its corresponding line. Thus the lines AB and CD, and therefore also their point of intersection E, will coincide with their corresponding elements. The row AB has thus three points A, B, E coincident with their corresponding points, and is therefore identical with it (§ 10). As there are six lines which join two and two of the four points A, B, C, D, there are six lines such that each point in either coincides with its corresponding point. Every other line will thus have the six points in which it cuts these, and therefore all ints, coincident with their corresponding points. The proof of tlii; second part is exactly the same. It follows§ 14. If two projective figures, which are not identical, lie in the same plane, then not more than three points which are not in a line, or three lines which do not pass through a point, can be coincident with their corresponding points or lines.
If the figures are in perspective position, then they have in common one line, the axis, with all points in it, and one point, the centre, with all lines through it. No other point or line can therefore coincide with its corresponding point or line without the figures becoming identical.
It follows also that-The
correspondence between two projective planes is completely determined if there are given-either to four points in the one the corresponding four points in the other provided that no three of them lie in a line, or to any four lines the corresponding lines provided that no three of them pass through a point.
To show this we observe first that two planes 1r, 7l', ' may be made projective in such a manner that four given points A, B, C, D in the one correspond to four given points A', B, C', D' in the other; for to the lines AB, CD will correspond the lines A'B' and C'D', and to the intersection E of the former the point E' where the latter meet. The correspondence between these rows is therefore determined, as we know three pairs of corresponding points. But this determines a correspondence (by § 12). To prove that in this case and also in the case of § 12 there is but one correspondence possible, let us suppose there were two, or that we could have in the plane ir' two figures which are each projective to the figure in vr and which have each the points A'B'C'D' corresponding to the points ABCD in -ir. Then these two figures will themselves be projective and have four corresponding points coincident. They are therefore identical by § 13.
Two projective planes will be in perspective if one row coincides with its corresponding row. The line containing these rows will be the axis of projection.
As in this case every point on s coincides with its corresponding point, it follows that every row a meets its corresponding row a on s where corresponding points are united. The two rows a, a are therefore perspective (G. § 30), and the lines joining corresponding points will meet in a point S. If r be any one of these lines cutting a, a' in the points A and A' and the line s at K, then to the line AK corresponds A'K, or the ray r corres onds to itself. The points B, B' in which r cuts another pair b, b' ofpcorresponding rows must therefore be corresponding points. Hence the lines joining corresponding points in b and b' also pass through S. Similarly all lines joining corresponding points in the two planes 1r and -lr' meet in S; hence the planes are perspective. The following proposition is proved in a similar way 1-Two projective planes will be in perspective position if one pencil coincides with its corresponding one. The centre of these pencils will be the centre of perspective.
In this case the two planes must of course coincide, whilst in the first case this is not necessary.
§ 15. We shall now show that two planes which are projective according to definition (§ 12) can be brought into perspective position, hence that the new definition is really equivalent to the old. We use the following property: If two coincident planes 1r and 1r' are perspective with S as centre, then any two corresponding rows are also perspective with S as centre. This therefore is true for the rowj and j' and for i and i', of which i and j' are the lines at infinity in the two planes. If now the plane vr' be made to slide on ir so that each line moves parallel to itself, then the point at infinity in each line, and hence the whole line at infinity in 1r', remains fixed. So does the point at- inhnity on j, which thus remains coincident with its corresponding point on j, and therefore the rows j and j remain perspective, that is to say the rays joining corresponding points in them meet at some point T. Similarly the lines joining corresponding points in i and i' will meet in some point T . These two points T and T' originally coincided with each other and with S.
Conversely, if two projective planes are placed one on the other, then as soon as the lines j and i are parallel the two points T and T' can be found by joining correspond ing points in j and 7', and also in i and i'. If now a point at infinity is called A as a point in ir and B' as a point in 1r', then the oint A' will lie on i' and B on j, so that the line AA' passes througii T' and BB' through T. These two lines are parallel. If then the plane 1r' be moved parallel to itself till T' comes to T, then these two lines will coincide with each other, and with them will coincide the lines AB and A'B'. This line and similarly every line through T will thus now coincide with its corresponding line. The two planes are therefore according to the last theorem in § 14 in perspective position. l
It will be noticed that the plane 1r' may be placed on ir in two different ways, viz. if we have placed ir' on rr we may take it off and turn it over in space before we bring it back to ir, so that what was its upper becomes now its lower face. For each of these positions we get one pair of centres T, T', and only one pair, because the above process must give every perspective position. It follows-In two projective planes there are in general two and only two pencils in either such that angles in one are equal to their corresponding angles in the other. If one of these pencils is made coincident with its corresponding one, then the planes will be perspective. This agrees with the fact that two perspective planes in space can be mad; ct;incident by turning one about their axis in two different ways 8 .
In the reasoning employed it is essential that the lines j and are finite. If one lies at infinity, say j, then i and j coincide, hence their corresponding lines i' and j' will coincide; that is, i' also lies at infinity, so that the lines at infinity in the two planes are corresponding lines. If the planes are now made coincident and perspective, then it may happen that the lines at infinity correspond point for point, or can be made to do so by turning the one plane in itself. In this case the line at infinity is the axis, whilst the centre may be a Hnite point. This gives similar figures (see § 16). In the other case the line at infinity corresponds to itself without being the axis; the lines joining corresponding points therefore all coincide with it, and the centre S lies on it at infinity. The axis will be some finite line. This gives parallel projection (see § 17). For want of space we do not show how to find in these cases the perspective position, but only remark that in the first case any pair of corresponding points in -/r and vr' may be taken as the points T and T, whilst in the other case there is a pencil of parallels in -/r such that any one line of these can be made to coincide point for point with its corresponding line in 1r', and thus serve as the axis of projection. It will therefore be possible to get the planes in perspective position by first placing any point A' on its corresponding point A and then turning ir' about this point till lines joining corresponding points are parallel.
§ 16. Similar Figures.-If the axis is at infinity every line is parallel to its corresponding line. Corresponding angles are therefore equal. The figures are similar, and (§ IO) the ratio of similitude of any two corresponding rows is constant.
If similar figures are in perspective position they are said to be similarly situated, and the centre of projection is called the centre of similitude. To place two similar figures in this position, we observe that their lines at infinity will coincide as soon as both figures are put in the same plane, but the rows on them are not necessarily identical. They are projective, and hence in general not more than txvo pp1intT(f;>n§ one)wilircoinc1gde Evith g1eir correspond§ 1ng pogits in t e ot er . 34 . o ma e t em 1 entica it is eit er su cient to turn one figure in its plane till three lines in one are parallel to their corresponding lines in the other, or it is necessary before this can be done to turn the one plane over in space. It can be shown that in the former case all lines are, or no line is, parallel to its corresponding line, whilst in the second case there are two directions, at right angles to each other, which have the property that each lipe in eigher direction is parallel to its corresponding line. We a so see t at-
f in two similar jigures three lines, of which no two are parallel, are parallel respectively to their corresponding lines, then every line has this property and the two jigures are similarly situated; or Two similar figures are similarly situated as soon as two corresponding triangles are so situated.
If two similar figures are perspective without being in the same plane, their planes must be parallel as the axis is at infinity. Hence-Any plane figure is projected from any centre to a parallel plane into a similar figure.
If two similar figures are similarly situated, then corresponding points may either be on the same or on different sides of the centre. If, besides, the ratio of similitude is unity, then corresponding points will be equidistant from the centre. In the first case therefore the two figures will be identical. In the second case they will be identically equal but not coincident. They can be made to coincide by turning one in its plane through two right angles about the centre of similitude S. The figures are in involution, as is seen at once, and they are said to be symmetrical with regard to the point S as centre. If the two figures be considered as part of one, then this is said to have a centre. Thus regular polygons of an even number of fsides and parallelograms have each a centre, which is a centre o symmetry.
§ 17. Parallel Projection.-If, instead of the axis, the centre be moved to infinity, all the projecting rays will be parallel, and we get what is called parallel projection. In this case the line at inliéty passes thropgh the centre and therefore cprreéponds tr? itself ut not point or point as in the case o simi ar gures. 0 any point I at infinity corresponds therefore a point I' also at infinity ut different from the first. Hence to parallel lines meeting at I correspond parallel lines of another direction meeting at I . Further, in any two corresponding rows the two points at infinity are corresponding points; hence the rows are similar. This gives the principal properties of parallel projection:- To parallel lines correspond parallel lines; or