Projective geometry

Projective geometry is a non-metrical form of geometry. First developed by Desargues in the 17th century, it did not achieve prominence as a field of mathematics until the early 19th century through the work of Poncelet and others. Projective geometry originated from the principles of perspective art.

Description
Projective geometry is a non-Euclidean geometry that formalizes one of the central principles of perspective art: that parallel lines meet at infinity and therefore are to be drawn that way. In essence, a projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line". Thus, two parallel lines will meet on a horizon in virtue of their possessing the same direction.

Idealized directions are referred to as points at infinity, while idealized horizons are referred to as lines at infinity.

However, a projective geometry does not single out any point or line in this regard — they are all treated equally. Indeed, with the extension, the axiomatization becomes substantially simpler (based on Whitehead, "The Axioms of Projective Geometry"):
 * G1: Every line contains at least 3 points
 * G2: Every two points, A and B, lie on a unique line, AB.
 * G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).

The reason each line is assumed to contain at least 3 points is apparent when thinking of the original motivating example of a Euclidean space supplemented by the lines and points at infinity. The 3rd point is the line's direction. Axiom 2 is thus seen to embody a form of Euclid's 5th postulate (which makes the designation of Projective geometry as non-Euclidean ironic): given a point and a direction, there is a unique line containing the point lying in the given direction.

Because a Euclidean geometry is contained within a Projective geometry, with Projective geometry having a simpler foundation, general results in Euclidean geometry may be arrived at in a more transparent fashion. Moreover, as already seen with the preceding interpretation of Axiom 2, separate but similar theorems in Euclidean geometry may be handled collectively within the framework of projective geometry; for instance, parallel and nonparallel lines need not be treated as separate cases.

One can pursue axiomatization in greater depth by postulating a ternary relation, [ABC] to denote when three points (not all necessarily distinct) are collinear. A relatively simple axiomatization may be written down in terms of this relation as well: For two different points, A and B, the line AB is defined as consisting of all points C for which [ABC]. The axioms C0 and C1 then provide a formalization of G2; C2 for G1 and C3 for G3.
 * C0: [ABA]
 * C1: If A and B are two points such that [ABC] and [ABD] then [BDC]
 * C2: If A and B are two points then there is a third point C such that [ABC]
 * C3: If A and C are two points, B and D also, with [BCE], [ADE] but not [ABE] then there is a point F such that [ACF] and [BDF].

The concept of line generalizes to planes and higher dimensional subspaces. A subspace, AB...XY may thus be recursively defined in terms of the subspace AB...X as that containing all the points of all lines YZ, as Z ranges over AB...X. Collinearity then generalizes to the relation of "independence". A set {A, B,...,Z} of points is independent, [AB...Z] if {A, B,...,Z} is a minimal generating subset for the subspace AB...Z.

The axioms may be supplemented by further axioms postulating limits on the dimension of the space. The minimum dimension is determined by the existence of an independent set of the required size. For the lowest dimensions, the relevant conditions may be stated in equivalent form as follows. A projective space is of:
 * (L1) at least dimension 0 if it has at least 1 point,
 * (L2) at least dimension 1 if it has at least 2 distinct points (and therefore a line),
 * (L3) at least dimension 2 if it has at least 3 non-collinear points (or two lines, or a line and a point not on the line),
 * (L4) at least dimension 3 if it has at least 4 non-coplanar points.

The maximum dimension may also be determined in a similar fashion. For the lowest dimensions, they take on the following forms. A projective space is of: and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar lines intersect — the very principle Projective Geometry was originally intended to embody. Therefore, property (M3) may be equivalently stated that all lines intersect one another.
 * (M1) at most dimension 0 if it has no more than 1 point,
 * (M2) at most dimension 1 if it has no more than 1 line,
 * (M3) at most dimension 2 if it has no more than 1 plane,

It is generally assumed that projective spaces are of at least dimension 2. In some cases, if the focus is meant to be on projective planes, a variant of M3 may be postulated. The axioms of (Eves 1997: 111), for instance, include (1), (2), (L3) and (M3). Axiom (3) becomes vacuously true under (M3) and is therefore not needed in this context.

Additional properties of fundamental importance include Desargues' Theorem and the Theorem of Pappus. In projective spaces of dimension 3 or greater there is a construction that allows one to prove Desargues' Theorem. But for dimension 2, it must be separately postulated.

Under Desargues' Theorem, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically. The resulting operations will satisfy the axioms of a fields — except that the commutativity of multiplication will require Pappus's hexagon theorem. As a result, the points of each line are in one to one correspondence with a given field, F, supplemented by an additional element, W, such that rW = W, −W = W, r+W = W, r/0 = W, r/W = 0, W−r = r−W = W. However, 0/0, W/W, W+W, W−W, 0W and W0 remain undefined.

The only projective geometry of dimension 0 is a single point. A projective geometry of dimension 1 consists of a single line containing at least 3 points. The geometric construction of arithmetic operations cannot be carried out in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of Desargues' Theorem. The simplest 2-dimensional projective geometry has 3 points on every line, with 7 points and lines in all arranged with the following schedule of collinearities: with the coordinates A = {0,0}, B = {0,1}, C = {0,W} = {1,W}, D = {1,0}, E = {W,0} = {W,1}, F = {1,1}, G = {W, W}. For an image review the Fano plane. The coordinates in a Desarguesian plane for the points designated to be the points at infinity (in this example: C, E and G) will generally not be unambiguously defined.
 * [ABC]
 * [ADE]
 * [AFG]
 * [BDG]
 * [BEF]
 * [CDF]
 * [CEG]

Visualize $$\mathbb{P}^2$$
$$\mathbb{P}^2$$ is used to map a plane into a plane. In $$\mathbb{P}^2$$ space a point is represented by the homogeneous coordinate $$(x_1,x_2,x_3)$$. If we think of $$(x_1,x_2,x_3)$$ as a point in the $$\mathbb{R}^3$$ space and third value of the homogeneous coordinate as a value in the $$z$$ direction, the $$\mathbb{P}^2$$ space can be visualized as a $$\mathbb{R}^3$$ space.

Points, rays, lines, and planes
A line can be represented in $$\mathbb{P}^2$$ space by the equation $$ax+by+c=0$$. If we treat $$a$$, $$b$$ and $$c$$ as the column vector $$\mathbf{l}$$ and $$x$$, $$y$$,$$1$$ as the column vector $$\mathbf{x}$$ then the equation above can be written in matrix form as:

$$\mathbf{x}^T\mathbf{l}=0$$ or $$\mathbf{l}^T\mathbf{x}=0$$

Or using vector notation

$$\mathbf{x}\cdot\mathbf{l}=0$$ or $$\mathbf{l}\cdot\mathbf{x}=0$$

$$k(\mathbf{x}^T\mathbf{l})=0$$ sweeps out a plane that goes through zero in $$\mathbb{R}^3$$ and $$k(\mathbf{x})$$ sweeps out a ray ( a ray goes through zero).

The plane and ray are subspaces in $$\mathbb{R}^3$$. A subspace always goes through zero.

Ideal points


In $$\mathbb{P}^2$$ the equation of a line is $$ax+by+c=0$$ and this equation can represent a line on any plane parallel to the $$x, y$$ plane by multiplying the equation by $$k$$.

If $$z=1$$ we have a normalized homogeneous coordinate. All points that have $$z=1$$ create a plane. Let's pretend we are looking at that plane and there are two parallel lines drawn on the plane. From where we are standing we can see only so much of the plane (the area outlined in red). If we walk away from the plane along the z axis we can see more of the plane. In our field of view original points have moved. We can reflect this movement by dividing the homogeneous coordinate by a constant. In the image to the right we have divided by 2 so the $$z$$ value now becomes 0.5. If we walk far enough away what we are looking at becomes a point in the distance. As we walk away we see more and more of the parallel lines. The lines will meet at a line at infinity ( a line that goes through zero on the plane at $$z=0$$). Lines on the plane when $$z=0$$ are ideal points. The plane at $$z=0$$ is the line at infinity.

The homogeneous point $$(0,0,0)$$ is where all the real points go when you're looking at the plane from an infinite distance, a line on the $$z=0$$ plane is where parallel lines intersect.

Duality
In the equation $$\mathbf{x}^T\mathbf{l}=0$$  there are two column vectors. You can keep either constant and vary the other. If we keep the point constant $$\mathbf{x}$$ and vary the coefficients $$\mathbf{l}$$ we create new lines that go through the point. If we keep the coefficients constant and vary the points that satisfy the equation we create a line. We look upon $$x$$ as a point because the axes we are using are $$x, y$$ and $$z$$. If we instead plotted the coefficients using axis marked $$a, b, c$$ points would become lines and lines would become points. If you prove something with the data plotted on axis marked $$x, y$$ and $$z$$ the same argument can be used for the data plotted on axis marked $$a, b$$ and $$c$$. That is duality.

Lines joining points and intersection of lines (using duality)
The equation $$\mathbf{x}^T\mathbf{l}=0$$ calculates the inner product of two column vectors. The inner product of two vectors is zero if the vectors are orthogonal. To find the line between the points $$\mathbf{x}_1$$ and $$\mathbf{x}_2$$ you must find the column vector $$\mathbf{l}$$ that satisfies the equations $$\mathbf{x}_1^T\mathbf{l}=0$$ and $$\mathbf{x}_2^T\mathbf{l}=0$$, that is we must find a column vector $$\mathbf{l}$$ that is orthogonal to $$\mathbf{x}_1$$ and $$\mathbf{x}_2$$. The cross product will find such a vector. The line joining two points is given by the equation $$\mathbf{x}_1 \times \mathbf{x}_2$$.

To find the intersection of two lines you look to duality. If you plot $$\mathbf{l}$$ in the coefficient space you get rays. To find the point $$\mathbf{x}$$ that is orthogonal to the two rays you find the cross product. That is $$\mathbf{l}_1 \times \mathbf{l}_2$$.

Projective transformation
A projective Transformation in $$\mathbb{P}^2$$ space is an invertible mapping of points in  $$\mathbb{P}^2$$ to points in  $$\mathbb{P}^2$$ that maps lines to lines. A $$\mathbb{P}^2$$ projectivity has the equation: $$\mathbf{x'}=\mathbf{Hx}$$. Where $$\mathbf H$$ is an invertible $$3\times3$$ matrix. This is, a projectivity is any conceivable invertible linear transform of homogeneous coordinates.

Duality
For projective spaces of dimension N, there will exist a duality between the subspaces of dimension R and dimension N−R−1. For N = 2, this specializes to the most commonly known form of duality — that between points and lines.

In 1825, Joseph Gergonne noted the principle of duality characterizing projective plane geometry: given any theorem or definition of that geometry, substituting point for line, lie on for pass through, collinear for concurrent, intersection for join, or vice versa, results in another theorem or valid definition, the "dual" of the first.

Similarly in 3 dimensions, the duality relation holds between points and planes, allowing any theorem to be transformed by swapping "point" and "plane", "is contained by" and "contains". To establish duality only requires establishing the dual versions of the axioms for the dimension in question. Thus, for 3-dimensional spaces, one needs to show that (1*) every line lies in 3 distinct planes, (2*) every two planes intersect in a unique line and a dual version of (3*) to the effect: if the intersection of plane P and Q is coplanar with the intersection of plane R and S, then so are the respective intersections of planes P and R, Q and S (assuming planes P and S are distinct from Q and R).

The duality principle was also discovered independently by Jean-Victor Poncelet.

Whatever its precise foundational status, projective geometry did include basic incidence properties. That means that any two distinct lines L and M in the projective plane intersect at exactly one point P. The special case in analytic geometry of parallel lines is subsumed in the smoother form of a line at infinity on which P lies. The point is then that the line at infinity is a line like any other in the theory: it is in no way special or distinguished. (In the later spirit of the Erlangen programme one could point to the way the group of transformations can move any line to the line at infinity).

Projective geometry also includes a full theory of conic sections, a subject already very well developed in Euclidian geometry. There are clear advantages in being able to think of a hyperbola and an ellipse as distinguished only by the way the hyperbola lies across the line at infinity; and that a parabola is tangent to the same line. The whole family of circles can be seen as the conics passing through two given points on the line at infinity — at the cost of requiring complex number coordinates. Since coordinates were not "synthetic", one replaces them by fixing a line and two points on it, and considering the linear system of all conics passing through those points as the basic object of study. This approach proved very attractive to talented geometers, and the field was thoroughly worked over. An example of this approach is the multi-volume treatise by H. F. Baker.

History
Projective geometry originated through the efforts of a French artist and mathematician, Gerard Desargues (1591–1661), as an alternative way of constructing perspective drawings. By generalizing the use of vanishing points to include the case when these are infinitely far away, he made Euclidean geometry, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. The work of Desargues was totally ignored until Michel Chasles chanced upon a handwritten copy in 1845. Meanwhile, Jean-Victor Poncelet had published the foundational treatise on projective geometry in 1822. The non-Euclidean geometries discovered shortly thereafter were eventually demonstrated to have models, such as the Klein model of hyperbolic space, relating to projective geometry.

This early 19th century projective geometry was a stepping stone from analytic geometry to algebraic geometry. When treated in terms of homogeneous coordinates, projective geometry looks like an extension or technical improvement of the use of coordinates to reduce geometric problems to algebra, an extension reducing the number of special cases. The detailed study of quadrics and the "line geometry" of Julius Plücker still form a rich set of examples for geometers working with more general concepts.

The work of Poncelet, Steiner and others was not intended to extend analytic geometry. Techniques were supposed to be synthetic: in effect projective space as now understood was to be introduced axiomatically. As a result, reformulating early work in projective geometry so that it satisfies current standards of rigor can be somewhat difficult. Even in the case of the projective plane alone, the axiomatic approach can result in models not describable via linear algebra.

This period in geometry was overtaken by research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretched existing techniques, and then by invariant theory. Towards the end of the century the Italian school of algebraic geometry (Enriques, Segre, Severi) broke out of the traditional subject matter into an area demanding deeper techniques.

In the later part of the 19th century, the detailed study of projective geometry became less important, although the literature is voluminous. Some important work was done in enumerative geometry in particular, by Schubert, that is now seen as anticipating the theory of Chern classes, taken as representing the algebraic topology of Grassmannians.

Hermann von Baravalle has explored the pedagogical potential of projective geometry for school mathematics.

Forms of the living world
In the spirit of projective geometry's origins in synthetic geometry, some mathematicians have investigated projective geometry as a useful way of describing natural phenomena. The first research in this direction was stimulated by a suggestion by the philosopher Rudolf Steiner (not to be confused with the mathematician Jakob Steiner, mentioned above).

In the first half of the twentieth century, both George Adams, and Louis Locher-Ernst independently explored the tension between central forces and peripheral influences. Lawrence Edwards (1912–2004) discovered significant applications of Klein path curves to organic development. In the spirit of D'Arcy Thompson's On Growth and Form, but with more mathematical rigor, Edwards demonstrated that such forms as the buds of leaves and flowers, pine cones, eggs, and the human heart can be simply described by certain path curves. Varying a single parameter, lambda, metamorphoses the interaction of what are known in projective geometry as growth measures into surprisingly accurate representations of many organic forms not otherwise easily describable mathematically; negative values of the same parameter produce inversions representing vortices of both water and of air.