If one surface meets another at some angle, an intersection is produced. Either surface may be plane, or curved. If both are plane, the intersection is a straight line; if one is curved, the intersection is a curve, except in a few special cases; and if both are curved, the intersection is usually curved. In the latter case, the entire curve does not always lie in the same planes. If all points of any curve lie in the same plane, it is called a plane curve. A plane intersecting a curved surface must always give either a plane curve or a straight line.
Fig. 131. Intersection of Plane and Pyramid.
Fig. 132. Intersection of Plane and Pyramid.
In Fig. 131 a square pyramid is cut by a plane A parallel to the horizontal. This plane cuts from the pyramid a four-sided figure, the four corners of which will be the points where A cuts the four slanting edges of the solid. The plane intersects edge ob at point 4v in elevation. This point must be found in plan vertically above on the horizontal projection of line ob, that is, at point 4h. Edge oe is directly in front of ob, so is shown in elevation as the same line, and plane A intersects oe at point 1vm elevation, found in plan at 1*. Points 3 and 2 are obtained in the same way. The intersection is shown in plan as the square 1-2-3-4, which is also its true size as it is parallel to the horizontal plane. In a similar way the intersections are found in Figs. 132 and 133. It will be seen that in these three cases where the planes are parallel to the bases, the sections are of the same shape as the bases, and have their sides parallel to the edges of the bases.
Fig. 133. Intersection of Plane and Prism.
Fig. 134. Intersection of Plane and Cone.
It is an invariable rule that when such a solid is cut by a plane parallel to its base, the section is a figure of the same shape as the base. If then in Fig. 134 a right cone is intersected by a plane parallel to the base the section must be a circle, the center of which in plan coincides with the apex. The radius must equal od.
In Fig. 135 and Fig. 136 the cutting plane is not parallel to the base, hence the section will not be of the same shape as the base.
Fig. 135. Intersection of Plane and Square Pyramid.
Fig. 136. Intersection of Plano and Hexagonal Pyramid.
The intersections are found, however, in exactly the same manner as in the previous figures, by projecting the points where the plane intersects the edges in elevation, on to the other view of the same line.
1. Find the horizontal projection of a transverse section of the pyramid of Fig. 130, made by a plane perpendicular to the vertical, but inclined at an angle to the horizontal plane of projection; and let all the sides of the base be at an angle with M N, Fig. 137.
Having drawn the vertical S S', the center line of the figures, its point of intersection with the line M N is the center of the plan. Since none of the sides of the base are to be parallel with M N, draw a diameter A'D' making the required angle with MN, and from the points A' and D' proceed to set out the angular points of the hexagon, as in Fig. 130. Then join the angular points which are diametrically opposite and project the figure thus obtained upon the vertical plane, as shown.
Now, if the cutting plane be represented by the line ad in the elevation, it is obvious that it will expose, as the section of the pyramid, a polygon whose angular points being the intersections of the various edges with the cutting plane, will be projected in perpendiculars drawn from the points where it meets these edges respectively. From the points a, f, b, etc., raise the perpendiculars aa',ff', bb', etc., to meet the lines A' D', F' C', B' E', etc. When the contiguous points of intersection of these lines are joined, a six-sided figure will be formed which will represent the section required. The edges FS and ES being concealed in the elevation, but necessary for the construction of the plan, have been expressed in dotted lines, as is also the portion of the pyramid situated above the cutting plane which, though supposed to be removed, is necessary in order to draw the lines representing the edges.
2. Find the horizontal projection of the transverse section of a regular five-sided pyramid, cut by a plane perpendicular to the vertical, but inclined at an angle to the horizontal plane of projection; and let one edge of the pyramid, BS, be in a plane perpendicular to both the horizontal and the vertical planes of projection, as shown in Fig. 138.
Fig. 137. Frustum of Hexagonal Pyramid.
The plan of the pyramid is constructed by describing from the center S' a circle circumscribing the base, and from B' dividing the circumference into five equal parts, and joining the contiguous points of division by straight lines. These form the polygon A' B' C' D' E', whose angles, when joined to the center S', show the projections of the edges of the pyramid. Then, following the method above explained, the elevation and the horizontal projection of the section made by the plane ac are obtained. But that method will not suffice for the determination of the point b', because the perpendicular let fall from the corresponding point 6, in the elevation, coincides with the projection of the edge B S. Let the pyramid supposedly be turned a quarter of a revolution round its axis; the line B's' will then have assumed the position S' b2. Project the. point b2 to b3, and join Sb3. Then since the required point must also be conceived to have described a quarter of a circle in a plane parallel to the horizontal plane, and that its new position must be in the line Sb3, it is obvious that its vertical projection is the point b4, the intersection of a horizontal line drawn through 6 with the line Sb3. The distance bb4 may then be used to determine the distance from S' to b', and determines the position of the latter point in the plan; or, following a more methodical process, by projecting the point b4 to b5, and describing a circle from the center S' passing through b5, its intersection with B's' is the point sought.
Fig. 138. Frustum of Pentagonal Pyramid.