&1\\\ &\ &\ &\ \\0&0&-1&0\end{bmatrix}}{\begin{bmatrix}x\\.\ .\\y\\.\ .\\z\\.\ .\\1\end{bmatrix}}=0.\qquad \qquad (22)}$

A quadrilinear transformation should convert this to

${\displaystyle [T_{m}:T_{n}:T_{b}:1]{\begin{bmatrix}1&0&0&0\\\ &\ &\ &\ \\0&1&0&0\\\ &\ &\ &\ \\0&0&0&1\\\ &\ &\ &\ \\0&0&-1&0\end{bmatrix}}{\begin{bmatrix}T_{x}\\.\ .\\T_{y}\\.\ .\\T_{z}\\.\ .\\1\end{bmatrix}}=0\qquad \qquad (23)}$

where

${\displaystyle {\begin{bmatrix}T_{x}\\.\ .\\T_{y}\\.\ .\\T_{z}\\.\ .\\1\end{bmatrix}}={\begin{bmatrix}\alpha &\beta &\gamma &\delta \\\ &\ &\ &\ \\\iota &\kappa &\lambda &\mu \\\ &\ &\ &\ \\\nu &\xi &o&\rho \\\ &\ &\ &\ \\\epsilon &\zeta &\eta &\theta \end{bmatrix}}{\begin{bmatrix}x\\.\ .\\y\\.\ .\\z\\.\ .\\1\end{bmatrix}}.\qquad \qquad (24)}$

Equation (22) is equivalent to ${\displaystyle [m:n:b:1]{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}}{\begin{bmatrix}{\bar {\alpha }}&{\bar {\iota }}&{\bar {\nu }}&{\bar {\epsilon }}\\{\bar {\beta }}&{\bar {\kappa }}&{\bar {\xi }}&{\bar {\zeta }}\\{\bar {\gamma }}&{\bar {\lambda }}&{\bar {o}}&{\bar {\eta }}\\{\bar {\delta }}&{\bar {\mu }}&{\bar {\rho }}&{\bar {\theta }}\end{bmatrix}}{\begin{bmatrix}\alpha &\beta &\gamma &\delta \\\iota &\kappa &\lambda &\mu \\\nu &\xi &o&\rho \\\epsilon &\zeta &\eta &\theta \end{bmatrix}}{\begin{bmatrix}x\\.\ .\\y\\.\ .\\z\\.\ .\\1\end{bmatrix}}=0\qquad \qquad (25)}$ where

${\displaystyle {\bar {\alpha }}=\left|{\begin{matrix}\kappa &\lambda &\mu \\\xi &o&\rho \\\zeta &\eta &\theta \end{matrix}}\right|;\qquad {\bar {\beta }}=\left|{\begin{matrix}\lambda &\mu &\iota \\o&\rho &\nu \\\eta &\theta &\epsilon \end{matrix}}\right|,}$ etc.

Applying equation (24) to equation (25) yields

${\displaystyle [m:n:b:1]{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}}{\begin{bmatrix}{\bar {\alpha }}&{\bar {\iota }}&{\bar {\nu }}&{\bar {\epsilon }}\\{\bar {\beta }}&{\bar {\kappa }}&{\bar {\xi }}&{\bar {\zeta }}\\{\bar {\gamma }}&{\bar {\lambda }}&{\bar {o}}&{\bar {\eta }}\\{\bar {\delta }}&{\bar {\mu }}&{\bar {\rho }}&{\bar {\theta }}\end{bmatrix}}{\begin{bmatrix}T_{x}\\.\ .\\T_{y}\\.\ .\\T_{z}\\.\ .\\1\end{bmatrix}}=0.\qquad \qquad (26)}$

Combining equations (26) and (23) produces

${\displaystyle {\begin{bmatrix}{\bar {\alpha }}&{\bar {\beta }}&{\bar {\gamma }}&{\bar {\delta }}\\{\bar {\iota }}&{\bar {\kappa }}&{\bar {\lambda }}&{\bar {\mu }}\\{\bar {\nu }}&{\bar {\xi }}&{\bar {o}}&{\bar {\rho }}\\{\bar {\epsilon }}&{\bar {\zeta }}&{\bar {\eta }}&{\bar {\theta }}\end{bmatrix}}{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}}{\begin{bmatrix}m\\.\ .\\n\\.\ .\\b\\.\ .\\1\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}}{\begin{bmatrix}T_{m}\\.\ .\\T_{n}\\.\ .\\T_{b}\\.\ .\\1\end{bmatrix}}.}$

Solve for ${\displaystyle [T_{m}:T_{n}:T_{b}:1]^{T}}$,

${\displaystyle {\begin{bmatrix}T_{m}\\.\ .\\T_{n}\\.\ .\\T_{b}\\.\ .\\1\end{bmatrix}}={\begin{bmatrix}{\bar {\alpha }}&{\bar {\beta }}&{\bar {\delta }}&-{\bar {\gamma }}\\{\bar {\iota }}&{\bar {\kappa }}&{\bar {\mu }}&-{\bar {\lambda }}\\{\bar {\epsilon }}&{\bar {\zeta }}&{\bar {\theta }}&-{\bar {\eta }}\\-{\bar {\nu }}&-{\bar {\xi }}&-{\bar {\rho }}&{\bar {o}}\end{bmatrix}}{\begin{bmatrix}m\\.\ .\\n\\.\ .\\b\\.\ .\\1\end{bmatrix}}.\qquad \qquad (27)}$

Equation (27) describes how 3-space transformations convert a plane (m, n, b) into another plane (Tm, Tn, Tb) where

${\displaystyle T_{m}={{\bar {\alpha }}m+{\bar {\beta }}n+{\bar {\delta }}b-{\bar {\gamma }} \over -{\bar {\nu }}m-{\bar {\xi }}n-{\bar {\rho }}b+{\bar {o}}},}$
${\displaystyle T_{n}={{\bar {\iota }}m+{\bar {\kappa }}n+{\bar {\mu }}b-{\bar {\lambda }} \over -{\bar {\nu }}m-{\bar {\xi }}n-{\bar {\rho }}b+{\bar {o}}},}$
${\displaystyle T_{b}={{\bar {\epsilon }}m+{\bar {\zeta }}n+{\bar {\theta }}b-{\bar {\eta }} \over -{\bar {\nu }}m-{\bar {\xi }}n-{\bar {\rho }}b+{\bar {o}}}.}$