Projective geometry Totally Explained

Everything about Projective Geometry totally explained

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 didn't achieve prominence as a field of mathematics until the early 19th century through the work of Poncelet and others.

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 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 line lie in the plane at infinity. However, infinity is a metric concept, so a purely projective geometry doesn't 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's 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's 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 can't be carried out in either of these cases. For dimension 2, there's 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 = 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. 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 wasn't 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's 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.

Further Information

Get more info on 'Projective Geometry'.

External Link Exchanges

Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:

<a href="http://projective_geometry.totallyexplained.com">Projective geometry Totally Explained</a>

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.