PROBLEM 49.

The three corners of an equilateral triangle of 1.25' sides are •5", 75, 4" above the horizontal plane respectively; determine the plane by its trace containing the figure and also its apparent plan.

This problem is simply a modification of those previously given in this chapter, and in which we have, instead of the angle of inclination, given the heights of the corners above the ground. If the given heights are considerable, making the space required greater than what we can conveniently use, we may, without in the least affecting the accuracy of the working, subtract an equal height from each of the given ones, and solve the problem with the new data thus obtained.

Construct the angles of inclination of the given sides by drawing lines tangental to arcs, described with distances equal to the heights given from the angular points as centres, and then proceed in the usual manner previously indicated.

NOTE. In the problems on the projection of solids we have often had the data given that the figure is inclined to the horizontal plane at a certain angle, and we have then drawn the plan and elevation according to the given position. We could, however, if we had chosen, instead of moving the object, supposed the ground plane to be tilted up until its surface should make with the horizontal plane an angle equal to the angle which, by conditions of the problem, was to have existed between the horizontal plane and the base of the figure. The best way of accomplishing this is to project the solid first upon planes parallel to the surface in question or in the ordinary way; and afterwards, by the removal of X Y, to assume a fresh plane upon which the desired drawing can be made. In constructing any new elevation, we have to observe that all projectors are always perpendicular to X Y.

EXERCISES.

1. One end of a line is 75" below the horizontal plane, the other end is 1.5" above it. The length of its plan is 3". What is the true length and inclination of the line?

2. Draw the plan of a cube of 3" edge in any position you please as long as no edge is horizontal.

3. Draw the plan of a square of 3.25" side, when its two diagonals are inclined at 28° and 38°.

4. A pentagon A B C D E, side 1.75", revolves upon the line, joining A with the centre of the opposite side till its plane becomes inclined at 50°. Draw its plan.