How a Flat Cut Draws an Ellipse
A cone’s surface narrows toward its tip, so a cutting plane meets the sides at different distances from that tip. When the plane is level—perpendicular to the cone’s central axis—it meets the cone at one constant height. Every point of the cut is equally far from the axis, making a circle.
Tilt the plane, and one side of the cut passes through the cone higher up while the opposite side reaches lower down. The cut is then farther from the axis at one end than at the other. Its boundary remains a smooth, closed curve, but it is longer in one direction: an ellipse. A circle is the special case in which the ellipse has equal width in every direction.
The Angle Sets the Curve
The cut must be shallow enough to pass across just one side of the cone and form a closed loop. As the tilt increases, the ellipse becomes more elongated. At a particular angle, the plane runs parallel to a straight sloping edge of the cone; the intersection opens into a parabola instead. A steeper plane can intersect both halves of an extended, double cone, producing a hyperbola.
These shapes are called conic sections because they arise from slicing a cone with a plane. The same geometry can be pictured with a flashlight beam: the outline where a cone of light meets a wall changes with the wall’s angle. Tilt the wall, and the circular outline stretches—an ellipse made by the same simple intersection.
