481 



PERSPECTIVE. 



.PERSPECTIVE. 



422 



vanishing plane respectively, while the vanishing point of each such 

 intersection in the original plane will be a second point in the vanishing 

 line of the plane inclined to the former. Therefore this vanishing line 

 will be determined, it" these two points be found. 



1"2. For if aright conical surface be imagined formed round the 

 auxiliary radial as an axis, having the vertex for its apex, and its side 

 forming with that axis the complement to the angle the secondary 

 original planes form with the first; the vanishing planes of these 

 secondary original ones will touch that conical surface in a line, which 

 will be the principal radial of each such vanishing plane ; and there- 

 fore each vanishing plane will cut the first original plane in a line 

 tangential to the circular section of the conical surface by that original 

 plane. Again, the point in which these tangents to the circular section 

 cut the intersecting line of the firtt original plane, will be common to 

 the intersections with the plane of the picture of each vanishing plane 

 respectively, or to the vanishing line as above stated. 



103. Constructions 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 oil 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 Hues 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 and 

 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, its position with respect to the vertex and the 

 plane of the picture being also given or assumed. 



P I 



108. Draw T z * 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 c v' 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 T c' equal to the comple- 

 ment of that at which the pkne of the tetrahedron is assumed to be 

 inclined to the plane of the picture (96), T c' will be the principal 

 radial of the plane, and c' the centre of its vanishing line ; consequently 

 a line, P, v,, drawn through c', parallel to Y z (82), will be that vanish- 

 ing line. Draw v Q, perpendicular to T 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 R parallel to v c' for the intersection of the auxiliary 

 vanishing plane with the original one ; n therefore, in which z R meets 

 v y, will represent the point in which the auxiliary radial meets the 

 original plane. 



110. Makec'v'in that line equal to c'v, and through V draw a 

 parallel to Tz, which will represent the vertex and its parallel (81) I 

 Drought into the plane of the picture by the turning of the vanishing 

 plane on the vanishing lino p, r^ 



In tbli and some subsequent flg-urci, the bracket with a letter to it If 

 Intended to signify that that letter applies to the point of convergence of the 

 llnei thus Indicated j as for example in this figure, i t refers to the point In 

 which FJ r t , and r, , would meet. It must be observed also, that reference 

 ti in some places made to lines or points, not shown in the figures, to prevent 



111. Draw the equilateral triangle ADD 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.inthe same direction the vanishing plane was turned round in, 

 on p. p,. Through v 7 draw the radials of the sides of the triangle 

 parallel to them, and cutting the vanishing line in P,, P 2 , 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 lines of its plane, ABC would have been above 

 the former line, and its image abd below it ; if that plane be supposed 

 turned round in the same direction. 



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 poiut 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 

 line, will cut the image of the original line in those of the points A, n. 



Let T be the intersecting point of an orlcinnl line TA, and t Its vanishing 

 point, T T being its indefinite image; then if v A ami TV be drawn in nny, 

 whatsoever, but opposite direction*, parallel to each other, TA being made 



