Con. 1. The perspective representations of all lines pa rallel to the perspective plane, being parallel to their ori ginals, are parallel to each other.
Cott. 2. The perspective representation of a plane figure parallel to the perspective plane, is similar to its original, and has to it the same ratio which the square of the dis tance of the point of sight from the perspective plane, has to the square of the distance of the same point from the plane of the original figure. Thus, a b c d is similar to A BCD, for the sides of the triangle a c d being parallel to the sides of the triangle ACD, the triangles (E. llth Book) are equiangular ; for the same reason, the triangles a b d, ABD, are equiangular. Hence the figure a b d c is simi lar to ABDC ; andabdc: ABDC — a Alr = : The vanishing line, and the intersection or base line, of any plane, are parallel to each other.
This proposition is quite evident ; for the vanishing line and base line are the intersections of parallel planes by the perspective plane. See Def. 7th and 9th.
The vanishing points of all lines situated in any origi nal plane, are in the vanishing line of that plane.
For the original lines being all in one plane, their pa rallels drawn through the point of sight will also be in one plane ; and the poitits in which they intersect the picture will be in the line in which that plane intersects the pic ture: but the points above-mentioned are the vanishing points of the original lines ; and the plane of the parallels being parallel to the original plane, its intersection above mentioned, with the picture, is the vanishing line of that original plane. Hence the truth of the proposition.
Cott. 1. The vanishing point of the common intersec tion of two original planes, is the intersection of their va nishing lines.
Con. 2. The vanishing line of a plane perpendicular to the picture passes through the centre of the picture. For the straight line drawn from the point of sight to the cen tre being at right angles to the picture, every plane drawn through the point of sight at right angles to the picture, must pass through that line, and consequently its intersec tion with the picture must pass through the centre ; but the vanishing line of a plane perpendicular to the picture, is the intersection of a plane, drawn through the point of sight parallel to the original, and therefore at right angles to the picture. Hence the truth of the proposition.
The intersections with the picture of all lines situated in any original plane, are in the intersection of that plane. This is evident.
Cott. 1. The intersection with the picture of the com mon intersection of two original planes, is the point in which their intersections cut each other.
COR. 2. Planes, whose common intersection is parallel to the picture, have parallel intersections, and also paral lel vanishing lines. This corollary will appear evident from what follows : Suppose a straight line to be parallel to a plane. Through the straight line let any plane be drawn to intersect the given plane, and the intersection of the two planes will be parallel to the original straight line; for that intersection can never meet the given line, being in a plane parallel to it, and being also in the same plane with it, they must be parallel to each other. Hence, the intersections of the two planes, mentioned in the corol lary, with the perspective plane, are each parallel to the common intersection of the two planes, and are therefore parallel to each other. The vanishing lines being paral lel to parallels, are also parallel to each other.
Let AB (Fig. 5.) be any straight line, and AD, BC, two straight lines drawn from its extremities parallel to each other, and let DC, the straight line which joins their ex tremities, intersect A B in E. Through the points A and B draw A d, B c, any two straight lines, also parallel to each other, and either equal to AD, BC, or having to each other the ratio of AD to BC ; and either in the plane in which AD, BC, are placed, or in a different plane : then the straight line d r, or d' c', will intersect AB in the same point E in which DC intersects it. For DC, d c, and d' c', divide AB into parts, which have the ratio of AD to BC, of A d to B c, or of A d' to B c', but these ratios are equal. Hence the point of intersection is the same for all.