Planes With Cones Or Cylinders

Sections cut by a plane from a cone have already been defined as conic sections. These sections may be any of the following: two straight lines, circle, ellipse, parabola, hyperbola. All except the parabola and hyperbola may also be cut from a cylinder.

Methods have previously been given for constructing the ellipse, parabola, and hyperbola, without projections; it will now be shown that they may be obtained as actual intersections.'

Ellipse

In Fig. 139 the plane cuts the cone obliquely. To find points on the curve in plan take a series of horizontal planes xyz, etc., between points cv and dv. One of these planes, as w, should be taken through the center of cd. The points c and d must be points on the curve, since the plane cuts the two contour elements at these points. Contour elements are those forming the outline. The horizontal projections of the contour elements will be found in a horizontal line passing through the center of the base; hence the horizontal projection of c and d will be found on this center line, and will be the extreme ends of the curve.

The plane x cuts the surface of the cone in a circle, as it is parallel to the base, and the diameter of the circle is the distance between the points where x crosses the two contour elements. This circle, lettered x on the plan, has its center at the horizontal projection of the apex. The circle x and the curve cut by the plane are both on the surface of the cone, and their vertical projections intersect at the points 2-2. Point 2 on the elevation then represents two points which are shown in plan directly above on the circle x, and are points on the required intersection. Planes y and z, and as many more as may be necessary to determine the curve accurately, are used in the same way. The curve found is an ellipse. The student will readily see that the true size of this ellipse is not shown in the plan, for the plane containing the curve is not parallel to the horizontal.

Ellipse   Section from a Cone.

Fig. 139. Ellipse - Section from a Cone.

In order to find the actual size of the ellipse, it is necessary to place its plane in a position parallel either to the vertical or to the horizontal. The actual length of the long diameter of the ellipse must be shown in elevation, cvdv, because the line is parallel to the vertical plane. The plane of the ellipse then may be revolved about cvdv as an axis until it becomes parallel to V, when its true size will be shown. For the sake of clearness of construction, cvdv is imagined moved over to the position c'd', parallel to cvdv. The lines 1-1, 2-2, 3-3 on the plan show the true width of the ellipse, as these lines are parallel to H, but are projected closer together than their actual distances. In elevation these lines are shown as the points 1, 2,3, at their true distance apart. Hence if the ellipse is revolved around its axis cvdv the distances 1-1, 2-2, 3-3 may be laid off on lines perpendicular to cvdv, and the true size of the figure be shown. In Fig. 140 a plane cuts a cylinder obliquely. This is a simpler case, as the horizontal projection of the curve coincides with the base of the cylinder. To obtain the true size of the section, which is an ellipse, any number of points are assumed on the plan and projected down on the cutting plane, at 1,2,3, etc. The lines drawn through these points perpendicular to l'-7' are made equal in length to the corresponding distances 2'-2', 3'-3', etc., on the plan, because 2'-2' is the true width of curve at 2.

Ellipse   Section from a Cylinder.

Fig. 140. Ellipse - Section from a Cylinder.

Parabola

If a cone is intersected by a plane which is parallel to only one of the elements, as in Fig. 141, the resulting curve is the parabola, the construction of which is exactly similar to that for the ellipse, as given in Fig. 139. If the intersecting plane is parallel to more than one element, or is parallel to the axis of the cone, a hyperbola is produced.

Parabora   Section from a Cone.

Fig. 141. Parabora - Section from a Cone.

In Fig. 142, the vertical plane A is parallel to the axis of the cone. In this instance the curve when found will appear in its true size, as plane A is parallel to the vertical. Observe that the highest point of the curve is found by drawing the circle X on the plan tangent to the given plane. One of the points where this circle crosses the diameter is projected down to the contour element of the cone, and the horizontal plane X drawn. Intermediate planes Y, Z, etc., are chosen, and corresponding circles drawn in plan. The points where these circles are crossed by the plane A are points on the curve, and these points are projected down to the elevation on the planes y,z, etc.