18. Fig. 659 is the figured plan of a straight line AB and a sphere whose centre is C. Draw the traces of the planes which contain the line and are tangential to the sphere.

In reproducing the above diagrams the sides of the small squares are to be taken equal to a quarter of an inch. The unit for the indices is 0-1 inch.

19. Draw two circles, one 2 inches and the other 1 inch in diameter. The centres of the circles to be 2 inches apart. These circles are the plans of two spheres. The centre of the smaller sphere is 1 inch and the centre of the larger sphere is 2.5 inches above the horizontal plane. Draw the scales of slope of all the planes which are tangential to the two spheres and are inclined at 70° to the horizontal plane.

20. AB is a straight line whose plan is 3 inches long. A is one inch and B is 2 inches above the horizontal plane. Represent a plane which passes between A and B, is inclined at 60° to the horizontal plane, and is at perpendicular distances of 1 inch and 1.5 inches from A and B respectively.

21. vh and he are two straight lines at right angles to one another and each 3 inches long. V is the vertex and VH the axis of a right circular cone whose vertical angle is 40°. H is on, and V is three inches above the horizontal plane. C is the centre of a sphere 2 inches in diameter which rests on the horizontal plane. Represent a plane which is tangential to the cone and sphere. 22. abc is a triangle. ab 1.5 inches, bc 1.3 inches, and ac =

=

2.5 inches, A is 2.5 inches, B is 1 inch, and C is 2-4 inches above the horizontal plane. AB is the axis of a right circular cylinder 1.6 inches in diameter. C is the centre of a sphere 1.4 inches in diameter. Draw the scale of slope of a plane which passes over the cylinder and under the sphere and is tangential to both.

23. A, B, and C are the centres of three spheres whose radii are, 0·8 inch, 0.4 inch, and 0.6 inch respectively. ab 1.7 inches, bc = 1.3 inches, and ac = 1·2 inches. The heights of A, B, and C above the horizontal plane are, 0·8 inch, 0.6 inch, and 1.6 inches respectively. Represent a plane which passes over the spheres A and C and under the sphere B and is tangential to all three.

24. The elevation of a solid of revolution, whose axis MN is parallel to the vertical plane of projection, is given in Fig. 660. Draw the horizontal trace of a vertical cylinder which envelops this solid, and show the elevation of the curve of contact. s' is the elevation of a point on the front surface of the solid. Represent a plane tangential to the surface at S.

25. The circle ab (Fig. 661) is the plan of a prolate spheroid whose axis is vertical and 3 inches long, its lower end being on the horizontal plane. cd is the plan of a straight line, C being 4 inches above and D on the horizontal plane. A cylinder whose axis is parallel to CD envelops the spheroid. Show in plan and elevation the curve of contact between the cylinder and the spheroid. r is the plan of a point on the lower surface of the spheroid. Draw the traces of the plane which is tangential to the spheroid at R.

26. Taking the spheroid and the point C as in the preceding exercise, draw in plan and elevation the curve of contact between the spheroid and a cone

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

enveloping it, the vertex of the cone being at C. Draw also the traces of a plane containing the point C, touching the spheroid, and inclined at 70° to the horizontal plane.

27. In Fig. 663, aa, bb, and cc are the plans and a'a', 'b', and c'c' are the elevations of three straight lines. pp' is a point on AA. Determine a point qq' on BB such that the line PQ shall be 1.5 inches long and sloping downwards from P to Q. Find a point r in CC such that PR + QR = 3.5 inches.

28. acb (Fig. 662) is the plan of a straight line inclined at 45° to the horizontal plane and sloping upwards from A to B. o is the plan of a vertical axis. A twisted surface of revolution is generated by the revolution of ACB about the vertical axis to which it is connected by a hori- X zontal line of which oc is the plan for the position of ACB shown. Draw in plan and elevation 16 positions of the generating line at equal intervals. r is the plan of a point on the lower part of the surface generated. Draw the traces of the plane which is tangential to the twisted surface at R. de is the plan of a horizontal line whose height above the horizontal plane is 1.5 inches. Show in plan and elevation the points of intersection of DE with the twisted surface.

0.5 a

20

45°

b

60° 70% 50%

1.2

la

20

29. ab, and cd (Fig. 664) are the plans of two straight lines. The points B and D are on the horizontal plane. A is 3 inches and C is 2 inches above the horizontal plane. An hyperboloid is generated by the revolution of CD about AB. Show the true shape of the section of the hyperboloid by a vertical plane of which HT is the horizontal trace. r is the plan of a point on the lower surface of the hyperboloid. Taking Hd as a ground line draw the traces of the plane which is tangential to the hyperboloid at R.

FIG. 663.

H

C

30. The axes of two hyperboloids A and B which are in rolling line contact are horizontal. The axis of B is 1.2 inches above that of A. The angular velocity of B is 1.5 times that of A. The ends of A are each 2 inches distant from its throat circle. Draw the plan of the hyperboloids and an elevation on a plane perpendicular to their line of contact.

FIG. 664.

31. The throat of a cross-roll (Fig. 646, p. 336) is 3.5 inches in diameter. The effective length of the roll is 7 inches. The cylinder is 2.5 inches in diameter and

its axis is inclined at 45° to the axis of the roll. Draw, half size, a plan of the roll and cylinder, their axes being horizontal, showing the curve of contact.

=

32. OA, OB, and OC are the semi-axes of an ellipsoid. OA = 1.75 inches, OB 1.25 inches, and OC = 1 inch. OA is horizontal and OB is perpendicular to the vertical plane of projection. The centre O is 1 inch above the horizontal plane. The plan is shown in Fig. 665. Draw a projection of the ellipsoid on a plane parallel to the plane of a circular section.

mn (Fig. 665) is the plan of a straight line which passes through O and is inclined at 45° to the horizontal plane. A cylinder enveloping the ellipsoid has its generating line parallel to MN. Show in plan and elevation the curve of contact between the cylinder and ellipsoid.

Xm 60

n

FIG. 665.

-2Y

v (Fig. 665) is the plan of a point which is 3.5 inches above the horizontal plane. A cone having its vertex at V envelops the ellipsoid. Draw in plan and elevation the curve of contact between the cone and ellipsoid.

33. OA, OB, and OC are three straight lines mutually perpendicular, OC being vertical. OA = 1.25 inches, OB = 1 inch, and OC = 0·75 inch. OB and OC are the semi-conjugate axes of two hyperbolas having OA as a common_transverse axis. These hyperbolas are axial sections of an hyperboloid of two sheets lying between two planes perpendicular to OA and each 4.25 inches from O. Draw a plan of both sheets of the hyperboloid and an elevation on a plane parallel to OA. Draw also an elevation on a plane perpendicular to OA. On each projection show the elliptic sections by planes perpendicular to OA at intervals of, say, 0·75 inch. Show also on each projection a number of axial sections.

34. An elliptic paraboloid is given in Fig. 666 by two elevations on planes at right angles to one another. Draw these elevations to the dimensions given, and

from the elevation (v) project a plan. Show on each projection the parabolic and elliptic sections indicated. V.T. is the vertical trace of a plane which is perpendicular to the plane of the elevation (v); show on the plan and on the elevation (w) the section of the paraboloid by this plane.

35. An hyperbolic paraboloid is given in Fig. 667 by two elevations on planes at right angles to one another. Draw these elevations to the dimensions given and from the elevation (v) project a plan. Show on each projection the parabolic and hyperbolic sections indicated. Show also on each projection a number of straight lines, say ten, which lie on the paraboloid, half the number to belong to one system and the other half to belong to the other system of straight generators of the paraboloid.

απ

36. A tortuous curve ARC is shown in plan and elevation in Fig. 668. Draw the projections of the tangent to the curve at the point R. Show the traces of the normal and Osculating planes of the curve at R. Draw also the projections of the principal normal and the binormal to the curve at R.

-1·3

FIG. 668.

CHAPTER XXIV

DEVELOPMENTS

305. Development of a Surface.-A surface is said to be developed when it is laid out on a plane, and the figure so obtained is called the development of the surface. As has already been pointed out there are certain surfaces which cannot be developed, as for example the surface of a sphere, but approximate developments of such surfaces may be obtained by dividing them up into a number of parts. The determination of the developments of developable surfaces will first be studied and then the method of determining approximate developments of undevelopable surfaces will be considered.

In general when the surface of a prism, pyramid, cylinder, or cone is referred to, the bases or ends will be excluded, but these bases or ends, being plane figures, may easily be added to the development of the remainder of the surface, in order to complete the development, if required.

306. Development of the Surface of a Prism. First consider the case of a right prism. Draw the plan and an elevation of the prism when the ends are horizontal as shown at (a) and (a') in Fig. 669. The vertical faces of the prism are rectangles and these placed side by side in a plane form the required development which is a rectangle. This development is most conveniently drawn as shown. On the horizontal line LM parts 1 2, 2 3, 3 4, etc. are marked off equal to the sides 1 2, 2 3, 3 4, etc. respectively of the base. Lines through the points 1, 2, 3, etc., on LM at right angles to LM and equal in length to the height of the prism are the positions of the vertical edges of the prism on the development LN.

Now suppose this prism to be converted into an oblique prism by having for the elevations of its ends the lines c'd' and e'f'. The development of the surface lying between the ends of the new prism will be a portion of the rectangle LN obtained as follows. From the ends o the vertical edges of the new prism draw horizontals or parallels to LM to meet the lines on the development which are the positions of these edges on the development. For example, from the elevation of the upper end of the edge 3 draw the horizontal r'R to meet the vertical 3 on LN. Other points such as R are determined

in the same way and consecutive points being joined as shown, the required development is determined.

The student should now have no difficulty in following the construction shown for determining the part of the development which must be removed due to a hole in the prism, the elevation of the hole being the triangle s'p'q. Observe that six additional vertical lines are required on the faces of the prism; these lines pass through the points where the horizontal internal edges of the hole meet the faces of the prism, and their elevations coincide in pairs. The positions of these additional lines on the development are easily found, and they contain important points in the complete development.

If any line or figure be drawn on the development of the surface of the prism, it is obvious how the elevation of the line or figure when

the development is wrapped round the solid, may be obtained by working backwards from the development to the elevation of the prism.

Again, given, by their projections, two points on the surface of the prism, the projections of the shortest line lying on the surface of the prism and joining the given points, are easily obtained when it is noticed that the required line will on the development be a straight line. It is necessary to observe however that there are two ways of joining the given points by lines whose developments are straight lines, one line goes round the surface in one direction and the other in the other, but one of these will generally be shorter than the other.

307. Development of the Surface of a Pyramid.-Excluding the base, the surface of a pyramid is made up of a number of triangles having a common vertex, and if the true forms of these triangles be