Projectivity

This is defined as an invertible transformation of the projective plane onto itself, that preserves lines. In the standard representation of the projective plane as the set of lines [x₁, x₂, x₃] of R³, a projectivity is a class of invertible matrices [A], consisting of the non-zero multiples of a non singular 3x3 real matrix A. With such a representation, point x = [x₁, x₂, x₃] is transformed to [A][x] = [Ax]. Projectivities (some times I call them also homographies) are defined by giving four points {P_i} and their images {Q_i}, both quadrupples in general position, meaning, for each quadrupple, that each subset of three points is not collinear.
Algebraically, if P_i is represented by [p_i] = [p_1i, p_2i, p_3i]^t (column vector), and Q_i correspondingly with [q_i] = [q_1i, q_2i, q_3i]^t, then the corresponding A has the form A = QKP^-1, K being a diagonal matrix: K = diag{k₁, k₂, k₃} and P, Q being the matrices with columns {p_i}, {q_i} respectively, for i=1,2,3.
The arbitrary constants k_i are determined by the fourth point and the condition:
QKP^-1p₄ = k₄q₄.
This represents a system of 3 homogeneous equations with four unknowns (unknowns are k₁, ... , k₄), and rank three, thus accepting a solution depending on one parameter (k₁, ... , k₄) = t(k⁰₁, ..., k⁰₄). This determines a unique class [A] = [QKP^-1].

As an example, construct the projectivity fixing the points [1,0,1]^t, [-1,0,1]^t, [0, -1,1]^t and mapping p₄ = [0,1,1]^t to the point at infinity along (0,1) which is q₄ = [0,1,0]^t. The corresponding matrices are:

Equation QKP^-1p₄ = k₄q₄, takes the form:

And has the solutions (k₁, k₂, k₃, k₄) = t(1,1,2,2), thus completely determining A, which is:

The projectivity defined through that matrix maps the points (x,y) of the unit circle: x² + y² = 1, to:

(x', y') satisfying y' = x'² - 1, which is a parabola (shown above). The easily shown vanishing of the determinant

proves that this projectivity F has for every point X(x,y), the image-point F(X)=(x',y') and the point E(0,-1) collinear. Further, by inspecting [A], we see that (i) the x-axis and point E are the only fixed points of F, whereas the whole line y = 1 maps to the line at infinity. A further computation of the cross-ratio [E,X,F(X),Y], where Y the intersection point of line EX with the x-axis, shows that this is -1 (harmonic). F coincides not with the perspectivity with center at E and axis the x-axis, and homology coefficient equal to 2.
In this discussion I have consistently and often used the projectification of the euclidean plane, which identifies the point (x,y) of euclidean space with the point [x,y,1] of the projective space, and inversely it identifies point [x,y,z] of the projective space with point (x/z, y/z) of the euclidean space (provided [x,y,z] is not a point on the line at infinity, meaning that z is non-zero).
The above image dictates a procedure of generation of parabolas, studied in ParabolaProjectFromCircle.html .

Projectivity

See Also