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

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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[edit | edit source]


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

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

This implies the following four equations:

Substitute the first three equations into the last one:

Solve for λ1,

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:

This implies the following four equations:

The last equation can be solved for λ2,

which can then be substituted into the other three equations:

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

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

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

to obtain


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

Likewise, the denominator becomes

The numerator TyN, when expanded and then simplified, becomes

Likewise, the numerator TzN becomes

Quadrilinear fractional transformations[edit | edit source]


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

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

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

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

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

such that the submatrix

is orthogonal.

Properties of quadrilinear fractional transformations[edit | edit source]

Given a pair of quadrilinear fractional transformations T1 and T2, whose coefficient matrices are and , then the composition of these pair of transformations is another quadrilinear transformation T3 whose coefficient matrix is equal to the product of the first and second coefficient matrices,

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 , the inverse of this projectivity is another projectivity T−1 whose coefficient matrix is the inverse of T1′s coefficient matrix,


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

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

This subgroup of quadrilinear transformations all have the form

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[edit | edit source]

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


be the equation of a plane. This is equivalent to

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

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

A quadrilinear transformation should convert this to


Equation (22) is equivalent to where


Applying equation (24) to equation (25) yields

Combining equations (26) and (23) produces

Solve for ,

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