6)are all German mathematicians like Peter Roquette, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, teachi ng a Conic section is ellipse, when in truth it is an oval?
Oct 4, 2017 7:05 PM
Re: 6)are all German mathematicians like Peter Roque tte, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, te aching a Conic section is ellipse, when in truth it is an ov al?
Oct 4, 2017 7:05 PM
=> (1/ab)y(x)^2 + (4/h^2)(x - h/2)^2 = 1 ...equation of an ellipse
Now lets just look at some "properties" of this "curve" (i.e. the cone/cyliner section):
y(h/2 + x')^2 = ab - ab(2(h/2 + x')/h - 1)^2 = ab - ab(2x'/h)^2
=> y_1(h/2 + x') = sqrt(ab) * (sqrt(1 - (2x'/h)^2) ...symmetric relative to x = h/2 (y_1(x) ist the part of the curve with y >= 0)
=> y_2(h/2 + x') = -sqrt(ab) * (sqrt(1 - (2x'/h)^2) ...symmetric relative to x = h/2 (y_2(x) ist the part of the curve with y <= 0)
And clearly there's a "maximum" of y_1(x) and a "minimum" of y_2(x) at x = h/2.
With other words, x = h/2 is another axis of symmetry of the figure besides y = 0.
So it's clearly NOT an "oval" as defined by AP.
> ... it [the cone section] has two axes of symmetry. There are more subtle > reasons for that (to actually see it, you need an actual proof [...] > but one thing you overlooked is that the entire cross-sectional area > shifts sideways as you rotate the sectional plane. > > You assume the cone's centreline must intersect the cross-section in > its centre of symmetry but this is false. (It happens top be true in > the case of the cylinder which is probably what led you into this trap.) > > [...] All you have done is you've shown that a particular method does > not detect more than one axis of symmetry. In order to PROVE your claim > you have to PROVE there is no OTHER axis of symmetry. Your proof fails > to do that, it simply detects one and it does not DISPROVE the existence > of any other. > > In reality, of course, the cross-section figure does have a second axis > of symmetry. Your proof is simply too weak to detect it.