74. The point A, in which any original line cuts the plane of the picture, is termed its intersecting point ; and n, iu which it cuts the vertical plane, is termed the station point of that original line.
75. If a line v r be supposed to pass through the vertex, parallel to any original line B B", it will cut the plane of the picture, if the original line itself be not parallel to that plane. This line v r is termed the radial of B n", and the point r, in whichlthis radial cuts the plane of the picture, is the vanishing point of the original line.
76. If any original line pass through the vertex, its radial will coincide with it, and the point in which such a line cuts the plane of the picture will not only be its common intersecting and vanishing points, but also the common image of all points in the original line, and consequently of the entire line itself.
77. It follows from these theorems that the original line, its director, its radial, and its image, all lie in its projecting plane, and therefore its image must pass through its intersecting and vanishing points; while its director must pass through its station point : and that these four lines must form a parallelogram, unless the original line be parallel to the plane of the picture; in which case the director and radial will coincide in one line, lying in the vertical plane, parallel both to the original and to its image.
78. Let us now consider the projection or image of any point B in an original line B n", and the situation of that imago in the indefinite one of the line, according to the position of the point B.
If a be at A, the Intersecting point of B a" (74); B and its image b• will coincide. If a lie in that part of A n which ie on the contrary side of the plane of the picture to that on which the vertex is situated, its image 6 will lie between the Intersecting and vanishing points, A, r, of the line : and If B be supposed to recede farther and farther from the former, the nearer to the latter will its itriA0 approach ; so that the vanishing point Is the Grail of the successive images of points, farther and farther distant from the vertex, or It may be considered as the image of an infinitely distant point in the original line.
79. If the point u' be situated between the intersecting and station points of the line, its image b' will be on the contrary aide of the inter. setting point to that on which the vanishing point P Is situated ; and If a be the station point n of the original line, it can have no image, or it. image may be eonaidered at an infinite distance from the vanishing point in either direction.
SO. If the point a" lie on the contrary side of the vertical plane to that on which the plane of the picture is situated, its image 6" will lie on the contrary side of the vanishing point to that on which the inter secting point Is situated ; and, as before, the vanishing point may be considered as the limit of the images in this direction ; er as the image of a point In the original line at an infinite distance from the station point in either direction.
81. Let two or more original lines be conceived as lying in an original plane T Z, and suppose a plane w, which will be termed the mein/tine place of the original one, to pass through the vertex parallel to that original plane. The lines 'Tx, D E, in which an original plane cuts the plane of the picture and the vertical plane, are called the intersecting and station lines, respectively, of that plane ; and the lines w P, T V T, in which the vanishing plane cuts the same two planes, are called the vanishing use and parallel of the rertes to that original plane.
82. The intersecting, station, and vanishing lines, and the parallel of the vertex, are all parallel to each other, these four lines being the mutual intersections of two parallel planes by two other parallel planes.
83. The intersecting and station points (74) of any original lines, lying in one plane, are points in the intersecting and station lines of that plane : and the vanishing points of the same original lines lie in the vanishing line of that plane ; for the radials of the originals must lie in the vanishing plane of that in which the original lines lie : and these radials must form with each other, and with the parallel of the vertex, angles respectively equal to those which the original lines form with each other and with the intersecting or station lines.