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Topic 4.1 Cutting the Cone 1 Instruction Intersections of a plane with a double napped cone Overview: This unit involves the shapes that can be created by the cutting of a “double napped” cone. A double napped cone is formed by taking a generator line and revolving it about a given axis. Generator Line Important tips: Vertex ***assume all generator lines will extend to infinity… thus all cones are double napped. A right circular cone makes a 90° at the vertex. Axis 2 Instruction Animations on cutting the cone 3 The Circle When a plane passes through the cone perpendicular to the axis the intersection forms a circle 4 The Ellipse When a plane makes an angle with the axis that is greater than the angle the generator makes with the axis (but is less than 90°) the intersection forms an ellipse. 5 The Parabola When a plane passes through the cone at an angle parallel to the generator the intersection forms a parabola 6 The Hyperbola When a plane makes an angle with the axis that is less than the angle the generator makes with the axis (or parallel to the axis) the intersection forms a hyperbola. 7 4 Degenerate conic sections. (what happens if the plane passes through the vertex at different angles.) Degenerate Circle When a plane passes through the cone perpendicular to the axis and passes through the vertex the intersection forms a point Degenerate Ellipse When a plane makes an angle with the axis that is greater than the angle the generator makes with the axis and passes through the vertex the intersection forms a point. Degenerate Parabola When a plane passes through the cone at an angle parallel to the generator and passes through the vertex the intersection forms a line. Degenerate Hyperbola When a plane makes an angle with the axis that is less than the angle the generator makes with the axis and passes through the vertex the intersection forms intersecting lines 8 Special Case The special case occurs when the vertex of the cone is “stretched to infinity”. The result is that the generator line becomes parallel to the axis and the new shape results in a cylinder. 9 The Special Cases Case I Case II Case III Plane parallel to axis and inside the cylinder Plane parallel to axis and on the generator Plane parallel to axis and on the generator Parallel Lines Single Line No Locus 10 11