It is evident that two distinct geometrical views are necessary to convey a complete idea of the form of the object; an elevation to represent the sides of the body, and to express its height; and a plan of the upper surface to express the form horizontally.
It is to be observed that this body has an imaginary axis or center line, about which the same parts or segments of the body are equally distant; this is an essential characteristic of all symmetrical figures.
Draw a horizontal dotted line M N for the center line of the plan views, Fig. 130. Then draw a perpendicular ZZ' to M N.
In delineating the pyramid, it is necessary, in the first place, to construct the plan. The point S', where the line ZZ' intersects the line M N, is to be taken as the center of the figure, and from this point, with a radius equal to the side of the hexagon which forms the base of the pyramid, describe a circle, cutting M N in A' and D'. From these points, with the same radius, draw four arcs of circles, cutting the primary circle in four points. These six points being joined by straight lines, will form the figure A' B'C' D' E'F', which is the base of the pyramid;and the lines A' D', B' E', and C' F' will represent the projections of its edges foreshortened as they would appear in the plan. If this operation has been correctly performed, the opposite sides of the hexagon should be parallel to each other and to one of the diagonals; this should be tested by the application of the square or other instrument proper for the purpose.
By the help of the plan obtained as above described, the vertical projection of the pyramid may be easily constructed. Since it is directly under the plan, it must be projected vertically downward; therefore, from each of the points A', B', C', drop perpendiculars to AD, the base line of the pyramid in the elevation. The points of intersection, A, B, C, and D, are the true positions of all the angles of the base; and it only remains to determine the height of the pyramid, which is to be set off from the point G to S, and to draw S A, SB, S C, and SD, which are the only edges of the pyramid visible in the elevation. Of these it is to be remarked that SA and SD alone, being parallel to the vertical plane, are seen in their true length; and, moreover, that from the assumed position of the solid under examination, the points F' and E' being situated in the lines BB' and CC', the lines SB and S C are each the projections of two edges of the pyramid.
Fig. 130. Construction of Regular Hexagonal Pyramid.
Example 1, having its base set in an inclined position, but with its edges SA and SD still parallel to the vertical plane, Fig. 130.
It is evident, that with the exception of the inclination, the vertical projection of this solid is precisely the same as in the preceding example, and it is only necessary to show the same view of the pyramid in its new position. For this purpose, after having fixed the position of the point D, draw through this point a straight line DA, making with MN an angle equal to the desired inclination of the base of the pyramid. Then set off the distance DA, equal to that used in Example 1; erect a perpendicular on the center, and set off GS equal to the height of the pyramid. Transfer also from the first example the distance BG and CG to the corresponding points, and complete the figure by drawing the straight lines A S, BS, CS, and DS.
In constructing the plan of the pyramid in this position, it is to be remarked that since the edges S A and S D are still parallel to the vertical plane, and the point D remains unaltered, the projection of the points A, D, and S, will still be in the line M N. The position of A' is determined by the intersection of the perpendicular A A' with MN. The remaining points, B', C', etc., in the projection of the base, are found, in a similar manner, by the intersections raised from the corresponding points in the elevation, with lines drawn parallel to M N, at a distance (set off at o, p) equal to the width of the base. By joining all the contiguous points, the figure A' B'C' D' E'F' is obtained representing the horizontal projection of the base, two of its sides, however, being dotted, as they must be supposed to be concealed by the body of the pyramid. The vertex 8 having been similarly projected to S', and joined by straight lines to the several angles of the base, the projection of the solid is completed.