projective geometry

projective geometry

projective geometry, branch of geometry concerned with those properties of geometric figures that remain invariant under projection. The basic elements are points, lines, and planes, and the following statements are usually taken as assumptions: (1) two points lie in a unique line; (2) three points not on the same line determine a plane; (3) two lines in a plane intersect in a point; (4) two planes intersect in a line; (5) three planes not containing the same line intersect in a point. The basic elements retain their character under projection; e.g., the projection of a line is another line, and the point of intersection of two lines is projected into another point that is the intersection of the projections of the two original lines. However, lengths and ratios of lengths are not invariant under projection, nor are angles or the shapes of figures. The concept of parallelism does not appear at all in projective geometry; any pair of distinct lines intersects in a point, and if these lines are parallel in the sense of Euclidean geometry, then their point of intersection is at infinity. The plane that includes the ideal line, or line at infinity, consisting of all such ideal points, is called the projective plane. Two properties that are invariant under projection are the order of three or more points on a line and the harmonic relationship, or cross ratio, among four points, A, B, C, D, i.e., AC/BC : AD/BD. One important concept in projective geometry is that of duality. In the plane, the terms point and line are dual and can be interchanged in any valid statement to yield another valid statement, e.g., statements (1) and (3) above; in space, the terms plane, line, and point are interchanged with point, line, and plane, respectively, to yield dual statements (sometimes with slight changes in wording) as in statements (2) and (5) and statements (1) and (4) above. The origins of projective geometry are found in the work of Pappus, Gérard Desargues, and others. It first emerged as a discipline in its own right with the work of J. V. Poncelet (1822) and was placed on an axiomatic basis by K. G. C. von Staudt (1847), both these mathematicians adopting the pure, or synthetic, approach, in which algebraic and analytic methods are avoided and the treatment is purely geometric, in contrast to the approach of A. F. Möbius, Julius Plücker, and others. Projective geometry is more general than the familiar Euclidean geometry and includes the metric geometries (both Euclidean and non-Euclidean) as special cases.

Branch of mathematics that deals with the relationships between geometric figures and the images (mappings) of them that result from projection. Examples of projections include motion pictures, maps of the Earth's surface, and shadows cast by objects. One stimulus for the subject's development was the need to understand perspective in drawing and painting. Every point of the projected object and the corresponding point of its image must lie on the projection ray, a line that passes through the centre of projection. Modern projective geometry emphasizes the mathematical properties (such as straightness of lines and points of intersection) preserved in projections despite the distortion of lengths, angles, and shapes.

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Projective geometry is a non-metrical form of geometry, notable for its principle of duality. Projective geometry grew out of the principles of perspective art established during the Renaissance period, and was first systematically developed by Desargues in the 17th century, although it did not achieve prominence as a field of mathematics until the early 19th century through the work of Poncelet and others. Projective geometry involves affine, metrical (with similitude), and Euclidean geometries as its special and more restrictive cases. Consequently, the projective geometry is a natural unifying frame for a large class of geometries, a point of view emphasized by Felix Klein in his Erlangen program. The incidence structure and the cross-ratio are fundamental invariants under the projective transformations.

Description

Projective geometry is the most general and least restrictive in the hierarchy of fundamental geometries, i.e. Euclidean - metric (similarity) - affine - projective. It is an intrinsically non-metrical geometry, whose facts are independent of any metric structure. Under the projective transformations, the incidence structure and the cross-ratio are preserved. It is a non-Euclidean geometry. In particular, it 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 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" using homogeneous 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 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.

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.

According to Greenberg (1999) and others, 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:

  • [ABC]
  • [ADE]
  • [AFG]
  • [BDG]
  • [BEF]
  • [CDF]
  • [CEG]

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.

However this geometry is not sufficiently complex to be consistent with Coxeter's (2003) approach, where the simplest example has 31 points, 31 lines, and 6 points on each line, which he writes as PG[2,5].

In Coxeter's notation, a finite projective geometry is written PG[a,b] where:

a is the number of dimensions, and
given a point on a line, b is the number of other lines through the point.

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 geometry.

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

See main article - 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 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". More generally, 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. 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 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).

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

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). Those given here are 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.

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:

  • 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 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.

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. Desargues's study on conic sections drew the attention of 16-years old Blaise Pascal and helped him formulate Pascal's theorem. Important for the subsequent development of projective geometry are the works of Gaspard Monge at the end of 18 and beginning of 19 century. 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 main contributions of Poncelet were the separation of 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 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.

See also

References

  • Coxeter, H. S. M., 1995. The Real Projective Plane, 3rd ed. Springer Verlag.
  • Coxeter, H. S. M., 2003. Projective Geometry, 2nd ed. Springer Verlag.
  • Hartshorne, Robin, 2000. Geometry: Euclid and Beyond. Springer.
  • Edwards, Lawrence, 1985. Projective Geometry, Rudolph Steiner Inst. 2nd ed. 2003, Floris (ppbk).
  • Edwards, Lawrence, The Vortex of Life.
  • Howard Eves, 1997. Foundations and Fundamental Concepts of Mathematics, 3rd ed. Dover.
  • Greenberg, M.J., 1980. Euclidean and non-Euclidean geometries, 2nd ed. Freeman.
  • Hilbert, D. and Cohn-Vossen, S., 1999. Geometry and the imagination, 2nd ed. Chelsea.
  • Locher-Ernst, Louis, Space and Counterspace.
  • Oswald Veblen and J. W. A. Young, 1938–46. Projective Geometry, 2 vols. New York: Blaisdell.
  • Richard Hartley and Andrew Zisserman , 2003. Multiple view geometry in computer vision, 2nd ed. Cambridge University Press. ISBN 0-521-54051-8

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