It is not absolutely necessary that the parts 1 2, 23, 34, etc. on the plan be equal and it may happen that in the solution of some problem on the cone, say, the determination of its intersection with another cone, plane sections through the vertex of the cone may have been used giving a series of lines whose horizontal traces do not divide the horizontal trace of the surface of the cone into equal parts. In such a case it is not necessary to draw a fresh set of lines arranged as in Fig. 674, but the parts I 2, 23, 34, etc. on the curve ABC must be made equal respectively to the parts 1 2, 23, 34, etc. on the plan.

Fig. 674 also shows the curve EFH, which is the form and position of the outline on the development of a cylindrical section of the surface of the cone. The construction lines for two points R, and R2 only on this curve are shown. The construction is obvious and needs no description.

310. Approximate Developments of Undevelopable Surfaces. It has already been stated (Art. 273, p. 313) that a developable surface must be capable of generation by a straight line and that any two consecutive positions of the generating line must be in the same plane. A surface which is undevelopable may however be divided into parts of which approximate developments may be drawn, the degree of approximation being greater the more numerous the parts into which the surface is divided. Three examples will be taken to illustrate the method of determining approximate developments of undevelopable surfaces.

The sphere will be taken as the first example. The surface of a

sphere may be conveniently divided into zones.

A zone of the surface

of a sphere is the portion lying between two planes which are perpendicular to an axis of the sphere. The surface of a sphere may also be conveniently divided into lunes. A lune of the surface of a sphere is the portion lying between two planes which contain an axis of the sphere.

Fig. 675 shows in plan and elevation one-eighth of a sphere. The surface is divided into three zones. Each zone is assumed to be the surface of a frustum of a cone and its development A is found in the usual way as shown. The plan is shown divided into three equal sectors and these are the plans of three equal lunes. The figure B is the approximate development of the half of one of these equal lunes of

the surface of the sphere. In addition to the information given in Fig. 675, it is only necessary to state that the distances 1 2, 2 3, and 34 on the figure B are equal to the arcs 1 2, 2 3, and 3 4 on the elevation.

The second example (Fig. 676) is the quarter of an annulus. The surface is divided into zones and lunes which are developed as in the case of the sphere. Any surface of revolution may be treated in the

same way.

In the last example (Fig. 677) the surface is one traced by a straight line which moves in contact with a straight line ae, a'e' and a curved line fl, f'l'. The straight line ae, a'e' is divided into a number of equal parts at the points bb', cc', and dd'. The curved line is divided into the same number of equal parts at the points gg', hh', and kk'. The

straight lines joining the points on the straight line to the points on the curved line as shown divide the surface into a number of threesided figures whose true

forms are found in the

usual way and are put together to form the approximate development AELF.

A B

gumming together when the

or perforated at short intervals. Strips may also be left adjoining certain of the edges of the development to form overlaps for development is folded up to the form of the solid. In Fig. 678, which is the development of the surface of a cube, the strips referred to are shown at a, a ...

1. A plan and an elevation of a right prism are shown in Fig. 679. p'q' is the elevation of a plane section of the solid. Draw the development of the surface of the prism and show on it the boundary line of the section. Draw also the elevation of the shortest line lying on the surface of the prism and joining the points M and N.

a

a

a

a

a

a

a

FIG. 678.

2. Two intersecting prisms are shown in Fig. 680, one being vertical and the other inclined. Draw the development of the surface of that part of the inclined prism which lies outside the other.

In reproducing the above diagrams the sides of the small squares are to be taken equal to half an inch.

3. An oblique prism standing on a square base is shown in Fig. 681. There is a rectangular hole in this prism which is shown in the elevation only. Draw the development of the surface of the prism showing on it the outline of the hole.

4. Draw the development of the frustum of a square pyramid shown in Fig. 682.

5. A pyramid is shown in Fig. 683. p'q' is the elevation of a plane section of the solid. Draw the development of the surface of the pyramid with the outline of the section on it.

6. Fig. 684 shows a pyramid in plan and elevation. A square hole in the solid is shown in the elevation only. Draw the development of the surface of the pyramid with the outline of the hole on it.

FIG. 683.

FIG. 684.

+ m2

FIG. 685.

FIG. 686.

In reproducing the above diagrams the sides of the small squares are to be taken equal to half an inch.

7. Draw the development of the surface of the pyramid given in Fig. 685. r's' is the elevation of a plane section of the solid. Show the outline of the section on the development. Draw, in plan and elevation, the shortest line lying on the surface of the pyramid and joining the points M and N.

8. A solid is shown in plan and elevation in Fig. 686. The base of the solid is a square and all the other faces are triangles. Draw the development of the surface of this solid.

9. The shaded part of Fig. 687 is the elevation of a portion of a right circular cylinder whose axis is vertical. Draw the development of the surface of this portion of the cylinder.

10. Fig. 688 shows two views of a sheet metal pipe B with two sheet metal branches A and C. Draw, one quarter of full size, the development of the surface of B.

11. Two pipes A and C (Fig. 689), of circular cross section and having their axes parallel, are connected by a third pipe B as shown. Draw to a scale of 1 inch to 2 feet the development of the surface of the pipe B.

[B.E.]

14. Fig. 692 represents a pipe made from sheet metal. Draw, to a scale of, a development of the pipe showing the shape of the sheet from which it is bent. [B.E.]

15. A right circular cone is cut by a plane which bisects its axis and is inclined at 45° to it. Draw the development of the surface of the frustum. Diameter of base of cone 3 inches, altitude 3 inches.

16. The elevation of a can is given in Fig. 693. Show the shapes to which the sheet metal must be cut, when flat, to form the two conical parts of the can. Omit the allowances for overlap at the seams.

+4-8

FIG. 691.

4.8

6·4

FIG. 692.

17. Fig. 694 shows the elevation of a right circular cone, whose axis is vertical, penetrating two circular cylinders whose axes are perpendicular to the axis of the cone. Draw the development of the surface of that part of the cone which lies between the two cylinders.

18. Fig. 695 shows the elevation of two frusta of two cones of revolution enveloping the same sphere. Draw the developments of the surfaces of the frusta.

19. Two vertical circular pipes of different diameters are connected by another which is conical, as shown in plan and elevation in Fig. 696. Draw the development of the surface of the conical pipe.

20. The solid shown in plan and elevation in Fig. 697 has a base which is a quadrant of a circle. Of the remaining six faces, four are plane triangles and two are conical surfaces. Draw the complete development of the surface of this solid.

FIG. 693.