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EXAMPLES.

EX. 9.-Draw a plan and two elevations of your drawinginstrument box, with the lid open, at an angle of 45° with the box, the end elevation to be in section. Scale, 6′′ = 1'.

EX. 10.-A hexagonal right pyramid, side of base 11", height 3", stands on the HP. Draw the plan and make a section by a vertical plane, the HT of which is a line through one corner of the base, passing from the plan of the vertex. (S. & A. E., 1888.)

EX. 11.-A letter A is made of material " thick, it is 3′′ high and 3′′ wide at the base, the width of the material being g", and it stands in the HP parallel to the V P. Draw its plan and make an elevation on a line parallel to a diagonal of the rectangle at the top, and a sectional elevation on a line through the plan of a top corner and making 35° with the plan of the front face.

EX. 12.-A cone, 3′′ high, where base is 2" in diameter, has its axis horizontal. Draw an elevation on a plane inclined at 60° to the base, and a section of it by a horizontal plane ̋ above the axis. (Vict. Univ. Hon., 1890.)

EX. 13.-A hollow square block, 2′′ outside edge, 1′′ inside edge, 3′′ long, stands with one base in HP and a vertical face at 30° to V P. Draw a plan and true shape of the section made by a plane inclined at 45° to the ground passing through the centre of the axis of the block.

EX. 14.-A hexagonal pyramid, 3′′ axis, 11′′ edge of base, lies with one triangular face in the HP, its axis being parallel to the V P. Draw its plan, and the plan and true shape of a section made by a plane inclined at 20° to the ground passing through the centre of the elevation of the axis.

EX. 15.-A cylinder, 2′′ high, 24" diameter of base, stands with one base in the HP. Draw plan and true shape of a section made by a plane inclined 30° degrees to the ground passing through the elevation of the axis at a point 1ğ" from the base.

EX. 16.-A cylinder, 3′′ high, 23" diameter of base, lies in the HP with a base at 60° to the V P. Draw its elevation, and the elevation of a section made by a vertical plane parallel to the V P cutting the plan of the axis " from one base.

EX. 17.-Draw the plans and true shapes of the three sections of a cone made by cutting planes, as in Fig. 86a, b. The cone

to be drawn in each case 5′′ high and 3′′ diameter of base.

EX. 18.--Draw the elevation and true shape of the section of a sphere of 31′′ diameter made by a vertical plane inclined 45°

to the VP, and passing through the plan of the sphere" in front of its centre.

EX. 19.-A sphere of 4" diameter rests on the H P, and the top quarter of the sphere is completely removed. Draw a plan of the remainder.

EX. 20.-A conical vessel open at the top is 41′′ high, 31′′ diameter outside at the bottom and 3′′ diameter outside at the top, the thickness of the shell being ". Draw its plan and elevation, the elevation to be in section, and the plan to show a horizontal section midway up the vessel.

Projection of Solids generated by the Revolution of Surfaces. It has been pointed out that cylinders, cones, and spheres

A

и a

g

Y

Fig. 88.

are examples of solids generated by the revolution of certain surfaces about a fixed axis. But the number of such solids of revolution is infinite, and as previous constructions do not apply except to simple cases, it is desirable to consider a more general example.

PROBLEM LIV. (Fig. 88).—To draw the projections and section of a given solid of revolution.

Let uv be the plan and u'v' the elevation of an axis, and a' b′ g be the elevation of a surface revolving about u'v'. It is required to draw the plan of the solid as generated, and of the section made by a horizontal plane whose vertical trace is S'T'.

The revolution of the points b', c', e', f' will generate circles lying in planes perpendicular to the axis u'v', and, therefore, their plans can be drawn, as shown, exactly as in Problem xlvii. Find the points h' d' m', so that d' shall be at the point of the curve furthest from the axis, and h' m' at points in the curve nearest to the axis, then these points will also generate circles of radii equal to ď'1, h' 2, and m' 3 respectively, the plans of which can be found. The complete plan of the solid, so far as its outline is concerned, is then shown by the figure a, b, c, e, f, g, a. To obtain points in the plan of the section, we must proceed by taking cross-sections of the solid perpendicular to the axis, and then project these cross-sections upon the HP. For example, the true shape of the cross-section through d'1 is shown in the figure marked A by the circle r s t, the section plane cutting this circle in the points 6', 7, the plan of these points is 6, 7, and give two points in the plan of the section. Other points are found in the same way, thus completing the sectional plan as shown. It is necessary to take cross-sections at all points where the direction of the curve changes.

EXAMPLES.

EX. 21.—A semi-ellipse axes 34′′ and 21′′ revolves about its major axis as an axis. The axis is inclined at 45° to HP and is parallel to V P. Draw the plan of the solid generated by the revolution of the semi-ellipse, and the plan of a section made by a horizontal plane passing through the centre of the axis.

EX. 22.—A line is parallel to the HP and inclined 35° to V P. A surface similar to that of Fig. 88 revolves about this line as an axis. Draw the elevation of the solid thus generated, and the elevation of the section made by a vertical plane parallel to the V P passing through the centre of the axis.

[blocks in formation]

INTERPENETRATION AND DEVELOPMENTS OF SURFACES AND SOLIDS-SECTIONS OF SPECIAL SOLIDS-HELICES AND SCREW THREADS.

THERE are a number of problems of frequent occurrence in practical draughtsmanship which are best solved by the application of methods usually regarded as a part of solid geometry. Among such problems may be mentioned the drawing of an ordinary steam dome upon a cylindrical boiler, or of the semispherical ends of egg-ended boilers, and the finding of the true shape of the plates for such parts; the drawing of the contact lines of the cylindrical branches of cocks and valves with the main casing (see Fig. 166), and the drawing of the correct outline of such parts as at the junction of the crank-web and crankshaft, or the meeting of other flat and curved surfaces, as in connecting-rod ends and other similar parts.

These problems may generally be regarded as special cases of the interpenetration and development of surfaces and solids, as, for example, the steam and boiler may be treated as a case of the interpenetration of two cylinders, and the cock with its inlets and outlets as the interpenetration of a cylinder with a

cone.

It will be understood that in the case of the steam dome and boiler, and of many similar examples, it is necessary to develop the true shape of surfaces in order that the plates may be so cut when flat, so that they shall join up correctly when bent to their required form.

PROBLEM LV. (Fig. 89).-To draw the projections of the interpenetration of a horizontal and vertical cylinder and the development of their contact surfaces.

Let A be the plan, B the end elevation, and C the side elevation of the cylinders.

The line of interpenetration is evidently shown in plan by the circle 1, 7, and in the end elevation B by the arc d' l'. In order to find its side elevation on C, we imagine the vertical cylinder to have a number of stripes drawn upon it, and we then find the real length of each stripe from the top base of the cylinder to the point where it enters the horizontal cylinder. This is done as follows:-Divide the plan of the vertical cylinder, the circle on A, into a convenient number of equal parts, say 12,

and mark as shown, 1, 2, . . . 12. Draw the elevation of the stripes in each of the elevations B and C, and mark the stripes 1', 2', . . . 12' on B, and 1", 2", . . . 12" on C. Care must be taken not to confuse the marking of the stripes in the two elevations, notice that the outside stripes 1' and 7', on the end elevation B, are the centre stripes on the side elevation C, while the centre stripes 4' and 10', on the end elevation, are the outside stripes on the side elevation. The stripes are correctly obtained on the end elevation B, by projecting from the plan,

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and, on the side elevation O, by drawing and dividing the semicircle on the line m n as shown.

The real length of the stripes are shown in the end elevation B, therefore mark off on each stripe in the elevation C its real length as obtained from the end elevation, thus 1"p=1'd' or

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