PROJECTIVE GEOMETRY. Projective geometry is a branch of mathematics which originated in the study of projec tions as applied to problems of perspective in the drawing of pictures and in optical instruments. These practical applications are now studied under the captions of descriptive geometry (see DESCRIPTIVE GEOMETRY) and geometrical optics (q.v.), while pro jective geometry itself is cultivated for its intrinsic interest. It is a well co-ordinated and symmetrical theory of considerable aesthetic merit, which on the one hand enables us to view elemen tary geometry as a completed whole, and on the other hand serves as the starting point of the higher algebraic geometry. For a more detailed study the reader is referred to one or more of the text books listed at the end of this article. The article itself attempts merely to state some of the general ideas, definitions and theorems in an introductory way and without proof.
The notion of projection is defined in the article on that subject (see PROJECTION). Sup pore that the points of a line a are projected into the points of a line b, and the points of b into the points of a line c, the points of c into the points of a line d, and so on, ending up with the points of the line k. Every point A of the line a then corresponds to a definite point K of the line k. This correspondence is called a projective transformation or projectivity, and is sometimes indi cated symbolically by The set of all points of a line 1 is called a range (or a pencil or row) of points, so that in the books on projective geometry there is often talk of projective ranges. In case we wish to discuss the correspondence, not of all the points of a, but only of certain ones, say
A2, A3, A4, we may writ" The first theorem about projectivities is that any three points of a straight line can be projected into any three points of a straight line. In other words the statement is true whenever it is true, (I) that A
are distinct and col linear, and (2) that K1,K2,K3 are distinct and collinear. It is also
true that if the points
are on different lines the projectivity can be brought about by the intervention of not more than two intermediate projections. That is, if the points
are given on different lines it is possible to find a line b such that
can be projected respectively into the points
of b, and these points into K1,K2,K3, respec tively. If
are on the same line, three inter mediate projections may be needed.
A second theorem is that any projective transformation of a line is fully determined by the fate of three of the points of the line. In other words, if it is specified that
goes to K1, A2 to K2,
then for any point
of the line A
there is a uniquely determined point
of the line
goes. This theorem is often referred to as the fundamental theorem of pro jective geometry.
From the point of view of elementary geometry, there are certain exceptions to these theorems, due to the existence of parallel lines. For example in fig. I, the point Ao is not projected into any point of the line b because the line from Ao to the centre of projection is parallel to b. This circumstance complicates the statement of the theorems about projections if we adhere to the point of view of elementary geometry. In projective geometry it is avoided by introducing the conception of points at infinity, or ideal points. To every straight line a there is attributed one point at infinity and every line parallel to a is said to meet a in the point at infinity of a. In fig. I the point
is therefore said to be projected into the point at infinity of the line b. The points at infinity of the straight lines of a plane are said to constitute a straight line, the line at infinity of this plane. All the points at infinity of a three-dimensional space are said to constitute a plane which is called the plane at infinity.