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section on ST," and the plan, a sectional plan on the line S'T', or simply a section on S' T'."

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PROBLEM LIII. (Fig. 84).—To draw the plan and elevation of the sections of a hexagonal pyramid, and to find the true shape of the sections.

Let S T be the HT of a vertical section plane, and S' T' the V T of a section plane inclined to the ground and perpendicular to the V P.

elevation.

Mark the vertex of the solid v and the corners of the base a, b, c, d, e, f in both plan and Then to draw the vertical section made by the plane S T, draw projectors from each point 1, 2, 3, 4 of the section in plan to meet the elevation of the edges in the elevation, as, for example, the section plane cuts the sloping edge vb at the point 2, and, therefore, it must cut the elevation v'b' of the same edge at the point 2', found by drawing a projector from 2 to cut vb. Proceeding in this way, we find the points 1', 2′, 3',4' in the elevation, and by joining in the right order we obtain the elevation of the section, which should be section-lined as shown.

For the plan of the section on S'T' we adopt the same method,

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Fig. 84.

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as, for example, the section plane cuts the elevation v'b' of the sloping edge vb at the point 6', and, therefore, the plan of this point must be at the point 6, in the plan vb of the edge, found by drawing a projector through the point 6'. The plan of the section when completed is the irregular hexagon 5, 6, 7, 8, 9, 10, and should be section-lined as shown.

True Shape of Section.-We know from previous examples that the plan 5, 6, . . . 10 of the section on the line S' T' cannot be the true shape of the section, because it is the plan of an inclined figure. We also know that a plane figure is only shown its true shape when it is projected upon a plane parallel to its own plane--that is, it must be looked at in a direction perpendicular to its own plane. The vertical section 1', 2′, 3', 4' is the

true shape of the section, as the VP of projection is parallel to the section plane ST. To draw the true shape of the section made by the inclined plane, we may regard the trace of the plane S'T' as a new X Y, and the line 5' . . 8′ as the elevation of the plane figure marked in plan 5, 6, . . . 10, of which we require a new plan on S'T' as a ground line. We then proceed as in the last series of examples, and draw projectors from each

Fig. 85.

point 5'... 10' perpendicular to S' T', and set off along each its distance in front of the VP; thus 51 52 is equal to b' 5 and 81 82 to p8, and so on. This is exactly the same as supposing the section plane to be turned into the VP of projection about its trace S' T'as a hinge, taking with it the outline of the section, and the projectors from each corner of the outline to the vertical plane.

It will be seen that a sectional view of a solid is of no service for the practical purpose of showing its construction and form, unless it shows the true shape of the section; hence, we do not find that engineering drawings generally contain either plans or elevations of inclined sections, but only the projections of their

true shapes. The draughtsman chooses the projection of the sections in the positions most likely to add to the clearness of the drawing, and as a rule most sections on engineering drawings are either horizontal or vertical ones. But it often occurs that a section is taken through an inclined part, in which case the true shape of the section is required, and must be obtained on the principle of the last problem. It will be found that the true shape can often be drawn without first obtaining the plan or elevation of the section, but in many cases it is necessary to have either a part or the whole of the plan or elevation. The following are additional examples of plans, elevations, and true shapes of sections obtained in similar ways, of some simple solids which are of common application in practical construction :

Cylinder (Fig. 85).-A is the plan and B the true shape of a section of the cylinder made by the inclined section plane S' T', and D the true shape of a section

made by the vertical section plane ST. The view, D, might be termed a sectional end elevation. The cylinder must be striped as shown.

Cone (Figs. 86a, b).-Fig. 86a, A is the plan and B the true shape of a section of the cone made by the inclined section plane ST. In Fig. 866, C is the plan and D the true shape x of a section made by the plane S1 T1, and E is the plan and F the true shape of a section made by the plane S2 T2. The true shapes, B, D, and F, are the three conic sections-the ellipse, parabola, and hyperbola. The section of a cone by a plane, such as ST, which cuts all the positions of the generating line of the cone, is an ellipse, and is shown at B (Fig. 86a); the sec

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Fig. 86a.

tion by a plane, such as S1 T1, parallel to any one position of the generating line, is a parabola, and is shown at D (Fig. 86b); while the section by a plane, such as S2 T2, parallel to the axis of the cone, is a hyperbola, and is shown at F (Fig. 866).

The methods of obtaining the sections of a cone are as follows, and should be carefully mastered :—

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

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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 ef' 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 o p, 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.

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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

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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.

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