PROBLEM 26.

Given the traces of a plane, and the projections of a line to find the angle which the line makes with the plane.

Let mm, mn, be the traces of the plane, and a b, a' b' the projections of the line.

Now, if from any point in a b a line be drawn perpendicular to the given plane, this line will make with a b the complement* of the angle which the given line makes with the given plane. If from any point b' in a b' (See Fig. 24), a

*The complement of an angle, is the difference between it and an angle of 90°, e.g. the complement of an angle of 50° is an angle of 40°; an angle of 35° is the complement of an angle of 55°.

line bb be drawn perpendicular to a b, the angle a b' b, which this line makes with a b', is the complement of the angle which a b' makes with a b; so that if we determine the angle at b', the angle at a will be the difference between it and an angle of 90°.

From a
Find c',

If a line is perpendicular to a plane, the plan of the line will be at right angles to the horizontals of the plane. Therefore, from a, draw an indefinite straight line at right angles to m m', or, what is the same thing, parallel to B L. draw a' e perpendicular to mn, meeting BL in c. the plan of c, by drawing the projector c c'. The line a, c then, drawn from a perpendicular to the given plane, makes with ab an angle cab, which is the complement of the angle which ab makes with the given plane. Find, by Prob. 23, the real magnitude of this angle. To do this we must determine the plane containing the three points a, b, c', Prob. 15. Now, since in the lines a b, a c', the points b, c, have the same index, be will be one of the horizontals of the required plane; and since the points b, c meet the horizontal plane of projection, bc' will be the horizontal trace of the plane. Therefore, draw BL at right angles to be produced (See No. 3). Now, a is elevated above the horizontal plane a distance equal to a' h (No. 2); therefore, draw a a" at right angles to B L, making a" d equal to a' h, and join a" b" a" b" is the elevation of the plane containing the three points. "Construct" this plane (See Fig. 42, and Obs. Prob. 23). We thus obtain g. Draw g g' at right angles to B L to meet a g', drawn parallel to в L (No. 3), and join b g', c' g'. Then b g' c' is the real angle contained by a b, a c'; and since this angle is the complement of the angle required, draw g' i at right angles to b g', and c'g'i is the angle, which the line a b makes with the plane whose traces are m m', m n.

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Obs. 1°. The angle bgc may be found without going

through the construction shown at No. 3, thus :-Draw a o at right angles to be, and find its real length. This is done by constructing a right-angled triangle, making the base equal a o, and the perpendicular equal to a' h, the height of a above o, and setting off the length of the hypotenuse (along a o produced) from o to g′ (35).

Obs. 2°. The line ab intersects the given plane in the point s' (Prob. 17); and since the angle, which a straight line makes with a plane, is the angle contained by the straight line and its projection on that plane (34), we have simply to find the real length of a s' as referred, first, to the given plane, and, secondly, to the line a b.

Now, the real length of a s ́, in the first case, is s'p, found by drawing a rat right angles to a b, and making a p equal to xy, the difference of the indices, or heights above the plane of projection of a, s'. Again, the real length of a s', in the second case, is s'r, found by making a r equal to x a', the reason just given. The angle psr is the angle required.

PROBLEM 27.

for

To determine the angle between two planes, or, conversely, to draw a plane to make a given angle with a given plane, and to pass through a line in the first.

a above the plane of projection, and join a b. Draw x y at right angles to a b, and let it be considered as the trace of a plane, cutting the traces of the given planes in x, y.

Now, if x y be considered as the trace of a plane at right angles to the horizontal plane, it would cut the given planes in a triangular section. This triangle will be found by making me equal to m s (for m is elevated above the plane of projection a distance equal to m s), and joining x e, y e. Now, the solution of the problem consists in finding the section of the planes when cut by a third plane at right angles, not to the horizontal plane, but to a b the planes' intersection. Therefore, draw mo at right angles to ab; mo will be the elevation of the plane drawn at right angles to ab; and since this plane contains the lines which measure the angle between the given planes, we have only to "construct "" to find this angle. Make mp equal to m o, and join x p, y p; xpy is the angle between the planes (32). The angle x py is termed the dihedral angle, and the profile angle of the planes (33).

m o

(a) When the planes are given by their horizontal and vertical traces.

Fig. 50.

B.

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Let a b, db be the horizontal traces; and a c, d c, the vertical traces of the planes.

Find, by Prob. 12, cb the horizontal projection of the

intersection of the planes. Draw x y at right angles to cb, cutting it in m. Make c's equal to e' b, and join cs; cs is the elevation of the planes' intersection. Again, make c'n equal c'm; and from n draw n p at right angles to c s. Make mo equal to n p, and join xo, yo; xoy is the angle between the planes.

(b) The converse operation, viz.—to draw a plane to make a given angle with a given plane is performed thus:-Let abgh be the given plane, See Fig. 49, a b being the line in which the planes are to intersect each other. Find ab, the elevation of a b. Draw x y at right angles to ab, cutting it in m; and from m, draw mo at right angles to a b. Make mp equal to m o, and join y p. At the point p, in y p, make the angle y p x equal to the angle which the required plane is to make with the given plane a b g h. The point x, where p x meets x y, will be a point in the horizontal trace of the plane sought; and since b is another point in this trace, join b x, and produce it to c; b c is the horizontal trace of the required plane, which will be completed by drawing a d parallel to b c.

PROBLEM 28.

To determine two parallel planes, each of which shall contain a given line, the lines not being in the same plane.