113 



PERSPECTIVE. 



PERSPECTIVE. 



414 



given relations to each other, and from their conventional relations to 

 the supposed co-ordinate planes, which may in every case be conceived 

 to be so situated as to facilitate these constructions. Having thus got 

 the projections of known or given lines, he proceeds from these data to 

 ascertain the absolute magnitudes of lines and angles depending on 

 these given ones, by making the constructions alluded to, founded on 

 the supposition of projecting lines and planes being turned round on 

 the projections determined by them, till they coincide with the co- 

 ordinate planes. 



85. If a plane be turned round on its intersection with another, a 

 line in the former will make the same angle with that intersection, 

 when the two planes are brought into one, that the line made with 

 that intersection when the planes were in situ. The two linea which 

 are the intersections of the projecting plane of a point (5) with the 

 co-ordinate planes in situ, which lines have been shown to be equal 

 respectively to the projecting lines (4) of that point, will be both 

 perpendicular to Y z, and therefore will coincide in one line perpen- 

 dicular to that line YZ, when the two co-ordinate planes coincide 

 in one. 



36. The two co-ordinate planes in situ form four dihedral angles, 

 and an original point may be situated in any one of these ; that is to 

 say, of a system of related original points referred to those planes, 

 some one or more may be in different dihedral angles : it is essential 

 that the learner should know how to assign the relative situation of the 

 original points in space from the relative situation of their plans anil 

 elevations to T z. 



37. Let us distinguish the four dihedral angles thus : 



\. +x+y; the point F,p*ia n the dihedral angle No. 1. 



2. x + y; i,t No. 2. 



3. x y ; M,m No. 3. 



4. +x y; x,n No. 4. 



38. Our limits will only admit of two or three examples of ele- 

 mentary constructions to illustrate the subject of projections, referring 

 to the theorem on which each step of the construction is founded. 



fiirtn a point rp in a given line A P, ap, to draw a plant 

 through T,p perpendicular to the yiven Kntf 



39. Draw a line r q through r, perpendicular to the plan of the line, 

 for that of a line parallel to the co-ordinate plane, and lying in the 



plane onght ; then (20) p 7, parallel to T z, will be the elevation of 

 this parallel. The line'_p Q.^j meets the co-ordinate plane in Q,j(13) } 



The point T, p,' ' the line AB, o,' the plane i., xn,' .ignlfy the point, 

 line, or plane, the plans, derations, or traces of which are A, a AB at IM 

 *, respectively. 



t Thin is the form of enunciation of a prob. in olld geometry, and Is to be 

 thus interpreted : Oiren the projection! p and p, on two co-ordinate planes 

 opposed to be brought into one, of a point P situated In an original line of 

 which the corresponding projections, A r, ap, are glren; to draw the lines 

 (traces) which will represent the Intersections of t plane with thoae co-ordinate 

 planei, thlt plane being-supposed to paw through T, and to be perpendicular 

 to A P. 



therefore will be a point in one trace of the plane sought : and since 

 this trace must be perpendicular to the elevation of the line, M n drawn 

 through 5 perpendicular to a p will be that trace. The same con- 

 struction, applied, mutatis' mutandis, to the other projection of the 

 point, will furnish a point in the horizontal trace of the plane sought, 

 which trace must be drawn through R perpendicular to A P. The two 

 traces thus found will intersect each other in a point of Y z (22). 



Git-en a plane L M, M n, and a point A, a .- to draw a line through the 

 point perpendicular to the plane, and to determine the point in which 

 this line cuts the tjifen plane. 



40. Through A/a draw lines perpendicular to the given traces LM 

 for the indefinite projections of the perpendicular sought (29) : from 

 the point N, in which AN cuts YZ, draw N perpendicular to Y z for the 

 intersection with the other co-ordinate plane of the plan-projecting 

 plane of the perpendicular (9) ; and from L, in which A r cuts ML, draw 

 L I perpendicular to Y z : the point I is the elevation of the point in 

 which the plan-projecting plane of the perpendicular cuts the trace 

 I.M, M I ; and is that in which the same plane cuts the trace M . 

 Consequently n I is the elevation of the intersection of the same plan- 

 projecting plane with the original plane. Now it is obvious that the 

 point sought must lie in this intersection ; consequently the point p, 

 in which ap cuts n I, must be the elevation of the point in which the 

 perpendicular intersects the given plane, 



41. The plan P of the point may be obtained by drawing fp 

 perpendicular to Y Z (35), to cut A N, the indefinite plan of the line ; or 

 by applying the foregoing construction, mutatis mutandis, to the other 

 projections. 



To draw a line through a given point r,p, to make any propoied anglt 

 with a yiven- line A B, a 6. 



42. If the proposed line is to be parallel to the given one, lines 

 drawn through the projections of the given point, parallel respectively 

 to those of the given line, will be the projections of the line sought 

 (23) ; but if the lines are not to be parallel, join t,p with any two 

 points A, a, B,6, taken at pleasure in the given line. ABPwill therefore 

 be the plan, and a bp the elevation of the triangle thus formed. Find 

 the traces, ML, in, of the plane of this triangle, by finding the points in 

 which any two of its sides intersect the co-ordinate planes (19), since 

 these points must lie in the traces required.* 



43. Draw M n perpendicular to Y z, to cut the traces anywhere at 

 pleasure in points M, n ; the line M TO, n is consequently the traces of a 

 plane assumed at pleasure as perpendicular to both co-ordinate planes 

 (22), and cutting the given plane M L n in a line, the projections of 

 which, of course, coincide with the traces of the plane. The length of 

 this line, or the real distance between the points M n when in situ, is 

 obtained by making mm' in Y z equal to mu ; then the hypothenuse 

 TO' is the intersection of the given plane with the assumed plane, 

 brought into the plane of projection by the rotation on m n of this 

 assumed plane. 



44. From M and L, as centres, with m' n, L n for radii respectively, 



* The i(le of the triangle may meet the co-ordinate planes in different 

 dihedral angles ; the projections of two of these point*, through which the same 

 trace must pass, may therefore lie on contrary sides of Y z. The traces of all 

 pianes should be drawn indefinitely extended on each side of this line YZ, or 

 are to be conceived as to extended when circumstances do not admit of their 

 being shown 10 



