An oblique parallel projection of the surface which has just been considered is shown in Fig. 554. The student should have no difficulty in making such a projection after he has studied Chapter XXI. This projection may be very readily drawn on squared paper, taking the axis for d at 45° to the axis for v.

When the student draws the oblique parallel projection shown in Fig. 554 he should put in the additional contours suggested with reference to Fig. 553.

Exercises XX

Note. Unless otherwise stated the unit for the indices is for 01 of an inch, so that an index 25 denotes a height of 2.5 inches above the horizontal plane.

1. Two points a and b in the plan of a straight line are 2 inches apart. The index of a is 3 and the index of b is 11. Determine, (1) the index of a point c in ab which is inch from a; (2) a point d., in the plan of the line; (3) a point e1 in the plan of the line; (4) a point f- in the plan of the line; (5) the true length of EF. 2. Draw a triangle a, b, c12 (ab 2.7 inches, bc 2.2 inches, ca = 1.7 inches). Find a point d in bc whose index is 8, and show the figured plans of two straight lines passing through B and C, and parallel to AD.

=

=

3. A straight line making an angle of 30 with the ground line is both horizontal and vertical trace of a plane. Show the scale of slope of this plane.

4. Show the scales of slope of two parallel planes inclined at 50°, the distance between the planes being 0-7 inch.

5. Determine the scale of slope of the plane of the triangle given in exercise 2, also the true shape of the triangle.

6. An equilateral triangle of 2.5 inches side has its angular points indexed 3, 12, and 15. Show the figured plans of the bisectors of the angles of the triangle of which the given triangle is the plan. Draw also the ellipse which is the plan of the circumscribing circle of the triangle.

7. Draw the scale of slope of a plane whose inclination is 50° and show the figured plan of a triangle ABC which lies in this plane. The inclinations of AB and BC are 30° and 45 respectively, the indices of a, b and c are 3, 24, and 8 respectively.

8. A plane is inclined at 35° to the horizontal, the lines of steepest upward slope going due East. A second plane has an inclination of 52°, the direction of the lines of upward slope being due North. Represent the planes by scales of slope, unit for heights 01 inch. Show the figured plan of the intersection of the planes. Find and measure the angle between the planes.

[B.E.]

-140

-120

100

80

60

40

20

κα

FIG. 555.

9. Fig. 555 is a plan, drawn to a scale of 1 inch to 200 feet, showing at a, b two places A, B on a hill-side, the surface of the ground being an inclined plane represented by a scale of slope, heights being indexed in feet. Draw Fig. 555 to a scale of 1 inch to 100 feet and add the plan of a zigzag path connecting A and B, made up of three straight lengths of a constant inclination equal to half that of the hill, and deviating equally on each side of a straight line through A and B. Ascertain and measure in feet the total length of this path. [B.E.]

10. Two lines AB and CD are given by their figured plans in Fig. 556. Draw the scale of slope of a plane which will contain the line CD and make an angle of 15° with the line AB.

11. The plans of six points a,, bs, C10, dis, Cao, and fas, taken in order, are situated at the angular points of a regular hexagon of 14 inches side. Find the intersection of the plane containing A, C, and F, with the plane containing B, D, and E, and state its inclination.

12. Draw a square a, bis co do of 2 inches side. Determine the plan of the sphere on whose surface the points A, B, C, and D are situated.

αν

030

d's

C35

FIG. 556.

13. Draw the figured plan of the common perpendicular to the straight lines AC and BD of the preceding exercise.

14. aso beg Cao dso (Fig. 557) is the figured plan of a straight roadway which is to be made, partly by cutting, and partly by embankment, on the ground

whose surface is given by its contoured plan. The sloping faces of the cutting and embankment are to be inclined at 40°. Draw the plan of the intersections of the faces of the cutting and embankment with the surface of the ground.

Show also, on EF as ground line, vertical sections at LL, MM, and NN.
N.B.-Fig. 557 is to be enlarged four times. The unit in this exercise is 1 foot.

15. The surface of a piece of ground is given by contours at vertical intervals of 5 feet (Fig. 558), the linear scale of the plan being 1 inch to 100 feet. A road is to be cut at the given heights, the face of the cutting on each side having a slope of 38° to the horizontal. Draw Fig. 558 to a scale of 1 inch to 50 feet and complete the plan of the finished earthwork. [B.E.]

16. You are given, in Fig. 559, the plan of the surface of a piece of ground, contoured in feet, the linear scale being 1 inch to 100

feet. The curve pp is the centre line of a road, 20 feet wide, which is to be made partly by cutting and partly by embankment, the former having a slope of 45° and the latter one of 38° to the horizontal. Draw Fig. 559 to a scale of 1 inch to 50 feet and complete the plan of the finished earthwork within the limits of the data. [B.E.]

17. Represent by contour lines, as in the example worked out in Art. 255, the surface whose equation is 1·8√/ข

M

=

t-60

The axes to be arranged as shown at (a), (b),

and (c), Fig. 560. v denotes velocity in feet per minute, and denotes temperature in degrees Fahrenheit. Scales. -For v, 1 inch to 50 feet per minute. For t, 1 inch to 20°. For M, 2 inches to 1. The limits of v and t are shown at (a) and (b).

M

(a)

0

→ 400

70t

t

힙 (6)

20

P

CHAPTER XXI

PICTORIAL PROJECTIONS

256. Pictorial Projections.-Since the orthographic projection of a line only shows its true length when the line is parallel to the plane of projection, it is usual, in making working drawings of an object, to arrange it so that as many of its lines as possible are parallel to at least one of the co-ordinate planes. The object is then in a simple position. Fig. 561 shows a plan and two elevations of a rectangular solid when the solid is in the simplest possible position in relation to the planes of projection. These projections are very easily drawn, but, although they represent the solid completely, they are not at all "pictorial," and a special training is necessary before the observer can form from them a correct mental picture of the object.

Now let the solid be tilted over as shown by the plan a and elevation a', Fig. 562, and let a new elevation a,' be drawn in the manner explained in Chap. XIV. It will now be observed that the elevation a, by itself gives a much better idea of the form of the object than any of the other projections shown in Figs. 561 and 562, but unless a much simpler method of drawing such a pictorial projection than that shown in Fig. 562 can be devised, and unless some simple method of