DRAWING PLATE, TITLE: CONIC SECTIONS. 47. This plate shows the different forms of the curves formed by the intersection of a cone or cylinder by a plane. If the plane of intersection is perpendicular to the axis of the cone or cylinder, the curve of the intersection will be a circle; but if it is inclined to the axis, it will be an ellipse in the case of a cylinder, and an ellipse, hyperbola, or parabola in the case of a cone, according to the angle of inclination. Fig. 1 is a cone cut by a plane which does not intersect the base of the cone. When the cutting plane does not intersect the base, or the new base of the cone when the cone is extended, the curve of intersection is an ellipse. Draw the plan and front elevation of a right cone whose altitude is 3 inches and whose base is 3 inches in diameter. Cut this cone by a plane a b, making an angle of 52° with the base. See figure. Divide the circle which represents the base of the cone in the plan into any number of parts, in this case 24, and, through the points of division A, E, H, etc., draw the radii O A, O E, O H, etc. to the center O. Draw also from these points straight lines A A', EE', HH', B B', etc., parallel to the axis of the cone O'n, and cutting the base A'B' in the points E', H', etc. From these points, draw lines to the apex O' of the cone, and cutting the base A'B' in points E', H', etc. From these points, draw lines to the apex O' of the cone, as E'O', H'O', etc., cutting the plane a b in the points D', F', etc. From these points D', F', etc., draw straight lines F'FF", D'DD", etc., parallel to the axis O'n of the cone, and intersecting the radii O A, O E, OH, OB, etc., in the points C, D, F, K, F", D", etc., and through these points of intersection draw the ellipse by aid of an irregular curve. Fig. 2 is a cone of the same size as in the preceding problem; but the cutting plane a b is, in this case, parallel to one of the elements* of the cone, and intersects the base. The *Any straight line drawn on the surface of a cone and passing through the apex (as O 4', Fig. 1. or O'A', Fig. 2, etc.) is called an element. |