Conic Sections: Slicing a Cone
Problem
Slice a double cone at different angles to produce a circle (horizontal cut), ellipse (tilted cut), parabola (cut parallel to edge), and hyperbola (steep cut through both cones). Show why they are called "conic sections."
Explanation
The four curves — circle, ellipse, parabola, hyperbola — are called "conic sections" because they are the cross-sections you get when you slice a double cone (two cones tip-to-tip) with a flat plane at different angles:
- Circle: Plane perpendicular to the cone's axis
- Ellipse: Plane tilted, but not as steep as the cone's side
- Parabola: Plane parallel to exactly one edge of the cone
- Hyperbola: Plane steeper than the cone's side (cuts both halves)
This geometric origin was known to the ancient Greeks (Apollonius of Perga, ~200 BC) and connects all four curves as members of one family.
Try it in the visualization
Drag the "cut angle" slider to tilt the slicing plane. Watch the cross-section morph from circle through ellipse to parabola and then hyperbola. The 3D view rotates to show the geometry clearly.
Interactive Visualization
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