Projective geometry

6:45 PM | BY ZeroDivide EDIT
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, in a given dimension, and that geometric transformations are permitted that move the extra points (called "points at infinity") to traditional points, and vice versa.
Properties meaningful in projective geometry are respected by this new idea of transformation, which is more radical in its effects than expressible by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometric language is adequate to the novel situation? It is not possible to talk about angles in projective geometry as it is inEuclidean geometry, because angle is an example of a concept not invariant under projective transformations, as is seen clearly inperspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions.
While the ideas were available earlier, projective geometry was mainly a development of the nineteenth century. A huge body of research made it the most representative field of geometry of that time. This was the theory of complex projective space, since the coordinates used (homogeneous coordinates) were complex numbers. Several major strands of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme leading to the study of theclassical groups) built on projective geometry. It was also a subject with a large number of practitioners for its own sake, under the banner of synthetic geometry. Another field that emerged from axiomatic studies of projective geometry is finite geometry.
The field of projective geometry is itself now divided into many research subfields, two examples of which are projective algebraic geometry (the study of projective varieties) andprojective differential geometry (the study of differential invariants of the projective transformations).

Overview[edit]

Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study ofconfigurations of points and lines. That there is indeed some geometric interest in this sparse setting was seen as projective geometry was developed by Desargues and others in their exploration of the principles of perspective art.[1] In higher dimensional spaces there are considered hyperplanes (that always meet), and other linear subspaces, which exhibit the principle of duality. The simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" (i.e. the line through them) and "two distinct lines determine a unique point" (i.e. their point of intersection) show the same structure as propositions. Projective geometry can also be seen as a geometry of constructions with a straight-edge alone.[2] Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels, and no concept of intermediacy.[3] It was realised that the theorems that do hold in projective geometry are simpler statements. For example the different conic sections are all equivalent in (complex) projective geometry, and some theorems about circles can be seen as special cases of these general theorems.
In the early 19th century the work of Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics .[3] Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa and Gino Fano late in the 19th century.[4] Projective geometry, like affineand Euclidean geometry, can also be developed from the Erlangen program of Felix Klein; projective geometry is characterized by invariants under transformations of theprojective group.
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-ratioare 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 ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine andEuclidean geometry.[8][9] Projective geometry is not "ordered"[3] and so it is a distinct foundation for geometry.

History[edit]

The first geometrical properties of a projective nature were discovered in the third century A.D. by Pappus of Alexandria.[3] Filippo Brunelleschi (1404–1472) started investigating the geometry of perspective in 1425[10] (see the history of perspective for a more thorough discussion of the work in the fine arts that motivated much of the development of projective geometry). Johannes Kepler (1571–1630) and Gérard Desargues (1591–1661) independently developed the pivotal concept of the "point at infinity".[11] Desargues developed 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 madeEuclidean geometry, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. Desargues's study on conic sections drew the attention of 16-year old Blaise Pascal and helped him formulate Pascal's theorem. The works of Gaspard Monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. The work of Desargues was 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. Poncelet separated the projective properties of objects in individual class and establishing a relationship between metric and projective properties. The non-Euclidean geometries discovered shortly thereafter were eventually demonstrated to have models, such as theKlein 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 fashionable, although the literature is voluminous. Some important work was done inenumerative geometry in particular, by Schubert, that is now seen as anticipating the theory of Chern classes, taken as representing the algebraic topology of Grassmannians.
Paul Dirac studied projective geometry and used it as a basis for developing his concepts of Quantum Mechanics, although his published results were always in algebraic form. See a blog article referring to an article and a book on this subject, also to a talk Dirac gave to a general audience in 1972 in Boston about projective geometry, without specifics as to its application in his physics.

Description[edit]

Projective geometry is less restrictive than either Euclidean geometry or affine geometry. It is an intrinsically non-metrical geometry, whose facts are independent of any metric structure. Under the projective transformations, the incidence structure and the relation of projective harmonic conjugates are preserved. A projective range is the one-dimensional foundation. Projective geometry formalizes one of the central principles of perspective art: that parallel lines meet at infinity, and therefore are 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 meet on a horizon line 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. In turn, all these lines lie in the plane at infinity. However, infinity is a metric concept, so a purely projective geometry does not single out any points, lines or plane in this regard—those at infinity are treated just like any others.
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, where separate but similar theorems in 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 – we single out some arbitrary projective plane as the ideal plane and locate it "at infinity" usinghomogeneous coordinates.
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 satisfy the axioms of a field—except that the commutativity of multiplication requires 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.
Projective geometry also includes a full theory of conic sections, a subject already very well developed in Euclidean 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 distinguished only by being tangent to the same line. The whole family of circles can be seen as conics passing through two given points on the line at infinity—at the cost of requiring complex coordinates. Since coordinates are 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.
There are many projective geometries, which may be divided into discrete and continuous: a discrete geometry comprises a set of points, which may or may not be finite in number, while a continuous geometry has infinitely many points with no gaps in between.
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 Fano plane is the projective plane with the fewest points and lines.
According to Greenberg (1999) and others, the simplest 2-dimensional projective geometry is the Fano plane, which has 3 points on every line, with 7 points and lines in all arranged with the following schedule of collinearities:
  • [ABC]
  • [ADE]
  • [AFG]
  • [BDG]
  • [BEF]
  • [CDF]
  • [CEG]
with the affine 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}. The coordinates in a Desarguesian plane for the points designated to be the points at infinity (in this example: C, E and G) are generally not unambiguously defined.[clarification needed]
In standard notation, a finite projective geometry is written PG(a,b) where:
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 sometimes used 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 incidence property 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 line at infinity is thus 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).
Given a line l and a point P not on the line, the elliptic parallel property contrasts with the Euclidean and hyperbolic parallel properties as follows:
Elliptic :any line through P meets l in just one point.
Euclidean :just one line through P may be found, which does not meet l.
Hyperbolic :more than one line through P may be found, which do not meet l.
The elliptic parallel property is the key idea which leads to the principle of projective duality, possibly the most important property which all projective geometries have in common.

Duality[edit]

For more details on this topic, see Duality (projective geometry).
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 onfor 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. More generally, for projective spaces of dimension N, there is 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. The duality principle was also discovered independently by Jean-Victor Poncelet.
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 conic curve (in 2 dimensions) or a quadric surface (in 3 dimensions). A commonplace example is found in the reciprocation of a symmetrical polyhedron in a concentric sphere to obtain the dual polyhedron.

Axioms of projective geometry[edit]

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[edit]

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 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.
One can add further axioms restricting the dimension or the coordinate ring. For example, Coxeter's Projective Geometry,[12] references Veblen[13] 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[edit]

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 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].
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.
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 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[edit]

Main article: Projective plane
In incidence geometry, most authors[14] give a treatment that embraces the Fano plane PG(2, 2) as the minimal finite projective plane. An axiom system that achieves this is as follows:
  • (P1) Any two distinct points lie on a unique line.
  • (P2) Any two distinct lines meet in a unique point.
  • (P3) There exist at least four points of which no three are collinear.
Coxeter's Introduction to Geometry[15] gives a list of five axioms for a more restrictive concept of a projective plane attributed to Bachmann, adding Pappus's theorem to the list of axioms above (which eliminates non-Desarguesian planes) and excluding projective planes over fields of characteristic 2 (those that don't satisfy Fano's axiom). The restricted planes given in this manner more closely resemble the real projective plane.

See also[edit]

Notes[edit]

  1. Jump up^ Ramanan 1997, p. 88
  2. Jump up^ Coxeter 2003, p. v
  3. ^ Jump up to:a b c d Coxeter 1969, p. 229
  4. Jump up^ Coxeter 2003, p. 14
  5. Jump up^ Coxeter 1969, pp. 93, 261
  6. Jump up^ Coxeter 1969, pp. 234–238
  7. Jump up^ Coxeter 2003, pp. 111–132
  8. Jump up^ Coxeter 1969, pp. 175–262
  9. Jump up^ Coxeter 2003, pp. 102–110
  10. Jump up^ Coxeter 2003, p. 2
  11. Jump up^ Coxeter 2003, p. 3
  12. Jump up^ Coxeter 2003, pp. 14–15
  13. Jump up^ Veblen 1966, pp. 16, 18, 24, 45
  14. Jump up^ Bennett 1995, pg. 4, Beutelspacher & Rosenberg 1998, pg. 8, Casse 2006, pg. 29,Cederberg 2001, pg. 9, Garner 1981, pg. 7, Hughes & Piper 1973, pg. 77, Mihalek 1972, pg. 29, Polster 1998, pg. 5 and Samuel 1988, pg. 21 among the references given.
  15. Jump up^ Coxeter 1969, pp. 229–234

References[edit]

  • F. Bachmann, 1959. Aufbau der Geometrie aus dem Spiegelungsbegriff, Springer, Berlin.
  • 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. New York: Oxford University Press. ISBN 0-19-929886-6.
  • Cederberg, Judith N. (2001). A Course in Modern Geometries. New York: Springer-Verlag. ISBN 0-387-98972-2.
  • Coxeter, H. S. M., 1995. The Real Projective Plane, 3rd ed. Springer Verlag.
  • 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. New York: John Wiley & Sons. ISBN 0-471-50458-0.
  • Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8,MR 0233275
  • Howard Eves, 1997. Foundations and Fundamental Concepts of Mathematics, 3rd ed. Dover.
  • Garner, Lynn E. (1981). An Outline of Projective Geometry. New York: North Holland. ISBN 0-444-00423-8.
  • Greenberg, M.J., 2007. Euclidean and non-Euclidean geometries, 4th ed. Freeman.
  • Richard Hartley and Andrew Zisserman, 2003. Multiple view geometry in computer vision, 2nd ed. Cambridge University Press. ISBN 0-521-54051-8
  • Hartshorne, Robin, 2009. Foundations of Projective Geometry, 2nd ed. Ishi Press. ISBN 978-4-87187-837-1
  • Hartshorne, Robin, 2000. Geometry: Euclid and Beyond. Springer.
  • Hilbert, D. and Cohn-Vossen, S., 1999. Geometry and the imagination, 2nd ed. Chelsea.
  • D. R. Hughes and F. C. Piper, 1973. Projective Planes, Springer.
  • 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. New York: Springer-Verlag. ISBN 0-387-98437-2.
  • Ramanan, S. (August 1997). "Projective geometry". Resonance (Springer India) 2 (8): 87–94. doi:10.1007/BF02835009. ISSN 0971-8044.
  • Samuel, Pierre (1988). Projective Geometry. New York: Springer-Verlag. ISBN 0-387-96752-4.
  • Veblen, Oswald; Young, J. W. A. (1938). Projective geometry. Boston: Ginn & Co. ISBN 978-1-4181-8285-4

External links[edit]

Wikimedia Commons has media related to Projective geometry.
projective geometry, branch of mathematics that deals with the relationships between geometric figures and the images, ormappings, that result from projecting them onto another surface. Common examples of projections are the shadows cast by opaque objects and motion pictures displayed on a screen.
Projective geometry has its origins in the early ItalianRenaissance, particularly in the architectural drawings of Filippo Brunelleschi (1377–1446) and Leon Battista Alberti (1404–72), who invented the method of perspectivedrawing. By this method, as shown in the figure, the eye of the painter is connected to points on the landscape (the horizontal reality plane, RP) by so-called sight lines. The intersection of these sight lines with the vertical picture plane (PP) generates the drawing. Thus, the reality plane is projected onto the picture plane, hence the name projective geometry. See also geometry: Linear perspective.
Although some isolated properties concerning projections were known in antiquity, particularly in the study of optics, it was not until the 17th century that mathematicians returned to the subject. The French mathematicians Girard Desargues (1591–1661) and Blaise Pascal (1623–62) took the first significant steps by examining what properties of figures were preserved (or invariant) underperspective mappings. The subject’s real importance, however, became clear only after 1800 in the works of several other French mathematicians, notably Jean-Victor Poncelet (1788–1867). In general, by ignoring geometric measurements such as distances and angles, projective geometry enables a clearer understanding of some more generic properties of geometric objects. Such insights have since been incorporated in many more advanced areas of mathematics.

Parallel lines and the projection of infinity

A theorem from Euclid’s Elements (c. 300 bc) states that if a line is drawn through a triangle such that it is parallel to one side (see the figure), then the line will divide the other two sides proportionately; that is, the ratio of segments on each side will be equal. This is known as the proportional segments theorem, or the fundamental theorem of similarity, and for triangle ABC, with line segment DE parallel to side AB, the theorem corresponds to the mathematical expression CD/DA = CE/EB.
Now consider the effect produced by projecting these line segments onto another plane as shown in the figure. The first thing to note is that the projected line segments A′B′ and D′E′ are not parallel; i.e., angles are not preserved. From the point of view of the projection, the parallel lines AB and DE appear to converge at the horizon, or at infinity, whose projection in the picture plane is labeled Ω. (It was Desargues who first introduced a single point at infinity to represent the projected intersection of parallel lines. Furthermore, he collected all the points along the horizon in one line at infinity.) With the introduction of Ω, the projected figure corresponds to a theorem discovered by Menelaus of Alexandriain the 1st century ad:C′D′/D′A′ = C′E′/E′B′ ∙ ΩB′/ΩA′.Since the factor ΩB′/ΩA′ corrects for the projective distortion in lengths, Menelaus’s theorem can be seen as a projective variant of the proportional segments theorem.

Projective invariants

With Desargues’s provision of infinitely distant points for parallels, the reality plane and the projective plane are essentially interchangeable—that is, ignoring distances and directions (angles), which are not preserved in the projection. Other properties are preserved, however. For instance, two different points have a unique connecting line, and two different lines have a unique point of intersection. Although almost nothing else seems to be invariant under projective mappings, one should note that lines are mapped onto lines. This means that if three points are collinear (share a common line), then the same will be true for their projections. Thus, collinearity is another invariant property. Similarly, if three lines meet in a common point, so will their projections.
The following theorem is of fundamental importance for projective geometry. In its first variant, byPappus of Alexandria (fl. ad 320) as shown in the figure, it only uses collinearity:
Let the distinct points A, B, C and D, E, F be on two different lines. Then the three intersection points—x of AE and BD, y of AF and CD, and z of BF and CE—are collinear.
The second variant, by Pascal, as shown in the figure, uses certain properties of circles:
If the distinct points A, B, C, D, E, and F are on one circle, then the three intersection points x, y, and z (defined as above) are collinear.
There is one more important invariant under projective mappings, known as the cross ratio (see thefigure). Given four distinct collinear points A, B, C, and D, the cross ratio is defined asCRat(A, B, C, D) = AC/BC ∙ BD/AD.It may also be written as the quotient of two ratios:CRat(A, B, C, D) = AC/BC : AD/BD.
The latter formulation reveals the cross ratio as a ratio of ratios of distances. And while neither distance nor the ratio of distance is preserved under projection, Pappus first proved the startling fact that the cross ratio was invariant—that is,CRat(A, B, C, D) = CRat(A′, B′, C′, D′).However, this result remained a mere curiosity until its real significance became gradually clear in the 19th century as mappings became more and more important for transforming problems from one mathematical domain to another.

Projective conic sections

Conic sections can be regarded as plane sections of a right circular cone (see the figure). By regarding a plane perpendicular to the cone’s axis as the reality plane (RP), a “cutting” plane as the picture plane (PP), and the cone’s apex as the projective “eye,” each conic section can be seen to correspond to a projective image of a circle (see the figure). Depending on the orientation of the cutting plane, the image of the circle will be a circle, an ellipse, a parabola, or a hyperbola.
A plane Ω passing through the apex and parallel to PP defines the line at infinity in the projective planePP. The situation of Ω relative to RP determines the conic section in PP: If Ω intersects RP outside the base circle (the circle formed by the intersection of the cone and RP), the image of the circle will be anellipse. If Ω is tangent to the base circle (in effect, tangent to the cone), the image will be a parabola. If Ω intersects the base circle (thus, cutting the circle in two), a hyperbola will result.
Pascal’s theorem, quoted above, also follows easily for any conic section from its special case for the circle. Start by selecting six points on a conic section and project them back onto the base circle. As given earlier, the three relevant intersection points for six points on the circle will be collinear. Now project all nine points back to the conic section. Since collinear points (the three intersection points from the circle) are mapped onto collinear points, the theorem holds for any conic section. In this way the projective point of view unites the three different types of conics.
Similarly, more complicated curves and surfaces in higher-dimensional spaces can be unified through projections. For example, Isaac Newton (1643–1727) showed that all plane curves defined by polynomials in x and y of degree 3 (the highest power of the variables is 3) can be obtained as projective images of just five types of polynomials.
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