425 



PERSPECTIVE. 



PERSPECTIVE. 



426 



cylinder by a plane be obtained, lines drawn tangents to this image 

 through the vanishing point of the axis will give the straight parts of 

 the outline of the solid ; these outlines must also be tangents to every 

 other curve which is the image of any section of the original cylinder. 

 128. If a line pass through the vertex and the apex of a cone, and 

 meet the plane of its base, or any other plane cutting the cone, two 

 lines drawn through the point of intersection tangents to the curve of 



the section will be the intersections with that plane of two others 

 passing through the vertex and tangential to the surface of the solid, 

 and these two planes will touch the cone in lines which will be the 

 originals of the outline of its image. 



129. The ray just mentioned passing through the apex of a cone is 

 analogous to the radial of a cylinder passing through the vertex, the 

 cylinder being considered as a cone, with its apex infinitely distant. 



130. If the line through the vertex and the apex of a cone, or the 

 ray of that apex, be parallel to the plane of its base, or of any section, 

 the tangents to the base lying in its plane, or in that of such section, 

 must be drawn parallel to that ray, and the image of the apex will be 

 the vanishing point of these parallel tangents. 



131. Let c be the centre of the picture ; a b, bisected in e, being 

 given as the image of a diameter, parallel to the plane of the picture, 

 of a sphere * e, therefore being the image of its centre (88). Draw an 

 indefinite line through c and e, and c V perpendicular to it, equal to 

 the assumed distance of the picture ; take any point c at pleasure in 

 e c, but as far from e. as convenient ; draw o /8 through perpendicular 

 to c e , making t a, e 0, equal to e a, e b. Join v e and set off its length 

 each way from e to / and m along a Hue perpendicular to c e. 



132. By this construction I m is a vanishing line, of which e is the 

 centre, v c equal to its principal radial, and c e its auxiliary vanishing 

 line C.ij) ; I and m will obviously be the vanishing points of the 

 diagonals of every square, lying in original planes having Im for their 



ting line, the sides of that square being parallel and perpendicular 

 to the intersecting line of its plane ; accordingly the quadrilateral 

 />/ A i is the image of such a square, lying in such a plane, and the line 

 a /3 being made equal to the given image of a diameter of the sphere, 

 a /3 and a b are the images of equal original lines parallel to the pictvire 

 ami equally distant from it, or both lying in a plane parallel to that of 

 the picture. If therefore an ellipse be described in fg h i, touching the 

 sides in the points a0y&, and having its transverse axis in c e, this 

 ellipse will be the image of an original circle equal to a great one of 

 the sphere, and having its plane parallel to that passing through the 

 vertex and the centre of the sphere, or this original circle may be 

 regarded as the oblique plan, on a plane parallel to it, of the section of 

 the sphere by the vanishing plane, the projecting lines being parallel 

 to the plane of the picture. 



133. Draw v n ptrpendiciuar to v e, cutting e c in n, and through n 

 draw a vanishing line perpendicular to en, or having en also for its 

 auxiliary vanishing line; make no,np, each equal to the auxiliary 

 radial v n ; make e r, e ,t in I m, each equal to the temi-con jugate axis 

 of the ellipse last drawn, and complete the trapezium w x y z as the 

 image of a square having op for its vanishing line, and its sides 

 parallel and perpendicular to the intersecting line of its plane. An 

 ellipse described in tcxy :, having its transverse axis in e n, mil be the 

 outline of the sphere. 



134. For n being the auxiliary vanishing point of the plane of the 

 original of fy It i , o p is the vanishing line of all planes perpendicular to 

 that original plane, and intersecting it in lines parallel to the plane of 

 the picture. The original square of the quadrilateral ic x y z is there- 

 fore perpendicular to the plane of the original of fij h i, or to the 

 vanishing plane passing through the vertex and centre of the sphere. 

 Now it will be seen that the conjugate axis of the ellipse in /</ // /' is 

 the oblique plan (59) of the chord of the tangents from the vertex to 

 the section of the sphere by the vanishing plane, which chord of the 

 tangents must be a diameter of the small circle of the solid, consti- 

 tuting the original of its apparent outline ; this small circle being the 

 base of the cone of rays tangential to its surface (62), and having iU 

 plane perpendicular to that of the vanishing plane passing through the 

 rertex and centre of the sphere; wjryzis consequently the image of 



ah need not he perpendicular to the line re ; it is shown so in the figure, 

 to avoid unnecessary lines ; but as every diameter of the sphere which lies in a 

 plane parallel to that of the picture ii also parallel to that plane, at may make 

 ny angle whatever with ce. 



t The points ra, tb, arc not the same, though they cannot be distinguished in 

 the figure. 



the square circumscribing the circular base, and the inscribed ellipse 

 that of the circle itself, or this ellipse is the outline of the sphere. 



135. If the distance of the vertex (70) be supposed to be indefinitely 

 great, compared to the magnitude of the object to be represented, the 

 pyramid of rays may be conceived to become a prism, or the rays to be 

 parallel. On this supposition the vanishing points of the lines of the 

 original object would be indefinitely distant from the centre of the 

 picture, and the images of parallel original lines would be parallels. 

 The isometric projection of a parallelepiped (57) is obviously a limited 

 case of this kind, the limitation being necessary from the object in 

 view, which induces us to adopt that kind of projection. But there 

 are occasions on which it is desirable to delineate rectilinear objects 

 pictorially, which from their small relative size, and from other con- 

 siderations, do not require the application of perspective projection, 

 and which would not be adequately represented by an isometric one. 

 In such cases the draughtsman may readily accomplish his purpose by 

 combining the principles of projection on co-ordinate planes with per- 

 spective, as in the following example. 



13(5. Let a hexagonal figure, alcdffy, be drawn, with the condition 

 that each pair of opposite sides shall be parallel, and consequently 

 equal ; from the angles a, c, f draw lines parallel to the alternate sides, 

 and meeting in a point rf, and from the intermediate angles b, e , y draw 

 lines parallel to the remaining sides respectively, and meeting in h. 

 The figure thus formed will be the orthographic or orthogonal projection 

 of a cube, under certain unknown conditions of inclination of the plane 

 of projection to the projecting lines, and of these to the original planes 

 of the solid. 



137. The projections of the centres of each face of the cube, as q, 

 may be found by drawing the diagonals, as a c, b d, and if lines be 

 drawn through the centres of each pair of opposite faces, as p r, which 

 lines will obviously be parallel to the edges of the solid, and perpen- 

 dicular to the planes of the faces, they will pass through the vertices 

 of right pyramids placed on each face. By making the altitude of 

 these pyramids, as pij, equal to half the projection of the parallel edges 

 bf, &c., of the solid, we obtain the remaining angles, I, m, n, o, p, r, of 

 the solid termed a rhomboidal dodecahedron, one diagonal of each 

 face of which is one edge of the original cube. 



138. By previously constructing the projection of a cube in the 

 manner just described, the sides of which will furnish a scale of the 

 ratio of the projections of any lines parallel to the edges of that cube, 

 the projection of any parallelopiporl may be obtained, and from this 

 again the image of any ynnnotriod solid deduced. In this manner 



