All of the sloping edges of the pyramid are evidently of the same length, and the edges of the base are shown in plan in their true lengths; so the development may be constructed, laying down first the true size of the triangle cab, join ing to the edge o'-b' the next face obc, and then in turn the other two faces.
The square, representing the base, may be constructed on any short side, as at c'-d'. The most expeditious construction is to take a radius equal to strike an arc through a' from center o', and on this arc, with the dividers or compasses, step off the length ab-bb four times, obtaining b', c', d', and a". These points are then connected and joined with o'.
The lines on the development showing the intersection of the plane X are next found. The point 1, for example, will appear in development on the line o'-a' at its real distance from o'. This real distance from o may be found in ele vation by drawing a horizontal line from 1° to cut in point thus determining ov-r, as the real distance. The reason for this is that if the pyramid be revolved about its axis until o-ab takes the position o-b",, the V projection will then coincide with and the point 1 will appear on at the same height as before revolution. The length is then laid off from 0', giving 1'. The real length from 0 to 2 is similarly found by a horizontal line from 2" to 2v,. The lengths 0-4 and o-3 are respectively equal to those of o-1 and 0-2. The points 1', 2', 3', 4', and 1" are then located and joined, thus completing the required development.
In connection with the construction for finding the true length of the line it should be remarked that it is not really necessary to think of the entire pyramid as revolving, for the same result is reached if the line alone be re volved about a vertical axis through o.
83. A square pyramid like that of Fig. 63, is shown again in Fig. 65, but intersected by a plane Y perpendicular to II. Plane Y is there fore seen edgewise in plan. The points of inter section a, b, and c are projected to the elevation and connected, thus determining the lines and The plane in this position cuts two edges of the base and one sloping edge.
In Fig. 66, the same pyramid is shown again, but with the corner cut off. This gives rather a clearer idea of the intersection, or section, as it is called, cut by the plane Y.
84. A regular pentagonal pyrathid, with ele vation, plan, and end view, is given in Fig. 67.
The pyramid has its axis parallel to both V and H; hence its base is seen edgewise in both plan and elevation. The pyramid is placed in the first angle, and the end view is drawn first. In this view the base is drawn in its full size. From the end view, the front elevation and the plan are constructed, the elevation being a view in the direction of the arrow.
The construction of the elevation of the pyramid should be apparent from the figure, and the plan is drawn by the principles of Article 66, which may be again referred to here. As the plan is a top view, the width of the base will be the same as on the end view. Hence the process is to project in the end view the corners of the base upon any horizontal line, as A-B, and transfer this line with its points of division to the position A'-B', directly below the base in the elevation. The apex ob may then be located and joined with the corners of the base Let X be a plane at right angles to the base of the pyramid. Then the plane in this position will intersect the three faces 0-1-5, 0-1-2, 0-2-3, and also the base of the solid.
The plane cuts the edges in the points marked a, b, c, and d. These points may be found in elevation at the same height, by the use of the T-square. Point a, on the edge of the base, is located at a7, point b on 0°-1°, and point d on the base at dv.
To locate cv, however, it is first necessary to find ch in plan, then project for cv. The points in the end view are projected to A-B at points a„ and and these points are then spaced off in the plan at a', and clh. The points bh and ch are then found by T-square lines on 0-1 and 0-2 respectively. The section is lined in considering the part of the pyramid to the right of plane X as removed.
Intersection of a Plane with a Curved Surface 85. It has been shown that the intersection of a plane with a pyramid or prism is found by connecting points in which the plane cuts the edges of the solid. In the case of a solid with a curved surface, as a cone or cylinder, the only edges are the base edges. There may, however, be straight or curved lines drawn on the surface, and the intersection of the plane with these lines be found, thereby locating points on the re quired curved intersection.