Projective geometry

In

The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry (the study of projective varieties) and projective differential geometry (the study of differential invariants of the projective transformations).

Overview

Projective geometry is an elementary non-

After much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations. Projective geometry can be modeled by the affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary".^[5] An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates.^[6][7] On the other hand, axiomatic studies revealed the existence of non-Desarguesian planes, examples to show that the axioms of incidence can be modelled (in two dimensions only) by structures not accessible to reasoning through homogeneous coordinate systems.

In a foundational sense, projective geometry and

The first geometrical properties of a projective nature were discovered during the 3rd century by

The work of Poncelet, Jakob 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

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

Projective geometry later proved key to

Description

This section may lack focus or may be about more than one topic. In particular, "Description" is either vague or too broad.. Please help improve this article, possibly by splitting the section, or discuss this issue on the talk page. (March 2023)

Projective geometry is less restrictive than either

Because a Euclidean geometry is contained within a projective geometry—with projective geometry having a simpler foundation—general results in Euclidean geometry may be derived in a more transparent manner, where separate but similar theorems of Euclidean geometry may be handled collectively within the framework of projective geometry. For example, parallel and nonparallel lines need not be treated as separate cases; rather an arbitrary projective plane is singled out as the ideal plane and located "at infinity" using homogeneous coordinates.

Additional properties of fundamental importance include

Using

The smallest 2-dimensional projective geometry (that with the fewest points) is the Fano plane, which has 3 points on every line, with 7 points and 7 lines in all, having the following collinearities:

with homogeneous coordinates A = (0,0,1), B = (0,1,1), C = (0,1,0), D = (1,0,1), E = (1,0,0), F = (1,1,1), G = (1,1,0), or, in affine coordinates, A = (0,0), B = (0,1), C = (∞), D = (1,0), E = (0), F = (1,1)and G = (1). The affine coordinates in a Desarguesian plane for the points designated to be the points at infinity (in this example: C, E and G) can be defined in several other ways.

In standard notation, a

a is the projective (or geometric) dimension, and

b is one less than the number of points on a line (called the order of the geometry).

Thus, the example having only 7 points is written PG(2, 2).

The term "projective geometry" is used sometimes to indicate the generalised underlying abstract geometry, and sometimes to indicate a particular geometry of wide interest, such as the metric geometry of flat space which we analyse through the use of homogeneous coordinates, and in which Euclidean geometry may be embedded (hence its name, Extended Euclidean plane).

The fundamental property that singles out all projective geometries is the elliptic

Given a line l and a point P not on the line,

Elliptic

there exists no line through P that does not meet l

Euclidean

there exists exactly one line through P that does not meet l

Hyperbolic

there exists more than one line through P that does not meet l

The parallel property of elliptic geometry is the key idea that leads to the principle of projective duality, possibly the most important property that all projective geometries have in common.

Duality

In 1825,

To establish duality only requires establishing theorems which are the dual versions of the axioms for the dimension in question. Thus, for 3-dimensional spaces, one needs to show that (1*) every point 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).

In practice, the principle of duality allows us to set up a dual correspondence between two geometric constructions. The most famous of these is the polarity or reciprocity of two figures in a

Another example is Brianchon's theorem, the dual of the already mentioned Pascal's theorem, and one of whose proofs simply consists of applying the principle of duality to Pascal's. Here are comparative statements of these two theorems (in both cases within the framework of the projective plane):

• Pascal: If all six vertices of a hexagon lie on a conic, then the intersections of its opposite sides (regarded as full lines, since in the projective plane there is no such thing as a "line segment") are three collinear points. The line joining them is then called the Pascal line of the hexagon.
• Brianchon: If all six sides of a hexagon are tangent to a conic, then its diagonals (i.e. the lines joining opposite vertices) are three concurrent lines. Their point of intersection is then called the Brianchon point of the hexagon.

(If the conic degenerates into two straight lines, Pascal's becomes Pappus's theorem, which has no interesting dual, since the Brianchon point trivially becomes the two lines' intersection point.)

Axioms of projective geometry

Any given geometry may be deduced from an appropriate set of axioms. Projective geometries are characterised by the "elliptic parallel" axiom, that any two planes always meet in just one line, or in the plane, any two lines always meet in just one point. In other words, there are no such things as parallel lines or planes in projective geometry.

Many alternative sets of axioms for projective geometry have been proposed (see for example Coxeter 2003, Hilbert & Cohn-Vossen 1999, Greenberg 1980).

Whitehead's axioms

These axioms are based on Whitehead, "The Axioms of Projective Geometry". There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are:

• G1: Every line contains at least 3 points
• G2: Every two distinct 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 to eliminate some degenerate cases. The spaces satisfying these three axioms either have at most one line, or are projective spaces of some dimension over a division ring, or are non-Desarguesian planes.

Additional axioms

One can add further axioms restricting the dimension or the coordinate ring. For example, Coxeter's Projective Geometry,^[14] references Veblen^[15] in the three axioms above, together with a further 5 axioms that make the dimension 3 and the coordinate ring a commutative field of characteristic not 2.

Axioms using a ternary relation

One can pursue axiomatization by postulating a ternary relation, [ABC] to denote when three points (not all necessarily distinct) are collinear. An axiomatization may be written down in terms of this relation as well:

• C0: [ABA]
• C1: If A and B are distinct points such that [ABC] and [ABD] then [BDC]
• C2: If A and B are distinct points then there exists a third distinct point C such that [ABC]
• C3: If A and C are distinct points, B and D are distinct points, with [BCE], [ADE] but not [ABE] then there is a point F such that [ACF] and [BDF].

For two distinct 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.

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 projective 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:

• (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,

and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar lines intersect—the very principle that projective geometry was originally intended to embody. Therefore, property (M3) may be equivalently stated that all lines intersect one another.

It is generally assumed that projective spaces are of at least dimension 2. In some cases, if the focus is 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.

Axioms for projective planes

In incidence geometry, most authors^[16] give a treatment that embraces the Fano plane PG(2, 2) as the smallest finite projective plane. An axiom system that achieves this is as follows:

• (P1) Any two distinct points lie on a line that is unique.
• (P2) Any two distinct lines meet at a point that is unique.
• (P3) There exist at least four points of which no three are collinear.

Coxeter's Introduction to Geometry

While corresponding points of a perspectivity all converge at a point, this convergence is not true for a projectivity that is not a perspectivity. In projective geometry the intersection of lines formed by corresponding points of a projectivity in a plane are of particular interest. The set of such intersections is called a projective conic, and in acknowledgement of the work of Jakob Steiner, it is referred to as a Steiner conic.

Suppose a projectivity is formed by two perspectivities centered on points A and B, relating x to X by an intermediary p:

${\displaystyle x\ {\overset {A}{\doublebarwedge }}\ p\ {\overset {B}{\doublebarwedge }}\ X.}$

The projectivity is then ${\displaystyle x\ \barwedge \ X.}$ Then given the projectivity ${\displaystyle \barwedge }$ the induced conic is

${\displaystyle C(\barwedge )\ =\ \bigcup \{xX\cdot yY:x\barwedge X\ \ \land \ \ y\barwedge Y\}.}$

Given a conic C and a point P not on it, two distinct secant lines through P intersect C in four points. These four points determine a quadrangle of which P is a diagonal point. The line through the other two diagonal points is called the polar of P and P is the pole of this line.^[19] Alternatively, the polar line of P is the set of projective harmonic conjugates of P on a variable secant line passing through P and C.

See also

Notes

References

• Bachmann, F. (2013) [1959]. Aufbau der Geometrie aus dem Spiegelungsbegriff (2nd ed.). Springer-Verlag.
  ISBN 978-3-642-65537-1
  .
• Baer, Reinhold (2005). Linear Algebra and Projective Geometry. Mineola NY: Dover.
  ISBN 0-486-44565-8
  .
• Bennett, M.K. (1995). Affine and Projective Geometry. New York: Wiley.
  ISBN 0-471-11315-8
  .
• Beutelspacher, Albrecht; Rosenbaum, Ute (1998). Projective Geometry: From Foundations to Applications. Cambridge: Cambridge University Press.
  ISBN 0-521-48277-1
  .
• Casse, Rey (2006). Projective Geometry: An Introduction. Oxford University Press.
  ISBN 0-19-929886-6
  .
• Cederberg, Judith N. (2001). A Course in Modern Geometries. Springer-Verlag.
  ISBN 0-387-98972-2
  .
• ISBN 9781461227342
  .
• Coxeter, H.S.M. (2003). Projective Geometry (2nd ed.). Springer Verlag.
  ISBN 978-0-387-40623-7
  .
• Coxeter, H.S.M. (1969). Introduction to Geometry. Wiley.
  ISBN 0-471-50458-0
  .
• Dembowski, Peter (1968). Finite Geometries.
  MR 0233275
  .
• ISBN 978-0-486-13220-4
  .
• Garner, Lynn E. (1981). An Outline of Projective Geometry. North Holland.
  ISBN 0-444-00423-8
  .
• Greenberg, M.J. (2008). Euclidean and Non-Euclidean Geometries: Development and History (4th ed.). W. H. Freeman.
  ISBN 978-1-4292-8133-1
  .
• Halsted, G. B. (1906). Synthetic Projective Geometry. New York Wiley.
• Hartley, Richard; Zisserman, Andrew (2003). Multiple view geometry in computer vision (2nd ed.). Cambridge University Press.
  ISBN 0-521-54051-8
  .
• ISBN 978-4-87187-837-1
  .
• Hartshorne, Robin (2013) [2000]. Geometry: Euclid and Beyond. Springer.
  ISBN 978-0-387-22676-7
  .
• ISBN 978-0-8218-1998-2
  .
• Hughes, D.R.; Piper, F.C. (1973). Projective Planes. Springer-Verlag.
  ISBN 978-3-540-90044-3
  .
• Mihalek, R.J. (1972). Projective Geometry and Algebraic Structures. New York: Academic Press.
  ISBN 0-12-495550-9
  .
• Polster, Burkard (1998). A Geometrical Picture Book. Springer-Verlag.
  ISBN 0-387-98437-2
  .
• Ramanan, S. (August 1997). "Projective geometry". Resonance. 2 (8). Springer India: 87–94.
  S2CID 195303696
  .
• Samuel, Pierre (1988). Projective Geometry. Springer-Verlag.
  ISBN 0-387-96752-4
  .
• Santaló, Luis (1966) Geometría proyectiva, Editorial Universitaria de Buenos Aires
• Veblen, Oswald; Young, J. W. A. (1938). Projective Geometry. Boston: Ginn & Co.
  ISBN 978-1-4181-8285-4
  .

External links

Wikimedia Commons has media related to Projective geometry.

• Projective Geometry for Machine Vision — tutorial by Joe Mundy and Andrew Zisserman.
• Notes based on Coxeter's The Real Projective Plane.
• Projective Geometry for Image Analysis — free tutorial by Roger Mohr and Bill Triggs.
• Projective Geometry. — free tutorial by Tom Davis.
• The Grassmann method in projective geometry A compilation of three notes by Cesare Burali-Forti on the application of exterior algebra to projective geometry
• C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann" (English translation of book)
• E. Kummer, "General theory of rectilinear ray systems" (English translation)
• M. Pasch, "On the focal surfaces of ray systems and the singularity surfaces of complexes" (English translation)