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Some fascinating properties of the conic sections

This applet demonstrates some results from the paper Kepler's Laws and conic sections by Alexander Givental. A right circular cone with angle 45 degrees and vertex at the origin is intersected by a plane a. Theorem 1. The projection of a conic section to the -plane is a quadratic curve whose focus is the vertex of the cone, the directrix is the line of intersection of the cutting plane with the plane , and its eccentricity is equal to the tangent of the angle between the planes. In the applet below slider “Distance” controls the distance between the intersection of the cutting plane and the origin. Slider “t” controls the angle (in radians) between the cutting plane and the plane.
  • Click on the 'Projection' checkbox to demonstrate the projection of the conics to the plane. Change the angle and the distance to see how the curves change.
  • Click on the "Eccentricity" checkbox to see the eccentricity calculated for the points on the projected curve.
  • Click on the “Second Focus” checkbox to see a demonstration of the theorem below. Drag the two sliders to choose different angles and distances.
Theorem 2. Into the conical cup, inscribe a paraboloid of revolution so that it touches the secting plane. Then the projection of the point of tangency to the horizontal plane is the second focus of the projected conic section (the vertex of the cone being the first one). Note : In order to rotate the coordinate system you can click and drag anywhere in the 3D Graphics View.