If rays be drawn from v' through a, B, and n, they will be found to pass through the images a, b, d of those points, and recourse must fre quently be had to this mode of determining the image of a point in a line, when circumstances prevent the possibility of determining it by means of the image of another line, also passing through the original point. Or if the distances of any points in a line, as a, B, from its intersecting point, be set off from that point along the intersecting line, and the radial of the line be laid off along the vanishing line, from the vanishing point, of the original line ; then lines drawn from the former points in the intersecting line to the point in the vanishing IMe, will cut the image of the original line in those of the points a, D." 113. By one or other of these principles, the images of any definite right line., and therefore of any rectilinear figure, may be obtained. For one or more original lines may be always' assumed as passing through one or more points, the images of which are required ; so that the indefinite images of the Assumed lines will give those of the points sought, by its Intersections with the images of other lines, in which those points lie : and these assumed lines may be so taken as to define the images with more precision, or to obviate the necessity of drawing radials of lines but little inclined to the intersecting lines of the planes in which they lie. z n is the perpendicular distance of the point in which the auxiliary radial cute the original plane from its intersecting line ; n therefore is the centre of the circular section of the conical surface before alluded to (102). Make z n' in it v' equal to z n draw v a to make at v, with v n, the complement of the angle at which the faces of the tetrahedron are inclined to each other. From n', as it centre, with n s fora radius, describe a circle. Draw lines to touch this circle, parallel; respectively to A 13, a n, A n. Through the point p, in which the tangent parallel to A B cuts T 2, and through r„ the vanishing point of An, draw P. r,, the vanishing line of the face of the solid meeting the face A BD in A n : and on the same principles r, r,, the vanishing lines of the two remaining faces are found; then r, in which these vanishing lines intersect each other, will be the vanishing points (94) of the edges of the solid, and lines accordingly drawn from a,b,d to these points will complete the image of the tetrahedron.
114. Simple as is the construction above described, for finding the vanishing lines of planes making any proposed angle with a given plane, it may frequently be avoided by availing ourselves properly of the oymmetry of the solid to be delineated. Thus, in the example before us, after finding the image, a, b, d, of one face of the tetrahedron, we might have determined the image of the centre of that face by drawing those of the perpendiculars on each side of the triangle from the opposite angles; a line drawn through this centre and through Qwould be the image of one, perpendicular to the plane of the triangle (97); this line would pans through the vertex of the pyramid, or through the angular point in which the other three faces meet, by finding the image of this point, which can be easily done by first determining the intersecting point of the perpendicular, and the intersecting line of any plane in which it lies ; then lines drawn from a, 6, and d to this image e would complete the figure.
115. When a vanishing line is obtained, it is frequently requisite to determine its centre and distance, or its principal radial ; this is done by the construction employed to determine the vanishing line r, is,. Thus, to determine the centre, &c., of vanishing line r, P., draw
a parallel to it through c, making c v" equal c v, the distance of the picture ; also draw c v"' perpendicular to the vanishing line for its auxiliary one, cutting the former in c" its centre. Make c" v"' equal c" v", the principal radial ; then v'" r„ v"' r, being drawn, they will be the radials of the three shies, a b, eb, ea, of the face of the solid, and will be found, accordingly, to make angles of 60' with each other (S3). The radial v"' r, will also be found equal to v' r„ these lines representing one and the same line, only brought into the plane of the picture by the rotation of two different vanishing planes on their vnnis.hing lines.
116. The perspective projection of a curve may always be found by ineaus of the images of a sufficient number of points in the original, or by the projection of some inscribed or circumscribed polygon ; if the curve be ? plane one : . in this case the image of a tangent to the original curve will be a tangent to the image of that curve. For if the image of the tangent meet that of the curve in more than one point, these points must be the images of points in the original curve through which the original of the tangent must pass : which is contrary to the supposition. But there are sonic theorems regarding the perspective projection of it circle, and constructions founded on them, which ought to be well understood by the draughtsman.
117. The rays from the circumference of a circle, obviously, form a conical surface, the section of which, by the plane of the picture, will be one of the conic sections. If the original circle, or base of the cone of rays, be parallel to the plane of the picture, the image will be a circle, the radius of which will be to that of the original in the ratio of the distance of the picture (701 to the distance of the plane of the original circle from the vertex (Se).
118. If an original circle do not touch, or cut, the station line of its plane, its image will be an ellipse wherever the plane of the picture may be ; unless the section by the plane of the picture happen to be a subcontnary one, an exception to which we shall recur on rt subsequent occasion. If the station line be a tangent to the circle, its image will be a paraGe/a ; and if that line cut the circle, the image will be the opposite branches of an hyperbola, lying ou contrary sides of the vanishing line of the original plane (SO).
119. Let KNLM be an original circle, An being the station line (81) ; the image of the circle will in this instance, be an ellipse. Draw the diameter c n to the circle, perpendicular to A B ; and let c be the point in c n through which the chords of the tangents from all points in A B pass, according to the well known property of the circle. Let V represent the vertex, the vertical plane being supposed to be turned round on the station line A n, till it coincide with the piano of the circle ; v v' being the director perpendicular to the station line. Make D E, in DB, equal to the tangent to the circle drawn from D ; bisect v E by a perpendicular, cutting A 11 in F ; on P as a centre, with r• v or F E for a radius, intersect A n in A and n, and draw lines through these points and through c ; K L, at N will,be the originals of the axis of the elliptic image of the given circle, wherever the plane of the picture may be assumed, and at whatever angle that plane and the vertical one be inclined to the plane of the circle.