Projective transformation


Projective transformation

A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. It describes what happens to the perceived positions of observed objects when the point of view of the observer changes. Projective transformations do not preserve sizes or angles but do preserve incidence and cross-ratio: two properties which are important in projective geometry. A projective transformation can also be called a projectivity.

As important special cases, a projective transformation can be in the (real) one-dimensional projective line "RP"1, the two-dimensional projective plane "RP"2, and the three-dimensional projective 3-space "RP"3.

Transformations on the projective line

Let "X" be a point on the "x"-axis. A projective transformation can be defined geometrically for this line by picking a pair of points "P", "Q", and a line "m", all within the same "x-y" plane which contains the "x"-axis upon which the transformation will be performed.

Draw line "l" through points "P" and "X". Line "l" crosses line "m" at point "R". Then draw line "n" through points "Q" and "R": line "n" will cross the "x"-axis at point "T". Point "T" is the transform of point "X" [Paiva] .

Points "P" and "Q" represent two different observers, or points of view. Point "R" is the position of some object they are observing. Line "m" is the objective world which they are observing, and the "x"-axis is the subjective perception of "m".

Analysis

The above is a synthetic description of a one-dimensional projective transformation. It is now desired to convert it to an analytical (Cartesian) description.

Let point "X" have coordinates "(x0,0)". Let point "P" have coordinates (P_x,P_y) . Let point "Q" have coordinates (Q_x,Q_y) . Let line "m" have slope "m" ("m" is being overloaded in meaning).

The slope of line "l" is

: P_y over P_x - x_0,

so an arbitrary point "(x,y)" on line "l" is given by the equation

: {y over x - x_0} = {P_y over P_x - x_0} ,

: y = {P_y over P_x - x_0} (x - x_0). qquad qquad (1)

On the other hand, any point "(x,y)" on line "m" is described by

: y = m x + b. qquad qquad (2)

The intersection of lines "l" and "m" is point "R", and it is obtained by combining equations (1) and (2):

: m x + b = {P_y x over P_x - x_0} - {P_y x_0 over P_x - x_0}.

Joining the "x" terms yields

: left( {P_y over P_x - x_0} - m ight) x = b + {P_y x_0 over P_x - x_0}

and solving for "x" we obtain

: x_1 = {b (P_x - x_0) + P_y x_0 over P_y - m (P_x - x_0)}.

"x"1 is the abscissa of "R". The ordinate of "R" is

: y_1 = m left [ {b (P_x - x_0) + P_y x_0 over P_y - m (P_x - x_0)} ight] + b.

Now, knowing both "Q" and "R", the slope of line "n" is

: {y_1 - Q_y over x_1 - Q_x} .

We want to find the intersection of line "n" and the "x"-axis, so let

: (Q_x, Q_y) + lambda (x_1 - Q_x, y_1 - Q_y) = (x,0) qquad qquad (3)

The value of "λ" must be adjusted so that both sides of vector equation (3) are equal. Equation (3) is actually two equations, one for abscissas and one for ordinates. The one for ordinates is

: Q_y + lambda (y_1 - Q_y) = 0

Solve for lambda,

: lambda = {-Q_y over y_1 - Q_y} qquad qquad (4)

The equation for abscissas is

: x = Q_x + lambda (x_1 - Q_x)

which together with equation (4) yields

: x = Q_x - Q_y left( {x_1 - Q_x over y_1 - Q_y} ight) qquad qquad (5)

which is the abscissa of "T".

Substitute the values of "x1" and "y1" into equation (5),

: x = Q_x - Q_y left [ { {b (P_x - x_0) + P_y x_0 over P_y - m (P_x - x_0)} - Q_x over {m b (P_x - x_0) + m P_y x_0 over P_y - m (P_x - x_0)} + b - Q_y} ight] .

Dissolve the fractions in both numerator and denominator:

: x = Q_x - Q_y left [ {b (P_x - x_0) + P_y x_0 - Q_x P_y + m Q_x (P_x - x_0) over m b (P_x - x_0) + m P_y x_0 + b P_y - m b (P_x - x_0) - Q_y P_y + m Q_y (P_x - x_0) } ight] .

Simplify and relabel "x" as "t(x)":

: t(x) = Q_x - Q_y left [ { (P_x - x_0) (b + m Q_x) + P_y (x_0 - Q_x) over (P_x - x_0) m Q_y + P_y (m x_0 + b - Q_y) } ight] .

"t(x)" is the projective transformation.

Transformation "t"("x") can be simplified further. First, add its two terms to form a fraction:

: t(x) = { (m Q_x P_y - Q_y P_y + b Q_y) x_0 + (b Q_x P_y - b Q_y P_x) over m (P_y - Q_y) x_0 + (m P_x Q_y + P_y (b - Q_y)) } qquad qquad (6)

Then, define the coefficients "α", "β", "γ" and "δ" to be the following

: alpha = m Q_x P_y - Q_y P_y + b Q_y, : eta = b Q_x P_y - b Q_y P_x, : gamma = m (P_y - Q_y), : delta = m P_x Q_y + P_y (b - Q_y).

Substitute these coefficients into equation (6), in order to produce

: t(x) = { alpha x + eta over gamma x + delta }

This is the Möbius transformation or bilinear fractional transformation (so called because it has a linear numerator and a linear denominator. Actually, it is bilinear because the composition of projections is a binary linear operator, similar to matrix multiplication).

Inverse transformation

It is clear from the synthetic definition that the inverse transformation is obtained by exchanging points "P" and "Q". This can also be shown analytically. If "P" ↔ "Q", then "α" → "α′", "β" → "β′", "γ" → "γ′", and "δ" → "δ′", where: alpha' = m P_x Q_y - P_y Q_y + b P_y = delta, : eta' = b P_x Q_y - b P_y Q_x = - eta, : gamma' = m (Q_y - P_y) = - gamma, : delta' = m Q_x P_y + b Q_y - Q_y P_y = alpha.

Therefore if the forwards transformation is

: t(x) = {alpha x + eta over gamma x + delta}

then the transformation "t′" obtained by exchanging "P" and "Q" ("P" ↔ "Q") is:

: t'(x) = {delta x - eta over - gamma x + alpha }.

Then

: t'(t(x)) = {delta left( {alpha x + eta over gamma x + delta} ight) - eta over - gamma left( {alpha x + eta over gamma x + delta} ight) + alpha} .

Dissolve the fractions in both numerator and denominator of the right side of this last equation:

: t'(t(x)) = {alpha delta x + eta delta - eta gamma x - eta delta over - alpha gamma x - eta gamma + alpha gamma x + alpha delta} :: = {alpha delta x - eta gamma x over alpha delta - eta gamma} = x .

Therefore "t"′("x") = "t"−1("x"): the inverse projective transformation is obtained by exchanging observers "P" and "Q", or by letting α ↔ δ, β → −β, and γ → −γ. This is, by the way, analogous to the procedure for obtaining the inverse of a two-dimensional matrix:

: egin{bmatrix} alpha & eta \ gamma & delta end{bmatrix} egin{bmatrix} delta & - eta \ - gamma & alpha end{bmatrix} = Delta egin{bmatrix} 1 & 0 \ 0 & 1 end{bmatrix}

where Δ = α δ − β γ is the determinant.

Identity transformation

Also analogous with matrices is the identity transformation, which is obtained by letting α = 1, β = 0, γ = 0, and δ = 1, so that

: t_I(x) = x.

Composition of transformations

It remains to show that there is closure in the composition of transformations. One transformation operating on another transformation produces a third transformation. Let the first transformation be "t"1 and the second one be "t"2:

: t_1(x) = {alpha_1 x + eta_1 over gamma_1 x + delta_1 }, : t_2(x) = {alpha_2 x + eta_2 over gamma_2 x + delta_2 }.

The composition of these two transformations is

: t_2(t_1(x)) = {alpha_2 left( {alpha_1 x + eta_1 over gamma_1 x + delta_1} ight) + eta_2 over gamma_2 left( {alpha_1 x + eta_1 over gamma_1 x + delta_1 } ight) + delta_2 } ::: = {alpha_2 alpha_1 x + alpha_2 eta_1 + eta_2 gamma_1 x + eta_2 delta_1 over gamma_2 alpha_1 x + gamma_2 eta_1 + delta_2 gamma_1 x + delta_2 delta_1 } ::: = {(alpha_2 alpha_1 + eta_2 gamma_1) x + (alpha_2 eta_1 + eta_2 delta_1) over (gamma_2 alpha_1 + delta_2 gamma_1) x + (gamma_2 eta_1 + delta_2 delta_1)}.

Define the coefficients α3, β3, γ3 and δ3 to be equal to

: alpha_3 = alpha_2 alpha_1 + eta_2 gamma_1, : eta_3 = alpha_2 eta_1 + eta_2 delta_1, : gamma_3 = gamma_2 alpha_1 + delta_2 gamma_1, : delta_3 = gamma_2 eta_1 + delta_2 delta_1.

Substitute these coefficients into t_2(t_1(x)) to obtain

: t_2(t_1(x)) = { alpha_3 x + eta_3 over gamma_3 x + delta_3}.

Projections operate in a way analogous to matrices. In fact, the composition of transformations can be obtained by multiplying matrices:: egin{bmatrix} alpha_2 & eta_2 \ gamma_2 & delta_2 end{bmatrix} egin{bmatrix} alpha_1 & eta_1 \ gamma_1 & delta_1 end{bmatrix} = egin{bmatrix} alpha_2 alpha_1 + eta_2 gamma_1 & alpha_2 eta_1 + eta_2 delta_1 \ gamma_2 alpha_1 + delta_2 gamma_1 & gamma_2 eta_1 + delta_2 delta_1 end{bmatrix} = egin{bmatrix} alpha_3 & eta_3 \ gamma_3 & delta_3 end{bmatrix}. Since matrices multiply associatively, it follows that composition of projections is also associative.

Projections have: an operation (composition), associativity, an identity, an inverse and closure, so they form a group.

The cross-ratio defined by means of a projection

Let there be a transformation "ts" such that "ts"("A") = infty, "ts"("B") = 0, "ts"("C") = 1. Then the value of "ts"("D") is called the cross-ratio of points "A", "B", "C" and "D", and is denoted as ["A", "B", "C", "D"] "s":

: [A,B,C,D] _s = t_s(D).

Let

: t_s(x) = {alpha x + eta over gamma x + delta},

then the three conditions for "ts(x)" are met when

: t_s(A) = {alpha A + eta over gamma A + delta} = infty, qquad qquad (7) : t_s(B) = {alpha B + eta over gamma B + delta} = 0, qquad qquad (8) : t_s(C) = {alpha C + eta over gamma C + delta} = 1. qquad qquad (9)

Equation (7) implies that gamma A + delta = 0 , therefore delta = - gamma A .Equation (8) implies that alpha B + eta = 0 , so that eta = - alpha B .Equation (9) becomes

: {alpha C - alpha B over gamma C - gamma A} = 1,

which implies

: gamma = alpha {C - B over C - A}.

Therefore

: t_s(D) = {alpha D - alpha B over alpha left( {C - B over C - A} ight) D - gamma A} = {alpha (D - B) over alpha left( {C - B over C - A} ight) D - alpha left( {C - B over C - A} ight) A} :: = {D - B over C - B} {C - A over D - A} = {A - C over A - D} {B - D over B - C}. qquad qquad (10)

In equation (10), it is seen that "ts"("D") does not depend on the coefficients of the projection "ts". It only depends on the positions of the points on the "subjective" projective line. This means that the cross-ratio depends only on the relative distances among four collinear points, and not on the projective transformation which was used to obtain (or define) the cross-ratio. The cross ratio is therefore: [A,B,C,D] = {A - C over A - D} {B - D over B - C}. qquad qquad (11)

Conservation of cross-ratio

Transformations on the projective line preserve cross ratio. This will now be proven. Let there be four (collinear) points "A", "B", "C", "D". Their cross-ratio is given by equation (11). Let "S(x)" be a projective transformation:

: S(x) = {alpha x + eta over gamma x + delta}

where alpha delta e eta gamma . Then

: [S(A) S(B) S(C) S(D)] = alpha A + eta over gamma A + delta} - {alpha C + eta over gamma C + delta} over {alpha A + eta over gamma A + delta} - {alpha D + eta over gamma D + delta cdot alpha B + eta over gamma B + delta} - {alpha D + eta over gamma D + delta} over {alpha B + eta over gamma B + delta} - {alpha C + eta over gamma C + delta

:: = { [(alpha A + eta) (gamma C + delta) - (alpha C + eta) (gamma A + delta)] [(alpha B + eta) (gamma D + delta) - (alpha D + eta) (gamma B + delta)] over [(alpha A + eta) (gamma D + delta) - (alpha D + eta) (gamma A + delta)] [(alpha B + eta) (gamma C + delta) - (alpha C + eta) (gamma B + delta)] }

:: = { [alpha A delta + eta gamma C - alpha C delta - eta gamma A] [alpha B delta + eta gamma D - alpha D delta - eta gamma B] over [alpha A delta + eta gamma D - alpha D delta - eta gamma A] [alpha B delta + eta gamma C - alpha C delta - eta gamma B] }

:: = { [alpha delta (A - C) + eta gamma (C - A)] [alpha delta (B - D) + eta gamma (D - B)] over [alpha delta (A - D) + eta gamma (D - A)] [alpha delta (B - C) + eta gamma (C - B)] }

:: = {(alpha delta - eta gamma) (A - C) (alpha delta - eta gamma) (B - D) over (alpha delta - eta gamma) (A - D) (alpha delta - eta gamma) (B - C)}

:: = {A - C over A - D} cdot {B - D over B - C} Therefore [S(A) S(B) S(C) S(D)] = [A B C D] , Q.E.D.

Transformations on the projective plane

Two-dimensional projective transformations are a type of automorphism of the projective plane onto itself.

Planar transformations can be defined synthetically as follows: point "X" on a "subjective" plane must be transformed to a point "T" also on the subjective plane. The transformations uses these tools: a pair of "observation points" "P" and "Q", and an "objective" plane. The subjective and objective planes and the two points all lie in three-dimensional space, and the two planes can intersect at some line.

Draw line "l"1 through points "P" and "X". Line "l"1 intersects the objective plane at point "R". Draw line "l"2 through points "Q" and "R". Line "l"2 intersects the projective plane at point "T". Then "T" is the projective transform of "X".

Analysis

Let the "xy"-plane be the "subjective" plane and let plane "m" be the "objective" plane. Let plane "m" be described by

: z = f(x,y) = m x + n y + b

where the constants "m" and "n" are partial slopes and "b" is the "z"-intercept.

Let there be a pair of "observation" points "P" and "Q",

: P : (P_x, P_y, P_z), : Q : (Q_x, Q_y, Q_z).

Let point "X" lie on the "subjective" plane:

: X : (x,y,0).

Point "X" must be transformed to a point "T",

: T : (T_x, T_y, 0)

also on the "subjective" plane.

The analytical results are a pair of equations, one for abscissa "Tx" and one for ordinate "Ty":

: T_x = {x (-m Q_x P_z - n Q_z P_y + Q_z (P_z - b)) + (n y + b) (Q_z P_x - Q_x P_z) over (m x + n y) (Q_z - P_z) - (m P_x + n P_y) Q_z + (Q_z - b) P_z}, qquad qquad (12)

: T_y = {y (-n Q_y P_z - m Q_z P_x + Q_z (P_z - b)) + (m x + b) (Q_z P_y - Q_y P_z) over (n y + m x) (Q_z - P_z) - (n P_y + m P_x) Q_z + (Q_z - b) P_z }. qquad qquad (13)

There are (at most) nine degrees of freedom for defining a 2D transformation: "Px", "Py", "Pz", "Qx", "Qy", "Qz", "m", "n", "b". Notice that equations (12) and (13) have the same denominators, and that "Ty" can be obtained from "Tx" by exchanging "m" with "n", and "x" with "y" (including subscripts of "P" and "Q").

Trilinear fractional transformations

Let

: alpha = -m Q_x P_z - n Q_z P_y + Q_z (P_z - b), : eta = n (Q_z P_x - Q_x P_z), : gamma = b (Q_z P_x - Q_x P_z), : delta = m (Q_z - P_z), : epsilon = n (Q_z - P_z), : zeta = - (m P_x + n P_y) Q_z + (Q_z - b) P_z,

so that

: T_x = {alpha x + eta y + gamma over delta x + epsilon y + zeta}. qquad qquad (14)

Also let

: eta = m (Q_z P_y - Q_y P_z), : heta = -m Q_z P_x - n Q_y P_z + Q_z (P_z - b), : kappa = b (Q_z P_y - Q_y P_z),

so that

: T_y = {eta x + heta y + kappa over delta x + epsilon y + zeta}. qquad qquad (15)

Equations (14) and (15) together describe the trilinear fractional transformation.

Composition of trilinear transformations

If a transformation is given by equations (14) and (15), then such transformation is characterized by nine coefficients which can be arranged into a coefficient matrix

: M_T = egin{bmatrix} alpha & eta & gamma\ eta & heta & kappa \ delta & epsilon & zeta end{bmatrix}.

If there are a pair "T"1 and "T"2 of planar transformations whose coefficient matrices are M_{T_1} and M_{T_2} , then the composition of these transformations is another planar transformation "T"3,

: T_3 = T_2 circ T_1 ,

such that

: T_3(x,y) = T_2 ( T_1 (x,y) ).

The coefficient matrix of "T"3 can be obtained by multiplying the coefficient matrices of "T"2 and "T"1:

: M_{T_3} = M_{T_2} , M_{T_1}.

Proof

Given "T"1 defined by

: T_{1x} = {alpha_1 x + eta_1 y + gamma_1 over delta_1 x + epsilon_1 y + zeta_1}, : T_{1y} = {eta_1 x + heta_1 y + kappa_1 over delta_1 x + epsilon_1 y + zeta_1}, and given "T2" defined by: T_{2x} = {alpha_2 x + eta_2 y + gamma_2 over delta_2 x + epsilon_2 y + zeta_2}, : T_{2y} = {eta_2 x + heta_2 y + kappa_2 over delta_2 x + epsilon_2 y + zeta_2},

then "T"3 can be calculated by substituting "T"1 into "T"2,

: T_{3x} = T_{2x} ( T_{1x}, T_{1y} ) = { alpha_2 left( {alpha_1 x + eta_1 y + gamma_1 over delta_1 x + epsilon_1 y + zeta_1} ight) + eta_2 left( {eta_1 x + heta_1 y + kappa_1 over delta_1 x + epsilon_1 y + zeta_1} ight) + gamma_2 over delta_2 left( {alpha_1 x + eta_1 y + gamma_1 over delta_1 x + epsilon_1 y + zeta_1} ight) + epsilon_2 left( {eta_1 x + heta_1 y + kappa_1 over delta_1 x + epsilon_1 y + zeta_1} ight) + zeta_2}.

Multiply numerator and denominator by the same trinomial,

: T_{3x} = {alpha_2 (alpha_1 x + eta_1 y + gamma_1) + eta_2 (eta_1 x + heta_1 y + kappa_1) + gamma_2 (delta_1 x + epsilon_1 y + zeta_1) over delta_2 (alpha_1 x + eta_1 y + gamma_1) + epsilon_2 (eta_1 x + heta_1 y + kappa_1) + zeta_2 (delta_1 x + epsilon_1 y + zeta_1)}.

Group the coefficients of "x", "y", and 1:

: T_{3x} = { x (alpha_2 alpha_1 + eta_2 eta_1 + gamma_2 delta_1) + y (alpha_2 eta_1 + eta_2 heta_1 + gamma_2 epsilon_1) + (alpha_2 gamma_1 + eta_2 kappa_1 + gamma_2 zeta_1) over x (delta_2 alpha_1 + epsilon_2 eta_1 + zeta_2 delta_1) + y (delta_2 eta_1 + epsilon_2 heta_1 + zeta_2 epsilon_1) + (delta_2 gamma_1 + epsilon_2 kappa_1 + zeta_2 zeta_1)} = {alpha_3 x + eta_3 y + gamma_3 over delta_3 x + epsilon_3 y + zeta_3}.

These six coefficients of "T"3 are the same as those obtained through the product

: egin{bmatrix} alpha_2 & eta_2 & gamma_2 \eta_2 & heta_2 & kappa_2 \delta_2 & epsilon_2 & zeta_2 end{bmatrix} egin{bmatrix} alpha_1 & eta_1 & gamma_1 \eta_1 & heta_1 & kappa_1 \delta_1 & epsilon_1 & zeta_1 end{bmatrix} = egin{bmatrix} alpha_3 & eta_3 & gamma_3 \eta_3 & heta_3 & kappa_3 \delta_3 & epsilon_3 & zeta_3 end{bmatrix}. qquad qquad (16)

The remaining three coefficients can be verified thus

: T_{3y} = T_{2y} ( T_{1x}, T_{1y} ) = { eta_2 left( {alpha_1 x + eta_1 y + gamma_1 over delta_1 x + epsilon_1 y + zeta_1} ight) + heta_2 left( {eta_1 x + heta_1 y + kappa_1 over delta_1 x + epsilon_1 y + zeta_1} ight) + kappa_2 over delta_2 left( {alpha_1 x + eta_1 y + gamma_1 over delta_1 x + epsilon_1 y + zeta_1} ight) + epsilon_2 left( {eta_1 x + heta_1 y + kappa_1 over delta_1 x + epsilon_1 y + zeta_1} ight) + zeta_2}.

Multiply numerator and denominator by the same trinomial,

: T_{3y} = {eta_2 (alpha_1 x + eta_1 y + gamma_1) + heta_2 (eta_1 x + heta_1 y + kappa_1) + kappa_2 (delta_1 x + epsilon_1 y + zeta_1) over delta_2 (alpha_1 x + eta_1 y + gamma_1) + epsilon_2 (eta_1 x + heta_1 y + kappa_1) + zeta_2 (delta_1 x + epsilon_1 y + zeta_1)}.

Group the coefficients of "x", "y", and "1":

: T_{3x} = { x (eta_2 alpha_1 + heta_2 eta_1 + kappa_2 delta_1) + y (eta_2 eta_1 + heta_2 heta_1 + kappa_2 epsilon_1) + (eta_2 gamma_1 + heta_2 kappa_1 + kappa_2 zeta_1) over x (delta_2 alpha_1 + epsilon_2 eta_1 + zeta_2 delta_1) + y (delta_2 eta_1 + epsilon_2 heta_1 + zeta_2 epsilon_1) + (delta_2 gamma_1 + epsilon_2 kappa_1 + zeta_2 zeta_1)} = {eta_3 x + heta_3 y + kappa_3 over delta_3 x + epsilon_3 y + zeta_3}.

The three remaining coefficients just obtained are the same as those obtained through equation (16). Q.E.D.

Planar transformations of lines

The trilinear transformation given be equations (14) and (15) transforms a straight line

: y = m x + b

into another straight line

: T_y = n T_x + c

where "n" and "c" are constants and equal to

: n = {m (epsilon kappa - zeta heta) + b (delta heta - epsilon eta) + (delta kappa - zeta eta) over m (epsilon gamma - zeta eta) + b (delta eta - epsilon alpha) + (delta gamma - zeta alpha)}

and

: c = {m (eta kappa - gamma heta) + b (alpha heta - eta eta) + (alpha kappa - gamma eta) over m (eta zeta - gamma epsilon) + b (alpha epsilon - eta delta) + (alpha zeta - gamma delta) }.

Proof

Given "y = m x + b", then plugging this into equations (14) and (15) yields

: T_x = {alpha x + eta (m x + b) + gamma over delta x + epsilon (m x + b) + zeta} = {(alpha + eta m) x + (eta b + gamma) over (delta + epsilon m) x + (epsilon b + zeta)},

and

: T_y = {(eta + heta m) x + ( heta b + kappa) over (delta + epsilon m) x + (epsilon b + zeta) }.

If "Ty = n Tx + c" and "n" and "c" are constants, then

: {partial T_y over partial x} = n {partial T_x over partial x}

so that

: n = {partial T_y / partial x over partial T_x / partial y}. Calculation shows that: {partial T_x over partial x} = { (epsilon b + zeta) (alpha + eta m) - (eta b + gamma) (delta + epsilon m) over [(delta + epsilon m) x + (epsilon b + zeta)] ^2 }

and

: {partial T_y over partial x} = { (epsilon b + zeta) (eta + heta m) - ( heta b + kappa) (delta + epsilon m) over [(delta + epsilon m) x + (epsilon b + zeta)] ^2 }

therefore

: n = {partial T_y / partial x over partial T_x / partial y} = { (epsilon b + zeta) (eta + heta m) - ( heta b + kappa) (delta + epsilon m) over (epsilon b + zeta) (alpha + eta m) - (eta b + gamma) (delta + epsilon m) } .

We should now obtain "c" to be

: c = T_y - n T_x ::: = {(eta + heta m) x + ( heta b + kappa) - left [ { (epsilon b + zeta) (eta + heta m) - ( heta b + kappa) (delta + epsilon m) over (epsilon b + zeta) (alpha + eta m) - (eta b + gamma) (delta + epsilon m) } ight] cdot [ (alpha + eta m) x + (eta b + gamma) ] over (delta + epsilon m) x + (epsilon b + zeta) }. Add the two fractions in the numerator:: c = { left{ [(epsilon b + zeta) (alpha + eta m) - (eta b + gamma) (delta + epsilon m)] [(eta + heta m) x + ( heta b + kappa)] - [(epsilon b + zeta) (eta + heta m) - ( heta b + kappa) (delta + epsilon m)] [(alpha + eta m) x + (eta b + gamma)] ight} over [(delta + epsilon m) x + (epsilon b + zeta)] [(epsilon b + zeta) (alpha + eta m) - (eta b + gamma) (delta + epsilon m)] }.

Distribute binomials in parentheses in the numerator, then cancel out equal and opposite terms:

: c = { - (eta b + gamma) (delta + epsilon m) (eta + heta m) x + (epsilon b + zeta) (alpha + eta m) ( heta b + kappa) + ( heta b + kappa) (delta + epsilon m) (alpha + eta m) x - (epsilon b + zeta) (eta + heta m) (eta b + gamma) over [(delta + epsilon m) x + (epsilon b + zeta)] [(epsilon b + zeta) (alpha + eta m) - (eta b + gamma) (delta + epsilon m)] }.

Factor the numerator into a pair of terms, only one of them having the "numerus cossicus" ("x"). There is another numerus cossicus in the denominator. The objective now is to get both of these to cancel out.

: c = { left{ [( heta b + kappa) (alpha + eta m) - (eta b + gamma) (eta + heta m)] (delta + epsilon m) x + [(alpha + eta m)( heta b + kappa) - (eta + heta m) (eta b + gamma)] (epsilon b + zeta) ight} over [(delta + epsilon m) x + (epsilon b + zeta)] [(epsilon b + zeta) (alpha + eta m) - (eta b + gamma) (delta + epsilon m)] }.

Factor the numerator,

: c = { [( heta b + kappa) (alpha + eta m) - (eta b + gamma) (eta + heta m)] [(delta + epsilon m) x + (epsilon b + zeta)] over [(epsilon b + zeta) (alpha + eta m) - (eta b + gamma) (delta + epsilon m)] [(delta + epsilon m) x + (epsilon b + zeta)] }.

The terms with the numeri cossici cancel out, therefore

: c = { (alpha + eta m) ( heta b + kappa) - (eta b + gamma) (eta + heta m) over (alpha + eta m) (epsilon b + zeta) - (eta b + gamma) (delta + epsilon m) } is a constant. Q.E.D.

Comparing "c" with "n", notice that their denominators are the same. Also, "n" is obtained from "c" by exchanging the following coefficients:

: alpha leftrightarrow delta, eta leftrightarrow epsilon, gamma leftrightarrow zeta .

There is also the following exchange symmetry between the numerator and denominator of "n":

: alpha leftrightarrow eta, eta leftrightarrow heta, gamma leftrightarrow kappa .

The numerator and denominator of "c" also have exchange symmetry: { eta leftrightarrow delta, heta leftrightarrow epsilon, kappa leftrightarrow zeta }.

The exchange symmetry between "n" and "c" can be chunked into binomials:

: n leftrightarrow c equiv { (alpha + m eta ) leftrightarrow (delta + m epsilon ), (gamma + b eta ) leftrightarrow (zeta + b epsilon ) }.

All of these exchange symmetries amount to exchanging pairs of rows in the coefficient matrix.

Planar transformations of conic sections

A trilinear transformation such as "T" given by equations (14) and (15) will convert a conic section: A x^2 + B y^2 + C x + D y + E x y + F = 0 qquad qquad (17) into another conic section: A' T_x^2 + B' T_y^2 + C' T_x + D' T_y + E' T_x T_y + F' = 0. qquad qquad (18)

Proof

Let there be given a conic section described by equation (17) and a planar transformation "T" described by equations (15) and (16) which converts points "(x,y)" into points "(Tx,Ty)".

It is possible to find an inverse transformation "T′" which converts back points "(Tx,Ty)" to points "(x,y)". This inverse transformation has a coefficient matrix: M_{T'} = egin{bmatrix} alpha' & eta' & gamma' \eta' & heta' & kappa' \delta' & epsilon' & zeta' end{bmatrix}.

Equation (17) can be expressed in terms of the inverse transformation:

: A left( {alpha' T_x + eta' T_y + gamma' over delta' T_x + epsilon' T_y + zeta'} ight)^2 + B left( {eta' T_x + heta' T_y + kappa' over delta' T_x + epsilon' T_y + zeta'} ight)^2 + C left( {alpha' T_x + eta' T_y + gamma' over delta' T_x + epsilon' T_y + zeta'} ight) + D left( {eta' T_x + heta' T_y + kappa' over delta' T_x + epsilon' T_y + zeta'} ight) + E left( {alpha' T_x + eta' T_y + gamma' over delta' T_x + epsilon' T_y + zeta'} ight) left( {eta' T_x + heta' T_y + kappa' over delta' T_x + epsilon' T_y + zeta'} ight) + F = 0.

The denominators can be "dissolved" by multiplying both sides of the equation by the square of a trinomial:

: A (alpha' T_x + eta' T_y + gamma')^2 + B (eta' T_x + heta' T_y + kappa')^2 + C (alpha' T_x + eta' T_y + gamma') (delta' T_x + epsilon' T_y + zeta') + D (eta' T_x + heta' T_y + kappa') (delta' T_x + epsilon' T_y + zeta') + E (alpha' T_x + eta' T_y + gamma') (eta' T_x + heta' T_y + kappa') + F (delta' T_x + epsilon' T_y + zeta')^2 = 0.

Expand the products of trinomials and collect common powers of "Tx" and "Ty":

: egin{matrix} (A alpha'^2 + B eta'^2 + C alpha' delta' + D eta' delta' + E alpha' eta' + F delta'^2) T_x^2 \ +(A eta'^2 + B heta'^2 + C eta' epsilon' + D heta' epsilon' + E eta' heta' + F epsilon'^2) T_y^2 \ +(2 A alpha' gamma' + 2 B eta' kappa' + C (alpha' zeta' + gamma' delta') + D (eta' zeta' + kappa' delta') + E (alpha' kappa' + gamma' eta') + 2 F delta' zeta') T_x \ +(2 A eta' gamma' + 2 B heta' kappa' + C (eta' zeta' + gamma' epsilon') + D ( heta' zeta' + kappa' epsilon') + E (eta' kappa' + gamma' heta') + 2 F epsilon' zeta') T_y \ +(2 A alpha' eta' + 2 B eta' heta' + C (alpha' epsilon' + eta' delta') + D (eta' epsilon' + heta' delta') + E (alpha' heta' + eta' eta') + 2 F delta' epsilon') T_x T_y \ + (A gamma'^2 + B kappa'^2 + C gamma' zeta' + D kappa' zeta' + E gamma' kappa' + F zeta'^2) = 0. end{matrix} qquad qquad (19)

Equation (19) has the same form as equation (18).

What remains to do is to express the primed coefficients in terms of the unprimed coefficients. To do this, apply Cramer's rule to the coefficient matrix "MT" to obtain the primed matrix of the inverse transformation:

: M_{T'} = {1 over Delta} egin{bmatrix} left| egin{matrix} heta &kappa \ epsilon & zeta end{matrix} ight| &left| egin{matrix} epsilon & zeta \ eta & gamma end{matrix} ight| &left| egin{matrix} eta & gamma \ heta & kappa end{matrix} ight
\ quad & quad & quad \left| egin{matrix} kappa & eta \ zeta & delta end{matrix} ight| &left| egin{matrix} zeta & delta \ gamma & alpha end{matrix} ight| &left| egin{matrix} gamma & alpha \ kappa & eta end{matrix} ight
\ quad & quad & quad \left| egin{matrix} eta & heta \ delta & epsilon end{matrix} ight| &left| egin{matrix} delta &epsilon \ alpha & eta end{matrix} ight| &left| egin{matrix} alpha & eta \ eta & heta end{matrix} ight
end{bmatrix} qquad qquad (20) where "Δ" is the determinant of the unprimed coefficient matrix.

Equation (20) allows primed coefficients to be expressed in terms of unprimed coefficients. But performing these substitutions on the primed coefficients of equation (19) it can be noticed that the determinant "Δ" cancels itself out, so that it can be ignored altogether. Therefore: A' = A ( heta zeta - kappa epsilon)^2+ B (kappa delta - eta zeta)^2+ C ( heta zeta - kappa epsilon) (eta epsilon - heta delta)+ D (kappa delta - eta zeta) (eta epsilon - heta delta)+ E ( heta zeta - kappa epsilon) (kappa delta - eta zeta)+ F (eta epsilon - heta delta)^2

: B' = A (epsilon gamma - zeta eta)^2+ B (zeta alpha - delta gamma)^2+ C (epsilon gamma - zeta eta) (delta eta - epsilon alpha)+ D (zeta alpha - delta gamma) (delta eta - epsilon alpha)+ E (epsilon gamma - zeta eta) (zeta alpha - delta gamma)+ F (delta eta - epsilon alpha)^2

: C' = 2 A ( heta zeta - kappa epsilon) (eta kappa - gamma heta)+ 2 B (kappa delta - eta zeta) (gamma eta - alpha kappa)+ C [ ( heta zeta - kappa epsilon) (alpha heta - eta eta) +(eta kappa - gamma heta) (eta epsilon - heta delta)] + D [ (kappa delta - eta zeta) (alpha heta - eta eta) + (gamma eta - alpha kappa) (eta epsilon - heta delta) ] + E [ ( heta zeta - kappa epsilon) (gamma eta - alpha kappa) + (eta kappa - gamma heta) (kappa delta - eta zeta) ] + 2 F (eta epsilon - heta delta) (alpha heta - eta eta)

: D' = 2 A (epsilon gamma - zeta eta) (eta kappa - gamma heta)+ 2 B (zeta alpha - delta gamma) (gamma eta - alpha kappa)+ C [ (epsilon gamma - zeta eta) (alpha heta - eta eta) + (eta kappa - gamma heta) (delta eta - epsilon alpha) ] + D [ (zeta alpha - delta gamma) (alpha heta - eta eta) + (gamma eta - alpha kappa) (delta eta - epsilon alpha) ] + E [ (epsilon gamma - zeta eta) (gamma eta - alpha kappa) + (eta kappa - gamma heta) (zeta alpha - delta gamma) ] + 2 F (delta eta - epsilon alpha) (alpha heta - eta eta)

: E' = 2 A ( heta zeta - kappa epsilon) (epsilon gamma - zeta eta)+ 2 B (kappa delta - eta zeta) (zeta alpha - delta gamma)+ C [( heta zeta - kappa epsilon) (delta eta - epsilon alpha) + (epsilon gamma - zeta eta) (eta epsilon - heta delta)] + D [ (kappa delta - eta zeta) (delta eta - epsilon alpha) + (zeta alpha - delta gamma) (eta epsilon - heta delta)] + E [ ( heta zeta - kappa epsilon) (zeta alpha - delta gamma) +(epsilon gamma - zeta eta) (kappa delta - eta zeta)] + 2 F (eta epsilon - heta delta) (delta eta - epsilon alpha)

: F' = A (eta kappa - gamma heta)^2+ B (gamma eta - alpha kappa)^2+ C (eta kappa - gamma heta) (alpha heta - eta eta)+ D (gamma eta - alpha kappa) (alpha heta - eta eta)+ E (eta kappa - gamma heta) (gamma eta - alpha kappa)+ F (alpha heta - eta eta)^2

The coefficients of the transformed conic have been expressed in terms of the coefficients of the original conic and the coefficients of the planar transformation "T". Q.E.D.

Planar projectivities and cross-ratio

Let four points "A", "B", "C", "D" be collinear. Let there be a planar projectivity "T" which transforms these points into points "A′", "B′", "C′", and "D′". It was already shown that lines are transformed into lines, so that the transformed points "A′" through "D′" will also be collinear. Then it will turn out that the cross-ratio of the original four points is the same as the cross-ratio of their transforms:: [A B C D] = [A' B' C' D'] .

Proof

If the two-dimensional coordinates of four points are known, and if the four points are collinear, then their cross-ratio can be found from their abscissas alone. It is possible to project the points onto a horizontal line by means of a pencil of vertical lines issuing from a point on the line at infinity:: [A B C D] = [A_x B_x C_x D_x] . The same is true for the ordinates of the points. The reason is that any mere rescaling of the coordinates of the points does not change the cross-ratio.

Let: A : (x_1, m x_1 + b), : B : (x_2, m x_2 + b), : C : (x_3, m x_3 + b), : D : (x_4, m x_4 + b).

Clearly these four points are collinear. Let

: T_x (x,y) = {alpha x + eta y + gamma over delta x + epsilon y + zeta}

be the first half of a trilinear transformation. Then

: T_x(A) = {alpha x_1 + eta (m x_1 + b) + gamma over delta x_1 + epsilon (m x_1 + b) + zeta} = {(alpha + eta m) x_1 + (eta b + gamma) over (delta + epsilon m) x_1 + (epsilon b + zeta)}, : T_x(B) = {alpha x_2 + eta (m x_2 + b) + gamma over delta x_2 + epsilon (m x_2 + b) + zeta} = {(alpha + eta m) x_2 + (eta b + gamma) over (delta + epsilon m) x_2 + (epsilon b + zeta)}, : T_x(C) = {alpha x_3 + eta (m x_3 + b) + gamma over delta x_3 + epsilon (m x_3 + b) + zeta} = {(alpha + eta m) x_3 + (eta b + gamma) over (delta + epsilon m) x_3 + (epsilon b + zeta)}, : T_x(D) = {alpha x_4 + eta (m x_4 + b) + gamma over delta x_4 + epsilon (m x_4 + b) + zeta} = {(alpha + eta m) x_4 + (eta b + gamma) over (delta + epsilon m) x_4 + (epsilon b + zeta)}.

The original cross-ratio is

: [x_1 x_2 x_3 x_4] = {x_1 - x_3 over x_1 - x_4} cdot {x_2 - x_4 over x_2 - x_3}.

It is not necessary to calculate the transformed cross-ratio. Just let

: S(x) = {(alpha + eta m) x + (eta b + gamma) over (delta + epsilon m) x + (epsilon b + zeta)}

be a bilinear transformation. Then "S(x)" is a one-dimensional projective transformation. But "Tx(A)=S(A)", "Tx(B)=S(B)", "Tx(C)=S(C)", and "Tx(D)=S(D)". Therefore

: [T_x(A) T_x(B) T_x(C) T_x(D)] = [S(A) S(B) S(C) S(D)]

but it has already been shown that bilinear fractional transformations preserve cross-ratio. Q.E.D.

Example

The following is a rather simple example of a planar projectivity:: T_x = {1 over x}, qquad T_y = {y over x}. The coefficient matrix of this projectivity "T" is: M_T = egin{bmatrix} 0 & 0 & 1 \0 & 1 & 0 \1 & 0 & 0 end{bmatrix} and it is easy to verify that "MT" is its own inverse.

The locus of points described parametrically as ( cos heta, , sin heta ) describe a circle, due to the trigonometric identity: cos^2 heta + sin^2 heta = 1 which has the same form as the canonical equation of a circle. Applying the projectivity "T" yields the locus of points described parametrically by (sec heta,, an heta) which describe a hyperbola, due to the trigonometric identity: sec^2 heta - an^2 heta = 1 which has the same form as the canonical equation of a hyperbola. Notice that points (~-1,0) and (1,0) are fixed points.

Indeed, this projectivity transforms any circle, of any radius, into a hyperbola centered at the origin with both of its foci lying on the "x"-axis, and vice versa. This projectivity also transforms the "y"-axis into the line at infinity, and vice versa:: T : (0, y) ightarrow left( {1 over 0}, {y over 0} ight) = (pm infty, pm infty), : T: (pm infty, pm infty) ightarrow left( {1 over pm infty}, {pm infty over pm infty} ight) = (0, y). The ratio of infinity over infinity is indeterminate which means that it can be set to any value "y" desired.

This example emphasizes that in the real projective plane, "RP²", a hyperbola is a closed curve which passes twice through the line at infinity. But what does the transformation do to a parabola?

Let the locus of points (x,x^2) describe a parabola. Its transformation is: T : (x,x^2) ightarrow left( {1 over x}, {x^2 over x} ight) = (x', 1/x') which is a hyperbola whose asymptotes are the "x"-axis and the "y"-axis and whose wings lie in the first quadrant and the third quadrant. Likewise, the hyperbola: y = {1 over x} is transformed by "T" into the parabola: y = x^2 quad .

On the other hand, the parabola described by the locus of points (x, pm sqrt{x}) is transformed by "T" into itself: this demonstrates that a parabola intersects the line at infinity at a single point.

Transformations in 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 "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".

Analysis

Let: X : (x,y,z,0), : T : (T_x,T_y,T_z,0), : P : (P_x,P_y,P_z,P_t), : 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

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 - lambda_1) X + lambda_1 P = (R_x,R_y,R_z,m R_x + n R_y + k R_z + b).

This implies the following four equations:

: R_x = x + lambda_1 (P_x - x) : R_y = y + lambda_1 (P_y - y) : R_z = z + lambda_1 (P_z - z) : R_t = lambda_1 P_t = m R_x + n R_y + k R_z + b

Substitute the first three equations into the last one:

: (m x + n y + k z) + lambda_1 (m P_x + n P_y + k P_z - m x - n y - k z - P_t) + b = 0

Solve for "λ1",

: lambda_1 = {-(b + m x + n y + k z) 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 - lambda_2) R + lambda_2 Q = (T_x,T_y,T_z,0)

This implies the following four equations:

: T_x = R_x + lambda_2 (Q_x - R_x), : T_y = R_y + lambda_2 (Q_y - R_y), : T_z = R_z + lambda_2 (Q_z - R_z), : R_t + lambda_2 (Q_t - R_t) = 0.

The last equation can be solved for "λ2",

: 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 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 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 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 = {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)] - x Q_t over lambda_1 P_t - Q_t}, : 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)] - y Q_t over lambda_1 P_t - Q_t}, : 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)] - z Q_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 = {lambda_{1N} [P_t Q_x - Q_t (P_x - x)] - x Q_t lambda_{1D} over P_t lambda_{1N} - Q_t lambda_{1D} }, : T_y = {lambda_{1N} [P_t Q_y - Q_t (P_y - y)] - y Q_t lambda_{1D} over P_t lambda_{1N} - Q_t lambda_{1D} }, : T_z = {lambda_{1N} [P_t Q_z - Q_t (P_z - z)] - z Q_t lambda_{1D} over P_t lambda_{1N} - Q_t lambda_{1D} },

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

: lambda_{1N} = b + m x + n y + k z : lambda_{1D} = P_t + m (x - P_x) + n (y - P_y) + k (z - P_z)

to obtain

: T_x = {T_{xN} over T_{xD = {(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)] over 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)] }. : T_y = {T_{yN} over T_{xD ,: T_{yN} = (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)] , : 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_{xN} = 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)

Likewise, the denominator becomes

: T_{xD} = (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).

The numerator "TyN", when expanded and then simplified, becomes

: T_{yN} = 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).

Likewise, the numerator "TzN" becomes

: T_{zN} = 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).

Quadrilinear fractional transformations

Let

: alpha = m P_t Q_x + n P_y Q_t + k P_z Q_t + Q_t (b - P_t), : eta = n (P_t Q_x - P_x Q_t), : gamma = k (P_t Q_x - P_x Q_t), : delta = b (P_t Q_x - P_x Q_t), : epsilon = m (P_t - Q_t), : zeta = n (P_t - Q_t), : eta = k (P_t - Q_t), : heta = (m P_x + n P_y + k P_z) Q_t + P_t (b - Q_t), : iota = m (P_t Q_y - P_y Q_t), : kappa = m P_x Q_t + n P_t Q_y + k P_z Q_t + Q_t (b - P_t), : lambda = k (P_t Q_y - P_y Q_t), : mu = b (P_t Q_y - P_y Q_t), : u = m (P_t Q_z - P_z Q_t), : 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), : ho = b (P_t Q_z - P_z Q_t).

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

: T_x = {alpha x + eta y + gamma z + delta over epsilon x + zeta y + eta z + heta}, : T_y = {iota x + kappa y + lambda z + mu over epsilon x + zeta y + eta z + heta}, : T_z = { u x + xi y + o z + ho over epsilon x + zeta y + eta z + heta}.

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

: M_T = egin{bmatrix} alpha & eta & gamma & delta \iota & kappa & lambda & mu \ u & xi & o & ho \epsilon & zeta & eta & heta 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] ightarrow [alpha x + eta y + gamma z + delta : iota x + kappa y + lambda z + mu : u x + xi y + o z + ho : epsilon x + zeta y + eta z + heta ] .

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] ightarrow [alpha x + eta y + gamma z : iota x + kappa y + lambda z : u x + xi y + o z : 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_{AR} = egin{bmatrix} alpha & eta & gamma & 0 \iota & kappa & lambda & 0 \ u & xi & o & 0 \0 & 0 & 0 & 1 end{bmatrix} such that the submatrix: egin{bmatrix} alpha & eta & gamma \iota & kappa & lambda \ u & xi & o end{bmatrix} is orthogonal.

Properties of quadrilinear fractional transformations

Given a pair of quadrilinear fractional transformations "T"1 and "T"2, whose coefficient matrices are M_{T_1} and M_{T_2} , then the composition of these pair of transformations is another quadrilinear transformation "T"3 whose coefficient matrix M_{T_3} is equal to the product of the first and second coefficient matrices,

: (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} , the inverse of this projectivity is another projectivity "T"−1 whose coefficient matrix M_{T_{-1 is the inverse of "T"1′s coefficient matrix,

: (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

: egin{bmatrix} alpha & eta & 0 & delta \iota & kappa & 0 & mu \0 & 0 & 0 & 0 \epsilon & zeta & 0 & heta end{bmatrix}

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

: egin{bmatrix} alpha & eta & delta \iota & kappa & mu \epsilon & zeta & heta end{bmatrix}.

This subgroup of quadrilinear transformations all have the form

: T : (x, y, z) ightarrow left( {alpha x + eta y + delta over epsilon x + zeta y + heta} , {iota x + kappa y + mu over epsilon x + zeta y + heta}, 0 ight).

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

patial transformations of planes

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

be the equation of a plane. This is equivalent to

: m x + n y - z + b = 0. qquad qquad (21)

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

: [m n -1 b] egin{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 ] egin{bmatrix} 1 & 0 & 0 & 0 \ & & & \0 & 1 & 0 & 0 \ & & & \0 & 0 & 0 & 1 \ & & & \0 & 0 & -1 & 0 end{bmatrix} egin{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 ] egin{bmatrix} 1 & 0 & 0 & 0 \ & & & \0 & 1 & 0 & 0 \ & & & \0 & 0 & 0 & 1 \ & & & \0 & 0 & -1 & 0 end{bmatrix} egin{bmatrix} T_x \. . \T_y \. . \T_z \. . \1 end{bmatrix} = 0 qquad qquad (23)

where

: egin{bmatrix} T_x \. . \T_y \. . \T_z \. . \1 end{bmatrix} = egin{bmatrix} alpha & eta & gamma & delta \ & & & \iota & kappa & lambda & mu \ & & & \ u & xi & o & ho \ & & & \epsilon & zeta & eta & heta end{bmatrix} egin{bmatrix} x \. . \y \. . \z \. . \1 end{bmatrix}. qquad qquad (24)

Equation (22) is equivalent to [ m : n : b : 1 ] egin{bmatrix} 1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 0 & 1 \0 & 0 & -1 & 0 end{bmatrix}egin{bmatrix} ar{alpha} & ar{iota} & ar{ u} & ar{epsilon} \ar{eta} & ar{kappa} & ar{xi} & ar{zeta} \ar{gamma} & ar{lambda} & ar{o} & ar{eta} \ar{delta} & ar{mu} & ar{ ho} & ar{ heta} end{bmatrix}egin{bmatrix} alpha & eta & gamma & delta \iota & kappa & lambda & mu \ u & xi & o & ho \epsilon & zeta & eta & heta end{bmatrix} egin{bmatrix} x \. . \y \. . \z \. . \1 end{bmatrix} = 0 qquad qquad (25) where: ar{alpha} = left| egin{matrix} kappa & lambda & mu \xi & o & ho \zeta & eta & heta end{matrix} ight| ; qquadar{eta} = left| egin{matrix} lambda & mu & iota \o & ho & u \eta & heta & epsilon end{matrix} ight|, etc.

Applying equation (24) to equation (25) yields

: [ m : n : b : 1 ] egin{bmatrix} 1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 0 & 1 \0 & 0 & -1 & 0 end{bmatrix}egin{bmatrix} ar{alpha} & ar{iota} & ar{ u} & ar{epsilon} \ar{eta} & ar{kappa} & ar{xi} & ar{zeta} \ar{gamma} & ar{lambda} & ar{o} & ar{eta} \ar{delta} & ar{mu} & ar{ ho} & ar{ heta} end{bmatrix}egin{bmatrix} T_x \. . \T_y \. . \T_z \. . \1 end{bmatrix} = 0. qquad qquad (26) Combining equations (26) and (23) produces: egin{bmatrix} ar{alpha} & ar{eta} & ar{gamma} & ar{delta} \ar{iota} & ar{kappa} & ar{lambda} & ar{mu} \ar{ u} & ar{xi} & ar{o} & ar{ ho} \ar{epsilon} & ar{zeta} & ar{eta} & ar{ heta} end{bmatrix}egin{bmatrix} 1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 0 & -1 \0 & 0 & 1 & 0 end{bmatrix}egin{bmatrix} m \. . \n \. . \b \. . \1 end{bmatrix} = egin{bmatrix} 1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 0 & -1 \0 & 0 & 1 & 0 end{bmatrix} egin{bmatrix} T_m \. . \T_n \. . \T_b \. . \1 end{bmatrix}. Solve for [ T_m : T_n : T_b : 1 ] ^T ,: egin{bmatrix} T_m \. . \T_n \. . \T_b \. . \1 end{bmatrix} = egin{bmatrix} ar{alpha} & ar{eta} & ar{delta} & -ar{gamma} \ar{iota} & ar{kappa} & ar{mu} & -ar{lambda} \ar{epsilon} & ar{zeta} & ar{ heta} & -ar{eta} \-ar{ u} & -ar{xi} & -ar{ ho} & ar{o} end{bmatrix}egin{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 = {ar{alpha} m + ar{eta} n + ar{delta} b - ar{gamma} over - ar{ u} m - ar{xi} n - ar{ ho} b + ar{o, : T_n = {ar{iota} m + ar{kappa} n + ar{mu} b - ar{lambda} over - ar{ u} m - ar{xi} n - ar{ ho} b + ar{o, : T_b = {ar{epsilon} m + ar{zeta} n + ar{ heta} b - ar{eta} over - ar{ u} m - ar{xi} n - ar{ ho} b + ar{o.

ee also

* Fundamental theorem of projective geometry

References

* [http://www.math.poly.edu/~alvarez/teaching/projective-geometry/Inaugural-Lecture/page_3.html Introduction to Projective Transformations by J. C. Alvarez Paiva]


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