VI.

SOLIDS WITH THE INCLINATIONS OF TWO ADJACENT FACES GIVEN.

For these problems refer to Chapter I., Problem 25.

LIMITS. Case I. If the given planes are at right angles to each other, their inclinations being 6o and o, 4o must be not less than 90° - 0o.

Case 2. If the given planes are not at right angles to each other, refer to Chapter I., Problem 26.

I. Plan and elevation of a cube of 3 inches edge, when the inclinations, 45° and 60°, of the planes of two adjacent faces are given.

(1) Determine the traces of plane 45° assumed at right angles to the vertical plane.

(2) Determine the projections of a perpendicular to plane 45°, in order to determine plane 60°, since every plane containing this perpendicular is at right angles to the former plane.

(3) Determine horizontal trace of plane 60°. To do this, make the point at which the perpendicular pierces

plane 45° the vertex of a right cone, generatrix 60°, base in horizontal plane. Then because every plane touching this cone is inclined 60o, a tangent to the circular base of the cone will be the horizontal trace of such a plane, and if it be drawn as well through the horizontal trace of the perpendicular to plane 45o, it will be the horizontal trace required, i. e. of a plane perpendicular to plane 45° and inclined itself 60°.

(4) Determine the plan of the intersection of the two planes. One point in this plan will be at the intersection of the horizontal traces of the two planes, and a second point will be the plan of the vertex of the cone. The line drawn through these two points will be the plan of the intersection required. Construct' this line into the horizontal plane, draw the square face of the cube on it so 'constructed,' and determine plan of the face therefrom. Complete plan and elevation of the cube as in former problems.

2. Plan and elevation of a prism hexagon. Plane of one face inclined 48°.

with base a regular Base 68°.

(1) Determine the plan of the rectangular face of prism as for the face of the cube first determined in the last problem.

(2) On a vertical plane at right angles to plane 68°, i. e. with its ground line taken at right angles to the horizontal trace of plane 68o, determine the vertical trace of plane 68°, and construct' this plane into the horizontal plane. On AB the edge common to plane of face and plane of base so 'constructed,' describe the

regular hexagon. Determine plan therefrom and complete problem as in former cases.

3. Plan and elevation of a tetrahedron. Planes of adjacent faces 6o and po.

(1) Determine the angle ao between the two adjacent faces of the solid.

(2) Draw the traces of plane 6o.

(3) Determine a second plane inclined 4o and making an angle of a with plane 6o, Chapter I., Problem 26. The intersection of these two planes 0° and po is the indefinite edge of the solid. 'Construct' plane 6o or p°, and on the intersection so 'constructed' mark off the length of the given edge of the tetrahedron, describe the face upon it, and proceed as in former problems.

I.

VII.

SECTIONS BY OBLIQUE PLANES.

To find the section of a prism standing on one of its bases when cut by an oblique plane which is given by

its traces.

Assume auxiliary vertical planes with their horizontal traces parallel to the horizontal trace of the given cutting plane, and containing the edges of the prism. The points in which these edges are cut by the plane of section. will be in the intersection of this plane with the assumed vertical planes. The heights at which they are cut will be shown by the heights at which the vertical traces of the sectional plane and assumed planes cross. These heights can readily be transferred to the proper edges in elevation, and the section required completed by joining adjacent edges.

To find the true shape of the section.

This can be done by 'constructing' the plane of section, and with it the section itself, into the horizontal or vertical plane of projection. E. g. to 'construct' point pinto the horizontal plane. Through draw pm perpendicular to the horizontal trace and meeting it in m. Measure the hypothenuse of the right-angled triangle

which has pm for its base, and the height of P for its perpendicular, from m along mp produced. This will give point P 'constructed.' If the compasses be held from m to P and then turned down to P 'constructed,' the student will be able to follow the path of the point P and line Pm while being 'constructed.' If the other points of the section be similarly 'constructed' and joined, we have the section in true shape.

To draw the development of the frustum of the solid, i. e. to determine the plane surface which if wrapped round the solid would exactly cover it.

This

The base and the true shape of the section being known, it remains to find the true form of the faces. can be ascertained by setting along a straight line the true distances of the edges, in proper succession, erecting perpendiculars from these points for the edges themselves, and measuring the height of each edge on the line representing it, and then joining adjacent points.

To draw a projection of the frustum on its plane of section.

From and the other points of the elevation of the section, draw perpendiculars to the vertical trace, and mark on them from the vertical trace the actual distances of the points P, &c. from that trace, i. e. the hypothenuses of the right-angled triangles of which perpendiculars from p', &c. on vertical trace, and from p, &c. on xy, are the sides. Join the consecutive points for the section.

For the rest of the frustum, i. e. for the points A, B, &c. of the base,