Is an elipse derived from a cone or cylinder?

Question

I was taught that an ellipse was a section through a cylinder, therefore symmetrical. I have seen many sites stating that it is a 'conic section', including wikipedia. A section through a cone can never be elliptical as the part nearer the bottom of the cone has to be more broadly curved than that near the top. Have I miss-understood something?

Answer 1

The bottom of the section of the cone, which you think has to be more broadly curved because the 'horizontal' diameter of the cone there is wide, is cut at a shallower angle. At the limit, for an extremely angled cut, it is nearly parallel to the side of the cone, and thus the tip will have to come out as being very pointy.

This balances that, and in the end the ellipse is symmetric.

A cylinder is a special case of a cone, one with zero taper, so calling it a 'conic' section retains the most general formulation.

Answer 2

You are quite right about the cylinder - A right circular cylinder that is cut on a plane not perpendicular to its altitude but also but also not parallel to its altitude will yield an ellipse whose minor axis is the diameter of the cylinder.



However, counter-intuitive though it may seem, a cross section through a right circular cone that is not perpendicular to its altitude is also an ellipse. See: http://en.wikipedia.org/wiki/Conic_section



"The three types of conics are the ellipse, parabola, and hyperbola. The circle can be considered as a fourth type (as it was by Apollonius) or as a kind of ellipse. The circle and the ellipse arise when the intersection of cone and plane is a closed curve. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone as in the picture at the top of the page this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case, the plane will intersect both halves (nappes) of the cone, producing two separate unbounded curves."



See also: http://en.wikipedia.org/wiki/Ellipse



"In geometry, an ellipse (from Greek ἔλλειψις elleipsis, a "falling short") is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is perpendicular to the axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant.



Ellipses are closed curves and are the bounded case of the conic sections, the curves that result from the intersection of a circular cone and a plane that does not pass through its apex; the other two (open and unbounded) cases are parabolas and hyperbolas. Ellipses can also arise as images of a circle under parallel projection and some cases of perspective projection. It is also the simplest Lissajous figure, formed when the horizontal and vertical motions are sinusoids with the same frequency."



Although it may seem counterintuitive, the effect of the tilt of the intersecting plane is to generate equal curvatures at opposing points on the ellipse. Make a cone and see!

Answer 3

You have misunderstood a lot.