SIMILAR FIGITILE";. - If the axis is at infinity every line is parallel to its corresponding line. Corresponding angles are there-fore equal. The figures are similar, and (§ 10) the ratio of simili-tude 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 shnilitude. 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 two points on one will coincide with their corresponding points in the other (G. § 34). To make them identical it is either sufficient to turn one figure in its plane till three lines in one are parallel to their corresponding lines in the other, or it is ii..cessary before this can lie done to turn the one plane over in space. It can be shown that in the former ease all lines are, or uo line is, parallel to its cor-responding line, whilst in the second ease there are two directions, at right angles to each other, which have the property' that each line in either direction is parallel to its corresponding line. We also see that - If in Iwo similar figures three lines, of which no t WO are parallel, are parallel respectively to their corresponding lines, then CVery 11(18 is. properly and the ilVO figures are similarly situated ; or If two similar figures are perspective without being in the same plane, their planes must be parallel as the axis is at infinity. Pence A ay 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 corresi iondi lig points will be equidistant from the centre. In the first case therefore the two figures will be identical. In the second ease they will be identi-cally 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. l'he figures are in involution, as is seen at once, and they are said to be symmetrical with regard to the point S els centre. 11' 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 sides and parallelograms have each a centre, which is a centre of symmetry.