B

PROBLEM 9.

Given the plan of a triangular surface, to find its elevation.

Let a b c be the plan of the given surface. The elevation

elevated above b, c. It is also inclined to the vertical plane; and neither the plan nor elevation gives the real length of the lines a b, a c. The construction for determining this will be given in a subsequent problem.

26. In the preceding problems, the student has been made familiar with the projection of lines and surfaces, perpendicular, parallel, and inclined to, the planes of projection. This combination may be said to embrace the whole theory of Orthographic Projection. In the succeeding problems in Descriptive Geometry, he will only find a modification and extension of the principles already enunciated.

Before commencing the study of these problems, he is recommended to put to himself such questions as the following:

1. Under what circumstances does the projection of a line become a point? ((b), 17.)

2. Under what circumstances is the projection of a line equal to the original line, and under what circumstances is it less? ((a), 17, and Prob. 5.)

3. If the projection of a line is parallel to the base line B L, what is its relation to the other plane of projection? If the projection of a line is parallel to B L, it will also be parallel to the other plane of projection. The line a b, Fig. 7, parallel to B L, is parallel to the horizontal plane. Again, the line a' b', Fig. 11, parallel to B L, is also parallel to the vertical plane. (See also (a), Prob. 5.)

4. If both the projections of a line are parallel to B L, what is the relation of the line to the planes of projection? (Prob. 3.)

DESCRIPTIVE GEOMETRY.

27. The intersections of a plane with the planes of projection, are called the traces of the plane.

Note. In Military Drawings trace is another term for plan.

28. The intersections of a line with the planes of projection, are called the traces of the line (Obs. 3°, Prob. 2, and a, Fig. 7).

29. The traces are termed vertical, or horizontal, as they are referred to the vertical or horizontal plane.

30. When the traces of a plane are given the plane itself is given.

31. When the projections of a line are given, its traces may be found; and, vice versa, when the traces are given its projections may be found.

32. The angle between two planes is the angle contained by two straight lines, one drawn in each plane, from the same point of their common intersection, and at right angles to it.

33. The angle between two planes is called the dihedral angle, and the profile angle.

34. The angle between a straight line and a plane, is the angle contained by the straight line and its projection upon the plane.

B

PROBLEM 10.

Given the traces of a line to find its projections.

Let a, b,* be the traces of the given line, i.e., let a, b, be

Fig. 14.

angles to B L, and join a b'; a b' is the horizontal plane of the line in space.

projection upon the

To find the vertical projection, draw a a' at right angles to B L, and join a b; a' b is the projection upon the vertical plane of the line in space.

Neither the plan nor elevation expresses the real length of the line. To find this will form the subject of the next problem.

* In Descriptive Geometry, points in space are usually indicated by capital letters A, B, C, etc., and, their projections by small letters a, b, c, etc. It is necessary to preserve a consistent notation. Thus, if a is the projection of a point upon one of the two planes, its projection upon the other plane should be a', i.e., the same letter accented. In some works on the subject, the accented letters are confined to the vertical plane. In the following constructions, we shall represent points and lines first taken by the letters a, b, c, etc., whether they are in the horizontal or vertical plane, while the projections of these will be indicated by a', b', c', etc.

Note. The converse operation of this problem, viz., to determine the traces of a line when its projections are given, would be performed thus: -Let a' b, a b' be the given projections. From b', where the horizontal projection meets B L, draw b' b at right angles to B L, meeting a' b in b; b is the vertical trace of the line. Again, from a', where the vertical projection meets B L, draw a' a, meeting a b' in a; a is the horizontal trace of the line.*

The vertical trace b shows that b' is elevated above the horizontal plane a distance equal to b b'; while a, the horizontal trace, shows that a' is removed from the vertical plane a distance equal to a' a.

PROBLEM 11.

Given the projections of a line to find (1), its length, (2) the angles which it makes with the planes of projection.

Let a b, a b' be the projections of the given line.

with the horizontal plane. That the student may understand what is meant by the plane revolving upon a b', let him place his triangle with the bevelled edge on a b', and, keeping this edge in contact with the surface of the paper, turn the triangle down until it becomes horizontal. It will thus assume

* In determining the traces of a line of which neither the horizontal nor vertical projection meets B L, produce the projections until they do meet it, and then proceed as explained.

the position of the triangle a b' c. To construct this triangle, draw b'c at right angles to a b', and make it equal to b b', (for bb' expresses the height of b' above the horizontal plane of projection), and join a' c. The hypotenuse a'c is the real length of the line.

The line ac may be considered as the elevation of a' b', when viewed at right angles to the plane, passing through it at right angles to the horizontal plane, i.e., in the direction of the line cb'.

The construction may also be made in the vertical plane, thus:-Make b'd equal to a' b', and join b d; b d is the real length of the line. The triangle b d 3′ represents the vertical plane conceived to pass through a' b', after it has been made to coincide with the vertical plane of projection, by being moved through the arc a' d.

Secondly, to find the angles which the given line makes with the planes of projection, The angle made with the horizontal plane is c a' bor b d b' (34). The angle made with the vertical plane is a bg which is found thus:From a, draw a g at right angles to a b, and equal to a a'. Join b g.

35. To find the real length of a line whose projections are given, we have this practical rule:-Upon the given horizontal projection construct a right-angled triangle of which the height or perpendicular is equal to the difference of the altitudes of the extremities of the line above the plane of projection.

As referred to the vertical plane, we should make the vertical projection of the line the base of a right-angled triangle of which the perpendicular is equal to the difference of the distances of the extremities of the line from the vertical plane of projection.