GROUP). The properties of any figure which remain unaltered when this figure undergoes the transformations of this group are called projective properties. For example, the property of a curve of having a second degree equation, i.e., of being a conic section, is unaltered. Likewise the property of a point, line and conic sec tion, that the point is the pole of the line with regard to the conic, is a projective property. The theory of all projective properties of plane figures is the subject matter of the projective geometry of the plane, The set of all projective collineations of a plane which leave a particular straight line invariant is a group which is called an affine group, and the theory of those properties of plane figures which are unaltered when the figures undergo the transformations of this group is called affine geometry (see AFFINE GEOMETRY).
The invariant line is called the line at infinity of the affine geom etry. An affine group has several subgroups, the most interesting being the ones determined by requiring that two points of the line at infinity shall be invariant. If the two points are real, the subgroup is one studied in the special theory of relativity (q.v.) and includes the Lorentz transformations. If they are conjugate imaginaries, the group is the group of similarity transformations of Euclidean geometry.
The Euclidean geometry can be characterized as the group of theorems which state those properties of figures which are left unaltered by the group of similarity transformations. Thus it makes no essential distinction between a large triangle and a small one which is similar to it, though it does deal with the ratio of the two triangles. But the latter is an attribute of the figure composed of the pair of triangles, not of either triangle separately. The theorems of affine and projective geometry are all included in Euclidean geometry, because any property which is left invariant by the projective group is of course left invariant by all its sub groups. Thus affine geometry is a subclass of very general the orems of Euclidean geometry which it seems desirable to isolate from the rest and study together. Projective geometry is a still
smaller class of still more general theorems. We seem by leaving out some of the details which distract our attention in the more elementary way of looking at geometry to get a deeper insight and a better grasp of the subject as a whole. Afterwards we are able to return to these details and grasp them rapidly by studying the particular subgroup which determines the Euclidean metric_ A study of the latter group with special reference to the two inva riant points gives a rapid and comprehensive survey of Euclidean geometry. For example, the circles are the conic sections which pass through these two points—and the points are therefore called the circular points at infinity. Perpendicular lines are pairs of lines which meet the line at infinity in pairs of points which har monically separate the circular points. The four tangents to a real conic from the circular points meet, in general, in four other points. Two of these are real and are the foci of the conic, and so on.