On Monday, October 2, 2017 at 11:51:08 AM UTC-5, Dan Christensen wrote:
Mr Sawat Layuheem 11:01 PM (17 minutes ago)
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Dan is ellipse in beer bottle
level of math of these two
On Monday, October 2, 2017 at 10:36:22 AM UTC-5, Me wrote: > 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 | \ > > (cone: b < a; cylinder: a = b = r) > > r(x) = a - ((a-b)/h)x > d(x) = a - ((a+b)/h)x > > y(x)^2 = r(x)^2 - d(x)^2 = ab - ab(2x/h - 1)^2 = ab(1 - 4(x - h)^2/h^2 > > => (1/ab)y(x)^2 + (4/h^2)(x - h)^2 = 1 ...equation of an ellipse > > Some considerations: > > => y(h/2 + x')^2 = sqrt(ab - ab(2(h/2 + x')/h - 1)^2) = ab - ab(2x'/h)^2 > > => y(h/2 + x') = sqrt(ab) * (sqrt(1 - (2x'/h)^2) ...symmetric relative to h/2 (hence Ec = cF) > > => y(h/2) = sqrt(ab) (= Gc = cH) > > ====================================================== > > @Archie: Yes, this proves that (certain) cone sections "as depicted in my diagram" as well as (certain) cylinder sections (as described by you) are ellipses. qed > > Note, Archie, that there is no reference to Dandelin Spheres whatsoever. > > Still not convinced? Can you point out an error in my simple calculation (of the shape of the cone/cylinder section) above?