Sections of Cone by Stripes (Fig. 86a).-Divide the plan of the base into a number of equal parts, and join each point to the plan of the vertex. Imagine these lines to be stripes drawn down the cone, and draw their elevation. The lines representing the elevation of the stripes will cut the V T of the section plane in points, which are in the elevation of the section; therefore, the plan of each point is directly underneath its elevation, and upon the plan of the stripe whose elevation cuts the elevation of the section plane. Thus, in the figure, the dotted lines v a, vb, vc, vd, are the plans of 4 stripes, and v' a', v'b' (which fall in the same

line), and v'c', v'd' (which also fall in the same line) are their elevations. The stripe v' a' cuts the line of the section in front at e', and at the back at f'; therefore, a projector from e'f' meets the plan of the stripes va and vb in e and f, which are two points in the plan of the section. The same reasoning applies to the stripes vc and vd, which give two other points, g and h, in the plans. Thus, by taking a sufficient number of stripes the plan of the section can be drawn, and its true shape found as in Problem liii. But the method evidently fails for the stripes v m

and vn, and to obtain the points r and s we have to proceed as follows::

Sections of Cones by Cuts parallel to Base.—All plane sections of a cone parallel to its base are circles. If, then, we take any horizontal section of the cone in Fig. 86a, such as at the line op, its plan will be a circle of diameter op, and is shown drawn upon the plan of the cone. But the cut op will pass through the point e' in the line of the section in front and the point f' at the back, so that a projector through the point ef" will cut the circle in the two points e and f, which are evidently points in the plan of the section. Therefore, the distance vr or vs is equal to 'k' or 'l'. Any number of other points can be found by taking additional cuts at different heights. It is unnecessary to draw the whole of the circles. The sections of Fig. 866 are found in this way, for it is more convenient than the method of stripes.

A

It is more convenient to obtain the true shape of the section by drawing its centre as shown dotted (Figs. 85 and 86a, b), in any convenient position parallel to the trace of the section plane, and then mark off distances on each side of the centre line, the distances being taken from the centre line of the plan of the section to the extreme points of the section. Thus, in Fig. 86b, g2 e2 or g2f2 in the true shape of the section D, is equal to ge or gf in the plan of the section C.

Sphere (Fig. 87).—A is the elevation and B the true shape of the section of a sphere made by the vertical plane whose horizontal trace is ST. The true shape of

X

B

any plane section of a sphere is a circle, and its inclined projection an ellipse. The method is identical with the method of sections for a cone.

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 14", height 3", stands on the HP. Draw the plan and make a section by a vertical plane, the H T of which is a line through one corner of the base, passing 3" 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 ", 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 H P, 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 15" from the base.

EX. 16.-A cylinder, 3′′ high, 21" 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 V P, 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 4" 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

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' gbe 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 d' 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 33′′ 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.