On Monday, October 2, 2017 at 5:08:21 PM UTC+2, Archimedes Plutonium wrote:
> Analytic Geometry Proof, Cylinder Section [as well as cone section] is a[n] > Ellipse:
^ x E| -+- <= x = h .' | `. / | \ . | . 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]?
It turns out that a cylinder can be considered as a special case of a cone here. Actually, there's a simple proof which works for both cases, cone and cylinder.
(@Archie: This shows that there is no essential difference between these two cases.)
Cone/Cylinder (side view):
/ | \ /b | \ /---+---´ <= x = h / |´ \ / ´ | \ / ´ | \ x = 0 => ´-------+-------\ / a | \