45° to XY. The angle between the plans of the axes is 90°. Draw the plan, showing the intersection of the surfaces of the cylinders, (a) when the axes intersect, (b) when the axes are 0.25 inch apart, (c) when the axes are 0.5 inch apart.

4. The same as the preceding exercise except that the angle between the plans of the axes is 60° instead of 90°.

5. A vertical tube, having an external diameter of 3 inches and an internal diameter of 2 inches, has a cylindrical hole through it, 1.5 inches in diameter. The axis of the hole is inclined at 45° to the horizontal plane and its perpendicular distance from the axis of the tube is 0.25 inch. Draw an elevation of the tube on a vertical plane which is parallel to the axis of the hole.

6. The same as the preceding exercise except that the elevation is to be on a vertical plane which makes 45° with the plan of the axis of the hole.

d

7. AB (Fig. 755) is the axis of a cylinder whose horizontal trace is a circle 2.25 inches in diameter. CD is the axis of a cylinder whose horizontal trace is an ellipse (major axis 3 inches, minor axis 2 inches) whose minor axis is parallel to XY. The axes of the cylinders are parallel to the vertical plane. Show the intersection of the cylinders in plan and elevation in each of the following cases. (i) when h = 0, (ii) when h = inch, (iii) when h = { inch.

8. Same as example 3, p. 390, except that h (Fig. 740) is to be 0 instead of inch.

9. Same as example 3, p. 390, except that h (Fig. 740) is to be inch instead of inch.

X a 45°

60 CY

h

C

-b

3

FIG. 755..

d at

K

10. The axis of a cylinder 2 inches in diameter is inclined at 60° to the ground. The axis of a second cylinder 24 inches in diameter is inclined at 45° to the ground. The angle between the plans of the axes is 80°, and the common perpendicular to the axes has a true length equal to k. Draw the plan, and an elevation on a vertical plane parallel to the axis of the first cylinder, showing the intersection of the surfaces, for each of the following cases. (i) k = 0, (ii) k = 0-25 inch, (iii) k = 0.5 inch.

11. abd is an equilateral triangle of 2.5 inches side. c is a point within the triangle, 1.75 inches from 6 and 1.25 inches from d. a, b, c, and d are the plans of points whose heights above the ground are 0.5 inch, 1 inch, 2.5 inches, and 1.5 inches respectively. A circular cylinder has its axis parallel to the line AD, and its surface contains the four points A, B, C, and D. A second circular cylinder has its axis parallel to the line BC, and its surface also contains the four points A, B, C, and D. Draw the plan showing the intersection of the surfaces of the cylinders.

12. A right cone having a base 3.5 inches in diameter, and an altitude of 3.5 inches, stands with its base on the ground. A cylinder, 2 inches in diameter lies on the ground and penetrates the cone. The axis of the cone is at a distance h from the axis of the cylinder. Draw the plan, and an elevation on a vertical plane inclined at 30° to the axis of the cylinder, showing the intersection of the surfaces, (i) when h = 0, (ii) when h = 0.25 inch, and (iii) when h is such that the curved surface of the cone touches the curved surface of the cylinder.

13. A right circular cone passes through a cylindrical tube. The axis of the cone intersects the axis of the tube at right angles. Internal diameter of tube, 3 inches. Vertical angle of cone 15°. Diameter of cone at centre of tube 1.5 inches. Determine the development of the surface of that part of the cone which

is within the tube.

14. The elevation

of two horizontal

4 diam

4 diam

15. Determine the intersection of the given cone and hollow cylinder (Fig. 757), and draw the development of the surface of that part of the cone which is within the cylinder.

16. Same as example 2, p. 394, except that av is to make 25° with XY instead

of 23°.

17. abc is an equilateral triangle of 4 inches side. A circle is described on ab as diameter, and another is described on ac as diameter. These circles are the horizontal traces of two cones. cis the plan of the apex of the cone of which the circle ab is the horizontal trace, and bis the plan of the apex of the other cone. Each apex is at a height of 4 inches above the horizontal plane. Show the plan of the intersection of the surfaces of the two cones and add an elevation on a ground line parallel to ab.

18. Same as the example on p. 396, except that the distance h in Fig. 743 is

4 inches, standing on the ground. A circle, 25 inches in diameter, is the plan of a sphere also standing on the ground. The centres of these circles are at a distance h from one another. Show the plan of the solids and their intersection and an elevation on a vertical plane which makes 30° with the plane containing the axis of the cone and the centre of the sphere, (1) when h = 4 inch, (2) when h = inch, and (3) when h is such that the curved surfaces of the solids touch one another.

29. Work the example illustrated by Fig. 750, p. 402, when the axis of the cone is moved until it is 0.5 inch distant from the axis of the anchor ring.

30. Work the example illustrated by Fig. 751, p. 403, when the vertical angle of the cone is increased until the straight boundary lines of the plan of the cone are tangential to the plan of the surface of revolution.

31. abc is a triangle, ab = 2 inches, bc = 1.25 inches, ca = 1.5 inches. a is the plan of a point which is 0.5 inch below the H.P. b and care the plans of points which are 1.25 inches and 2 inches respectively above the H.P. Determine the plans of the two points which are 2.75 inches distant from each of the points A, B, and C, and state their distances from the

H.P.

32. The plan of a sphere with a vertical triangular hole in it is shown in Fig. 758. Draw an elevation on a ground line parallel to ab.

33. The equilateral triangle abc (Fig. 759) of 2 inches side is the horizontal trace of a pyramid of which v is the plan of the vertex. The height of the vertex above the H.P. is 4 inches. The square defg of 2 inches side is the horizontal trace of a prism. Draw the plan and elevation of the pyramid and prism showing the intersection of their surfaces,-(1) when the long edges of the prism are vertical, (2) when the long edges of the prism are parallel to the V.P. and inclined at 60 to the H.P., sloping upwards from right to left in the elevation.

34. ab (Fig. 760) is a regular hexagon of 14 inches side. cde is an equilateral triangle of 2 inches side. The

X

145

d

-1.5"

-1-4

60

FIG. 758.

1-2

b

21⁄2-1⁄2-ү 30 g

e

b

C

FIG. 759.

22

υ

hexagon is the horizontal trace of a prism whose long edges are parallel to af, a'f'. The triangle is the horizontal trace of a prism whose long edges are

parallel to dg, d'g'. Draw the plan and elevation of these solids showing the intersection of their surfaces.

35. ab (Fig. 761) is a regular hexagon of 14 inches side and whose centre is h. cd is a square of 2 inches side and whose centre is k. The hexagon is the horizontal

trace of a prism whose long edges are parallel to ae, a'e'. The square is the horizontal trace of a pyramid whose vertex V is 5 inches above the H.P. Draw the plan and elevation of these solids showing the inter

section of their surfaces.

36. Certain solids are shown in Figs. 762, 763, and 764 in plan and elevation. For each solid draw the plan as shown and add an elevation on a ground line inclined at 45° to the centre line ss without drawing the elevation given. Use the dimensions marked but make no measurements from the illustrations given.

37. Draw the curve of intersection of the given cone (Fig. 765) with the helical surface (of uniform pitch) generated by the revolution of the horizontal line VH about the axis of the cone, the line descending to the base during one anti-clockwise turn.

The given point P will lie on the required curve. Determine the tangent to the curve at P. Also draw the normal and osculating planes at P. [Β.Ε.]

h

Y

2-2

10

h

FIG. 765.