Conic Sections

This simulation cuts a double cone with a plane and draws the resulting curve. Students change the tilt from 0 to 90 degrees and the height of the plane, let Auto-sweep tilt run, and hide or show the cone. Readouts name the curve, give the eccentricity e and show the tilt together with the tilt at which a parabola appears. A level cut gives a circle, a steeper one an ellipse, a plane parallel to the cone side a parabola, and a steeper plane a hyperbola. Teachers can use it to connect geometry with algebra.

• Set Tilt to 0 degrees to get a circle, then raise it slowly to see an ellipse.
• Find the tilt where the Curve readout says parabola and read e.
• Increase Tilt past that value to see the two branches of a hyperbola.
• Change Plane height and check that the type of curve does not change.
• Turn Auto-sweep tilt on and watch the Eccentricity e readout.

1. 1Name the four conic sections and how each is made by slicing a cone.
2. 2Relate the tilt of the plane to the type of conic.
3. 3Describe eccentricity as 0 for a circle, 1 for a parabola and above 1 for a hyperbola.
4. 4Connect the geometry of the cone to the equations of conics.

Questions teachers ask