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CHAPTER XIII

PROJECTIONS OF SIMPLE SOLIDS IN SIMPLE POSITIONS

149. Projections of Solids. It has already been indicated (Chap. XI.) that one object of Descriptive Geometry is to convey to the mind an impression of the exact form and size of objects, which have length, breadth, and thickness, by means of representations of them on a surface, which has length and breadth only.

Now, a solid may be conceived to be made up of an immense number of small particles or material points, the relative positions of which may be represented by means of their projections on two planes, as explained in the two preceding chapters; but as in looking at a solid it is generally only the points on its external surface which are seen, all impressions of the form and size of objects are derived from the form and extent of their surfaces. It is therefore unnecessary in representing an object on paper to give the projections of points in its interior, that is, it is only necessary to represent the surface of the object.

Again, the form and extent of a surface are generally known when the forms, lengths, and relative positions of a sufficient number of lines on that surface are known. But it has been seen that by the method of projections the forms, lengths, and relative positions of lines may be represented on a single flat surface. Hence, a solid may be represented on paper by the projections of a sufficient number of lines on its surface.

When the solid has plane faces, the projections of the boundary lines of the faces are all that is necessary in order to represent them.

150. Definitions of Solids.-There are certain solids, of definite and simple form, which are of frequent occurrence and which will now be defined.

A polyhedron is a solid bounded entirely by planes. The edges of a polyhedron are the lines of intersection of its bounding planes. The sides or faces of a polyhedron are the plane figures formed by its edges.

A polyhedron is said to be regular when its faces are equal and regular polygons. Generally when a polyhedron is referred to a regular polyhedron is understood.

There are only five regular polyhedra, namely, the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.

The tetrahedron (Fig. 318) has four faces, all equilateral triangles.

The cube (Fig. 319)

has six faces, all squares.

The octahedron (Fig.

320) has eight faces, all equilateral triangles.

The dodecahedron (Fig. 321) has twelve faces, all pentagons.

The icosahedron (Fig. 322) has twenty faces,

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all equilateral triangles.

A prism is a poly

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hedron having two of its faces, called its ends or bases, parallel, and the remaining faces are parallelograms. Those faces of a prism which are parallelograms are generally called the sides of the prism.

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A parallelepiped is a prism whose bases are parallelograms.

A pyramid is a polyhedron having a polygon for its base, and for its sides it has triangles which have a common vertex and the sides of the polygon for their bases. The common vertex of the triangles is called the vertex or apex of the pyramid.

The axis of a prism is the straight line joining the centres of its ends; and the axis of a pyramid is the straight line joining its vertex to the centre of its base.

A right prism has its axis at right angles to its ends; and an oblique prism has its axis inclined to its ends.

A right pyramid has its axis at right angles to its base; and an oblique pyramid has its axis inclined to its base.

The altitude of a prism is the perpendicular distance between its ends; and the altitude of a pyramid is the perpendicular distance of its vertex from its base.

Prisms and pyramids are named from the form of their bases-as, triangular, square, pentagonal, hexagonal, etc. Examples of prisms and pyramids are shown in Figs. 323-326.

A cylinder resembles a prism. If the sides of the bases of a prism be continually diminished in length and increased in number, the ultimate form of the boundary lines of the bases will be curved lines,

and the ultimate form of the prism will be a cylinder. A right circular cylinder has its axis at right angles to its ends, which are equal circles.

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A right circular cylinder may also be defined as a solid described by the revolution of a rectangle about one of its sides, which remains stationary. The fixed line about which the rectangle revolves is the axis, and the circles described by the opposite revolving sides are the bases or ends of the cylinder.

The diameter of a right circular cylinder is the diameter of its circular ends.

RIGHT
CIRCULAR
CYLINDER.

RIGHT
CIRCULAR
CONE.

SPHERE.

A cone resembles a pyramid. If the sides of the base of a pyramid be continually diminished in length and increased in number, the ultimate form of the boundary line of the base will be a curved line, and the ultimate form of the pyramid will be a cone. A right circular cone has its axis at right angles to its base, which is a circle. A right circular cone may also be defined as a solid described by the revolution of a right angled triangle about one of the sides containing the right angle, which side remains stationary. The fixed line about which the triangle revolves is the axis, and the circle described by the other side containing the right angle is the base of the cone.

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FIG. 327.

FIG. 328.

FIG. 329.

A sphere is a solid every point on the surface of which is at the same distance from a point within it called its centre. A sphere may also be defined as a solid described by the revolution of a semicircle about its diameter, which remains stationary. The middle point of the diameter of the semicircle is the centre of the sphere.

151. Projections of a Prism.-The projections of a prism are simplest when the solid is so placed that its ends are parallel to one of the planes of projection, and whatever projections of the prism may be

required it is generally necessary to first draw the projections of the solid when so situated.

When the ends of the prism are parallel to one of the planes of projection, their projections on that plane will show their true form, and as the form of the ends is supposed to be given these projections

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are drawn first. If the prism is a right prism the projections of the two ends on the plane of projection to which they are parallel will evidently coincide. The projections of the ends on the other plane of projection will be straight lines parallel

to the ground line and at a distance apart equal to the altitude of the prism.

The foregoing observations are illustrated by Figs. 330, 331, and 332. Fig. 330 shows the plan and elevation of a right triangular prism when its ends are horizontal. The plan abc is first drawn, its position in relation to XY being arbitrarily chosen unless it is specified. The elevation is projected from the plan, as shown. The height of the elevation of the lower end above XY is equal to the height of that end above the horizontal plane, which may be arbitrarily chosen, unless it is specified. The distance between the horizontal lines which are the elevations of the ends is equal to the altitude of the prism, which would be given.

b

FIG. 332.

Fig. 331 is a pictorial projection of the prism and the planes of projection in their true positions.

Fig. 332 shows the plan and elevation of an oblique hexagonal prism when its ends are parallel to the vertical plane of projection, one end being in that plane. In this case the elevation is first drawn. The hexagons which are the elevations of the ends of the prism have the sides of the one respectively parallel to the sides of the other, their relative positions being otherwise determined from the specification of the prism. It will be observed that the hexagons have been placed with a diameter parallel to XY.

It will be noticed that in Fig. 330 the elevation of one of the edges of the prism is shown as a dotted line and that in Fig. 332 the elevations of five of the edges are so shown. The reason for this is that these edges are hidden by the solid when it is viewed from the front. In like manner if an edge of the solid is hidden when viewed from above the plan of that edge would be shown by a dotted line. When a projection of an edge which should be a full line coincides with a projection of an edge which should be a dotted line only the full line can be shown.

152. Projections of a Pyramid. The projections of a pyramid are simplest when the solid is so placed that its base is parallel to one of the planes of projection, and whatever projections of the pyramid may be required it is generally necessary to first draw the projections of the solid when so situated.

When the base of the pyramid is parallel to one of the planes of projection, its projection on that plane will show its true form, which

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is supposed to be given. This projection is therefore drawn first. the pyramid is a right pyramid the projection of its vertex on the plane of projection to which the base is parallel will be at the centre of the projection of the base on that plane. The projection of the base on the other plane of projection will be a straight line parallel

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