Let's consider the Sectioning of a Cylinder and a Cone.
^ x E| -+- .' | `. / | \ . | . G | +c | H . | . \ | / `. | ´ y <----------+ ´ F
> The above is a view of a ellipse with center c and is produced by the > Sectioning of a Cylinder as long as the cut is not perpendicular to the base, > and as long as the cut involves two points not larger than the height of the > cylinder walls. What we want to prove is that the cut is always a ellipse, > which is a [certain] plane figure of two axes of symmetry with a Major Axis > and Minor Axis and center at c. > > So, what is the proof that [cut] figure EGFH is always an ellipse in the > cylinder section [as well as in the cone section]?
Here's is an easy proof for it:
Cylinder (side view):
| | | |-------+-------+ <= x = h | | ´| | | ´ | | |´ | | ´ | | | ´ | | x = 0 => ´-------|-------| | r | |