the required line are C and d, where the traces of the given planes intersect. From those points respectively let fall C c and

X P

X

M

Fig. 8.

d D perpendicular to X X; join C D, cd; these will be the projections required.

28. To Find the Projections of the Point where a Straight Line Intersects a Plane (the traces of the line and of the plane being given), it is only necessary to draw the traces of two planes traversing the given line in convenient directions, and find the projections of the lines in which those two planes cut the given plane; the intersections of those projections will be the projections of the point required.

Fig. 9.

BX

29. Given (in fig. 10), the Traces of Two Planes, CA d, C B d,

to Find the Angle between them.-From either of the intersections of the traces (say d) let fall d D perpendicular to X X; draw DC, joining D with the other intersection of the traces. Through any convenient point, I, in D C, draw GI H perpendicular to D C,

=

cutting A C in G and BC in H. Along X X lay off D E DC, and Di = DI; join d E (this will be the length of the line of intersection of the planes). From i let fall ik perpendicular to dE; in IC take I K ik; join K G, KH; GK H will be the

angle required.

=

When the traces of the two given planes are inconveniently placed for the completion of the figure, we may substitute for either pair of traces another pair of traces parallel to them, and more conveniently placed.

30. Given (in fig. 10), the Traces, A ♂ and A C, of a Plane; also the Traces, d and C, of a Straight Line in that Plane; to Draw the Traces of a Planc which shall Cut the given Plane in that Line at a given Angle.—From either of the traces of the straight line, as d, let fall d D perpendicular to X X; draw the straight line D C, joining D with the other trace, C, of the straight line. Through any convenient point, I, in D C, draw IG perpendicular to DC, cutting CA in G. In X X lay off D E = D C and Di = DI; join d E, and on it let fall the perpendicular i k. In IC take ΙΚ ik; join K G. Then draw K H, making G K H = the given angle, and cutting G I, produced if necessary, in H. Draw CH, cutting X X in B, and join B d; these will be the traces of the plane required.

31. Given (in fig. 11), the Traces of a Plane, A B C, and the Projections of a Point, G, g, to Draw the Traces of a Plane Traversing the given Point, and Parallel to the given Plane.-Through either of the projections of the given point (say G) draw GH parallel to the corresponding trace of the given plane, and cutting XX in H. (This will be one of the projections of a line through the given point, parallel to the trace A B of the given plane.)

Through H draw HD perpendicular to X X; and through g draw g D parallel to X X, cutting HD in D (g D will be the projection and D one of the traces of the line before mentioned). Through D draw D E parallel to C B, cutting X X in E; and through E draw E F parallel to B A; D E F will be the traces of the required plane.

32. Given, the Traces of a Plane, E F, ED (in fig. 11), and One Projection of a Point in that Plane, to Find the other Projection of that Point. Suppose g, the vertical projection of the point, to be given. Draw g D parallel to X X, cutting E D in D. From D let fall D H perpendicular to X X. From g draw g G perpendicular to X X, and from H draw H G parallel to E F; the intersection of those lines, G, will be the required horizontal projection of the given point.

33. Given (in fig. 12), the Traces, A B C, of a Plane, and the Projections, D, d, of a Point, to Draw the Projections of a Perpendicular let Fall from the Point on the Plane. From one of the projections of the given point (say D) draw DEF perpendicular to the corresponding trace, B A, of the given plane, and cutting BA in E, and X X in F. From E let fall Ee perpendicular to

X X; from F draw F perpendicular to X X, cutting the trace BC in f; join fe; from d draw dg perpendicular to BC, cutting

A

E

Fig. 12.

of the point, draw dg parallel to draw E C perpendicular to a b,

fe in g; and from g draw g G perpendicular to X X, cutting DF in G. D G and d g will be the projections of the perpendicular required.

34. Given (in fig. 13), the Pro jections of a Point, D, d, and those of a Straight Line, A B, a b, to Draw the Traces of a Plane which shail Traverse the Point, and be Perpendicular to the Line.-Through one of the projections of the given point (say D) draw D G perpendicular to A B (the corresponding projection of the given line), cutting X X in G. Through G draw Gg perpendicular to X X; through d, the other projection XX, cutting Gg in g; through g cutting X X in C; and through C draw CF perpendicular

to A B.

ECF will

be the traces of the required plane.

35. Given, the Projections of a Point and of a Straight Line, to Draw the Projections of a Perpendicular let Fall from the Point upon the Straight Line.—

Find by the preceding rule the traces of a plane traversing the given point, and perpendicular to the given line; then, by Article 23, find the traces of a plane traversing the given point and line;

and finally, by Article 27, find the projection of the line of intersection of those two planes.

36. Given, the Projections of Two Straight Lines that are neither

Parallel nor Intersecting, to Find the Projections of their Common Perpendicular. By Article 23 A, find the traces of a plane traversing one of the lines and parallel to the other. Then, by Article 33, find the projections of a perpendicular let fall on that plane from any convenient point in the second line. Then through the projections of the foot of that perpendicular draw the projections of a straight line parallel to the second straight line; these will cut the projections of the first straight line at one end of the common perpendicular, whose projections will be parallel and equal to those of the perpendicular already found.

36A. Projections of a Circle.-When an instrument which draws ellipses accurately is at hand, it may be used for the purpose of drawing the project ons of a circle of a given radius, described about a given poin. in a given plane, and may thus facilitate much the solution of various problems. The following is the process for obtaining the projections of a circle:

Given (in fig. 14), the Traces of a Plane, A B C, and the Projections of a Point in that Plane, D, d, to Draw the Projections of a Circle of a given Radius, described in the given Plane and about the given Point.-For the vertical projection, describe about d a circle of the given radius, df = de, and draw the diameter ef parallel to the trace

C B; ef will itself be the vertical projection of one diameter of the circle. Draw dg perpendicular to ef. Find, by Article 24, the angle which the given plane makes with the vertical plane of projection, and lay off g d h equal to the angle so found. From h, in the circle, draw hk parallel to fe, and cutting dg in k; then dk will be the

vertical projection of a

radius of the circle perpendicular to ef. Then on the major axis, ef, and minor semi-axis, d k, describe an ellipse; that ellipse will be the required vertical projection of the circle.

The horizontal projection is obtained by a precisely similar process, the rule of Article 24 being now used to find the angle which the given plane makes with the horizontal plane of projection.

The two ellipses are both touched by a pair of tangents, M m,