Projective Geometry

We shall not attempt here to justify the conception of points at infinity. The question is dis cussed at length in the books on projective geometry. It may be remarked however that the ordi nary points are just as much idealized as are the points at infinity. No one has ever seen an actual point or realized it by an experi ment of any sort. Like the point at infinity it is an ideal creation which is useful for some of the purposes of science. With the aid of these conceptions we obtain great simplicity and symmetry in the statements of geometry. For example we have : (a) any two points have one and only one straight line in common; (b) any two straight lines in a plane have one and only one point in com mon; (c) any two planes have one and only one line in common; (d) a straight line and a plane have one and only one point in common, unless the line lies in the plane; (e) a straight line and a point are on one and only one plane, unless the point is on the line. Each of these statements would have to be split up into at least two propositions in order to state its full content in the language of elementary geometry; and each of the propositions would be more complicated to state than the more general projective proposition in which it is contained.

Principle of Duality.

If we restrict attention to the points and lines of a single plane there is a duality between the two propositions (a) and (b) in the list above. They can be written: (a) Two points are on one and only one line.

(b) Two lines are on one and only one point.

Either proposition remains true if the words point and line are interchanged. The same thing is true of every theorem of the projective geometry of the plane. If it is properly formulated it remains valid when the words point and line are interchanged. This statement is called the principle of duality in the plane. (See DUALITY.) Its exact meaning depends, of course, on what we mean by a theorem of projective geometry. This is explained be low with the aid of the notion of the projective group.

Af ter the principle of duality in the plane has been compre hended it is necessary only to state one of each pair of dual theorems. The other one goes without saying. Indeed, after we have arrived at a number of propositions from which we are going to deduce all the rest by logical processes without appeal to other knowledge, and of ter we have verified the duals of these funda mental propositions, we know in advance that the principle of duality will hold for all the theorems which we are going to derive. This way of dealing with the principle of duality requires that the material of projective geometry shall be organized as a distinct body of knowledge (see the remarks on axioms, etc.,

below).

The principle can also be established by showing the existence of "duality transformations" which carry every plane figure into a dual figure in which the points and lines of the first are replaced by the lines and points of the second.

There is also a principle of duality in space, according to which the propositions of three-dimensional projective geometry when properly stated remain valid when the words point and plane are interchanged. In the list above, the propositions (a) and (c) are dual, (d) and (e) are dual, and (b) is dual to the proposition that any two straight lines with a point in common are in a common plane. In like manner there are principles of duality in spaces of any number of dimensions.

Pencils of Lines.

We must now understand that the state ments which we have made about one-dimensional projectivities hold without exception, and therefore include all the special cases which arise when one or more of the points in question is a point at infinity. Glancing again at fig. 1, it is evident that there is a one-to-one correspondence between the points of the line a and the lines through the point 0 which are in the same plane with a, namely, each point corresponds to the line joining it to 0. Such a system of lines is called a pencil of lines and the point 0 in which all lines of the pencil meet is called the centre of the pencil.

The notions of projectivity and perspectivity are now extended so as to apply to pencils of lines as follows : The correspondence between a range of points and a pencil of lines in which each point lies on its corresponding line is called a perspectivity; and any correspondence between two pencils of lines, or between a pencil of lines and a range of points, or between two ranges of points which is the resultant of any number of perspectivities is called a projectivity. This definition includes the definition previously given of a projectivity between two ranges of points as a special case. Under this definition, in fig. I, the range of points on a is projective with the pencils of lines at 0 and P as well as with the pencil of parallel lines; and each of these pencils of lines is projective with the others; each of the projectivities being a one-to-one correspondence determined by the figure. Just as in the analogous case of ranges of points, if are lines of one pencil and bi,b2,b3 are lines of another pencil, there is one and only one projectivity in which ai,a2,a3 correspond to respectively.