# Projective Geometry

## ratio, complex, elements, homogeneous, cross, co-ordinates, element and set

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Invariants of Binary Forms.—This theorem opens the way for a study of projective geometry by means of algebra. In prac tice it is found advantageous to use homogeneous co-ordinates (see CO-ORDINATES). An element whose non-homogeneous co-ordinate is x is represented by any pair of numbers x1,x2 such that In case x2 = o, the pair of numbers represents the element. oo (represented by B in the discussion above). Homogeneous co ordinates give a multiplicity of names for the same element, for whenever and (yi,y2) represent the same element. Any homogeneous algebraic equation represents a finite number of elements. For example represents three elements, because there are three values of the ratio which satisfy this equation. The left-hand member of a homogeneous equation of this sort, of any degree, is called a binary form (binary because there are two variables : see FORMS, ALGEBRAIC).

A projective transformation is represented in homogeneous co ordinates by linear homogeneous equations of transformation provided the points in each set of f our are collinear. Such a ratio of two ratios is called a cross ratio or a double ratio or an anhar monic ratio.

A cross ratio of four collinear points is the same as the corre sponding cross ratio of any four points into which they are pro jected. This fact enables us to define the cross ratio of any four elements of any one-dimensional form, and in each case the cross ratio has a significant geometrical meaning. For example, we can define the cross ratio of four points of a conic as that of any four points of a straight line which are projective with the given four points of the conic. It then can be proved very simply that when ever the cross ratio ABCD is — I the tangents to the conic at A and B intersect on the line CD ; and conversely, if the tangents at A and B intersect on the line CD then the cross ratio ABCD is — I. Whenever the cross ratio ABCD of any four elements of a one-dimensional form is — i we say that the four elements form a harmonic set and that the elements AC harmonically separate the elements BD.

Co-ordinates in One-dimensional Forms.—If A,B,D are three and the projective geometry of one-dimensional forms reduces, algebraically, to the study of the effect of transformations of this sort upon binary forms. This study centres about the theory of invariants (see ALGEBRAIC FORMS) of binary forms, an in variant being a function which is unaltered except for a factor by the transformations In geometric applications it is found, in gen eral, that the vanishing of an invariant represents a geometric property which is unaltered by projectivities.

Two-dimensional Projective Geometry.--It is natural to call such collections of elements as the set of all points in a plane, or the set of all lines in a plane, or the set of all planes through a point, etc., two-dimensional forms. The concept of a projective transformation can be extended to two dimensions without diffi culty. It turns out that any projective transformation of a two dimensional form is fully determined by the fate of any four elements no three of which are in the same one-dimensional form.

Non-homogeneous co-ordinates are introduced in such a way that an element is named by an ordered pair of numbers (x,y, the first and second names of the element). If the elements are points, then these co-ordinates are ordinary cartesian co-ordinates (see ANALYTIC GEOMETRY). Homogeneous co-ordinates are sets of three numbers such that and projective transformations of the points of a plane into points of the same plane are given by equations of transformation of the type, = 12= g3 = in which the determinant of the coefficients, is not zero. These transformations are also called collineations because they trans form collinear points into collinear points.

## Geometry of the Complex Domain.

Since it is possible to perform all the operations of algebra with complex numbers (see COMPLEX NUMBERS)—that is to say with the numbers of the form -- I where a and v are real—it is possible to work out the algebraic theory of projective transformations on the assumption that the co-ordinates, and the coefficients of the equations of transformations, are complex numbers. The ob jects which are transformed are sets of triads of numbers where a given set contains all triads obtained by letting p be any real or complex number except zero. We call these objects complex points. The collection of complex points which satisfy a homogeneous first degree equation is called a complex line, and the collection of complex points which satisfy a homogeneous second degree equation a complex conic, and so on. The totality of complex points is called a complex pro jective plane. Thus there is built up, by simple algebraic processes, a complex projective geometry.

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