103. Constrictions can be made, founded on these general theorems, by which the perspective projection of plane figures, or solids, may be obtained on a plane, taken to represent the plane of the picture, and all other original or vanishing planes brought to coincide with it by being turned round ou their intersections with each other, and with the assumed plane of the picture.
104. If an original plane be supposed turned round on its intersect ing line till it coincide with the plane of the picture, the relations of lines in the plane to each other, and to the intersecting line, will not be affected by so doing. And if the vanishing plane of that original plane be also turned round on the vanishing line in the same direction, the same observation will apply to the radials of original lines in the original plane, which will preserve their original relative position to that vanishing line : these radials will be therefore parallel to the original lines respectively, when both they and the original lines are brought into the same plane.
105. Since the principal radial is perpendicular to the vanishing line, this radial will coincide with the auxiliary vanishing line when the vanishing plane is brought into the plane of the picture.
106. But if, as frequently occurs, the constructions must be made on the supposition that the original plane has been turned round on its intersecting line in one direction, and the corresponding vanishing plane turned round on its vanishing line in the contrary direction ; the radials will not be parallel to the original lines on this supposition, when the two planes coincide in one. These radials must therefore be drawn, making the same angles with the parallel of the vertex snd with the auxiliary vanishing line, that the original lines make with the intersecting line, and with lines perpendicular to it.
107. As an example of the application of the foregoing principles, let it be required to draw the perspective projection of a tetrahedron, of a given magnitude, ite position with respect to the vertex and the plane of the picture being also given or assumed.
108. Draw T at pleasure, to represent the intersecting line of one plane of the solid ; and take any point c for the centre of the picture. Through c draw nit' perpendicular to T Z for the auxiliary vanishing line of the plane (95), also draw c v parallel to T z, and equal to the given distance of the picture (70) ; this, and the following steps in the construction, being founded on the supposition of the auxiliary vanishing plane (95) being turned round on c v', till it coincide with the plane of the picture. Make the angle c v c' equal to the comple
ment of that at which the plane of the tetrahedron is assumed to be inclined to the plane of the picture (96), v c' will be the principal radial of the plane, and c' the centre of its vanishing line ; consequently a line, drawn through c', parallel to Y Z (82), will be that ing line. Draw v Q. perpendicular to v c', for the auxiliary radial, cutting v c' the auxiliary vanishing line, in q the auxiliary vanishing point of the plane of the original tetrahedron.
109. Draw z n parallel to at c' for the intersection of the auxiliary vanishing plane with the original one ; a therefore, In which z n meets v Q, will represent the point in which tins auxiliary radial meets the original plane.
110. Make c'v' in that line equal to c' v, and through v' draw a parallel to Y a, which will represent the 'vertex and its parallel (81) brought into the plane of the picture by the turning of the vanishing plane on the vanishing line re 111. Draw the equilateral triangle A BB for the face of the tetra hedron in its given or assumed position with respect to the intersecting line of its plane and the centre of the picture : this construction implies that the original plane of that face has been turned round on YZ, in the same direction the vanishing plane was turned round in, on B„ Through v' draw the radials of the sides of the triangle parallel to them, and cutting the vanishing line in P„ P,, the vanishing points of those sides ; the perspective images of which being drawn through the Intersecting and vanishing points of the sides respectively, will form the image a b d of the given face.
112. If the original triangle had been assumed as lying between the intersecting and station linen of its plane, a B C would have been above the former line, and its image a b d below it ; if that plane be supposed turned round in the same direction.