Topic 4.1 Cutting the Cone
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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
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Instruction
Animations on cutting the cone
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The Circle
When a plane passes through the cone perpendicular to the axis the intersection forms a circle
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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
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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
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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.
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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
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