Introduction
Conic sections are the curves obtained by intersecting a double cone with a plane. There are three types: parabolas, ellipses, and hyperbolas. In this section we give the focus–directrix and focal definitions of each, read the key features off the standard equations, and see a worked example together with a second example that changes the orientation.
Instructions.
The same cut, animated. Press “Play” to watch the plane \(z = m\,y + 1\) swing from horizontal to steep in one continuous motion: the circle stretches to an ellipse, opens into a parabola at the instant the plane becomes parallel to the gold ruling on the cone (\(m = 1\)), and then breaks into the two branches of a hyperbola once the plane is steep enough to reach the lower nappe. The reading at the lower left compares the tilt of the plane with the \(45^\circ\) slant of the cone, which is what decides the three cases. Stop the animation at any time and drag the tilt slider yourself. Drag the figure to change the viewpoint, or press “Spin” to let it turn.
