Four curves from one cut: circle, ellipse, parabola and hyperbola, set by the cut’s tilt against the cone’s side, the radius of a level cut, why the distance from the tip changes only the size, and what happens in a cut through the tip.
Step 1 of 5 · Four curves
Step 1 of 5
Four curves
A double cone, two cones meeting at their tips, is cut by a plane. The curve where the plane meets the cone is called a conic section, and the Greeks found more than two thousand years ago that there are four kinds. The kind is set by the cut’s tilt against the tilt of the cone’s side:
- a level cut gives a circle;
- a cut less steep than the side gives an ellipse, a closed oval;
- a cut parallel to the side gives a parabola;
- a cut steeper than the side gives a hyperbola, with a branch in each half of the cone.
Here the cone opens either side of its axis, so its side slopes at to the level, and the blue line marks it. Tilt of the cut is . Predict what will happen when it is parallel to the side.
Step 2 of 5
Hyperbola
A cut steeper than the side slices the upper cone, carries on below the tip’s level and slices the lower one too. Find such a tilt.
Step 3 of 5
Circle
In a level cut, a distance above the tip, every point of the curve is the same distance from the axis: a circle. Its radius, the height and the cone’s side make a right triangle, with an angle of at the tip. Work out the radius.
Step 4 of 5
Size and shape
Distance from the tip moves the cut parallel to itself. Try it, and answer what it changes.
Step 5 of 5
Through the tip
When the cut passes through the tip itself, it does not slice a curve out of the cone but touches it only at the tip, or along straight lines of it. These are the degenerate conic sections. The tilt here is , a hyperbola. Predict the cut when Distance from the tip is .
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Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.
Practice · Predict, then look
The cone’s side slopes at to the level, because it opens either side of its axis. What will the cut be when Tilt of the cut is , parallel to the side?
Practice · Predict, then look
The cone’s side slopes at to the level, because it opens either side of its axis. What will the cut be when Tilt of the cut is , parallel to the side?
Practice · Reach the target
Change Tilt of the cut so that the cut crosses both halves of the cone, then press Check.
Practice · Reach the target
Change Tilt of the cut so that the cut crosses both halves of the cone, then press Check.
Practice · Calculate
The cut is level, units above the tip, and the cone’s side opens from the axis. What is the circle’s radius? ()
Practice · Calculate
The cut is level, units above the tip, and the cone’s side opens from the axis. What is the circle’s radius? ()
Practice · Choose
The tilt stays at . What happens to the ellipse as the cut moves further from the tip? Try Distance from the tip, then answer.
Practice · Choose
The tilt stays at . What happens to the ellipse as the cut moves further from the tip? Try Distance from the tip, then answer.
Practice · Predict, then look
The tilt is , and the cut is a hyperbola. What will the cut be when Distance from the tip is , through the tip itself?
Practice · Predict, then look
The tilt is , and the cut is a hyperbola. What will the cut be when Distance from the tip is , through the tip itself?
Lesson complete. Every practice question here is done.