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Problem 17. To find the elevation of a hollow cylinder, its plan being given.

Let ABC be the plan of the given cylinder. It is required to find its elevation.

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From A, draw AA' at right angles to xy and making A'a' equal to the length of the cylinder, then draw A'C' parallel to the ground line (cy) and draw Cd perpendicular to xy.

Let D be the plan of the axis of the cylinder. Now, in order to represent the interior of the given cylinder, draw AC passing through D and parallel to xy. From the points E and F, erect perpendiculars ' and FF"; then the lines E'é' and F'f' being the elevations of E and Fare covered by the surface ABC, and thus are represented as dotted lines in the figure.

Note.—The line D'd' is called the axis of the cylinder. When the axis is perpendicular to the plane of its base, the cylinder is termed a right cylinder; but when it is inclined to the base, it is termed oblique.

Problem 18.
To find the plan of a cone, its elevation being given.

Let A'B'C' be the elevation of a cone. It is required to find its plan.

The line B'C' represents a circle on the horizontal plane, and the points B' and C will be the extremities of the diameter, and a' the centre. Obtain the projection of this line, by drawing a line parallel

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to xy, and draw projectors from B', a', and C' at right angles to xy to meet it in B, A, and C. From Á, with radius AB or AC, describe the circle BCD. The plan of the apex A' coincides with A, the centre of the circle.

NOTE.-A cone may be defined as a pyramid, having an infinite number of faces.

Problem 19. To find the elevation of a cone, its plan being given. Let ABC be the plan of a cone. It is required to find its elevation. From the point A, draw a line at right angles to xy, meeting it in A'; also from point C, draw a line perpendicular to xy, meeting it in the point C". Now, since A and C are points on the horizontal plane, and as ay represents that plane seen in elevation, the points A' and C on it are the elevations of those points. D is the plan of the

apex of the cone, to find the elevation of which we draw a line at right angles to xy from D, and elevate it above the line the required height of the point D above the horizontal plane.

Then join D'A', D'C"; and D' A'C' will be the elevation of the cone.

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e Note. —The axis of a cone is a line joining the centre of its base to the apex.

Problem 20.
To find both plan and elevation of a sphere.

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In either case, these will be represented by circles, whose diameters are equal to the diameter of the given sphere ; e.g., take the elevation of the sphere. The view of it from above representing the plan will be a circle of which the line B'C" is the elevation. Again, let the plan be given, the elevation of it will be a circle represented in plan by the straight line BC.

NOTE.—If a semicircle revolve upon its diameter, it generates the surface of a sphere.

SECTION IV.

TRACES OF LINES AND PLANES.

1. The horizontal and vertical planes of projection are from their

mutual relationship termed co-ordinate planes. 2. The points in which any line intersects the co-ordinate planes are

called the traces of that line, and these traces are termed horizontal or vertical, according as they are referred to the horizontal

or vertical plane. 3. The lines in which any plane intersects the co-ordinate planes are

termed the traces of that plane, and are distinguished as the horizontal or vertical trace, according to the plane of projection

in which it lies. 4. When the traces of a plane are given, the plane itself is given ;

and when the projections of a line are given, its traces may be found: or, conversely, the traces being given, its projections may

be found. 5. It has been stated that the intersection of the planes of projec

tion is called the ground line, or base line (wy). Now, if a plane be not parallel to the ground line, it must meet it in a point

common to both of its traces. 6. If a plane be parallel to the ground line, its traces are also parallel

to the ground line ; for as the base line is parallel to the plane, it cannot meet it, and therefore cannot meet the traces which are lines in the plane ; but each trace and the ground line are

in one plane, consequently they are parallel (Euc. I., Def. 35). 7. If a plane be perpendicular to the ground line, its traces are also

perpendicular to it (Euc. XI., Def. 3) ; and if a plane be parallel to one plane of projection, its trace upon the other is parallel to the ground line (Euc. XI., 16).

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