*
Anyone who has taught mathematics at the college freshman level (as I have at Georgia Tech as a TA) went through the fairly simple process, using analytic geometry to define what is a conic section. If you can follow this, just perform these steps: First, write the definition of a cone (in x, y, z space). It is not difficult. Then, write the equation of an inclined plane in three-space. Then, if you have done the work accurately, you can solve these two equations simultaneously. That gives the locus of all points common to the cone and the inclined plane. (This is the definition of a section.) The resulting equation will be an ellipse, a circle, a parabola, or a hyperbola, depending on the exact inclined plane you have chosen. By the way, this was first demonstrated in about 300 BC, even before the original Archmedes (not the Plutonium version.)

earle
*

On 10/23/2022 1:06 AM, Earle Jones wrote:
> *
> Anyone who has taught mathematics at the college freshman level (as I have at Georgia Tech as a TA) went through the fairly simple process, using analytic geometry to define what is a conic section. If you can follow this, just perform these steps: First, write the definition of a cone (in x, y, z space). It is not difficult. Then, write the equation of an inclined plane in three-space. Then, if you have done the work accurately, you can solve these two equations simultaneously. That gives the locus of all points common to the cone and the inclined plane. (This is the definition of a section.) The resulting equation will be an ellipse, a circle, a parabola, or a hyperbola, depending on the exact inclined plane you have chosen. By the way, this was first demonstrated in about 300 BC, even before the original Archmedes (not the Plutonium version.)
>
> earle
> *

Plutonium's argument is based on axes of symmetry. While his so-called
"proof" is rambling and in no way a valid math proof, its basic argument
is a tilted plane intersecting a cone will have the side nearest the
apex of the cone will be smaller than the side tilting away from the
apex, simply because the cone itself gets smaller near the apex and
larger away from it. Thus his "proof" is that the cone isn't symmetric
_around the axis of the cone_. However the cone's formula will have
something like (x-k)^2 in it, which is obviously symmetric around the
x=k plane.

I know if you look at a drawing, it doesn't look like it could be an
ellipse. I think of this as like a "mathematical optical illusion".

This is easier to visualize if the cone is tilted around the y axis,
with its apex at the point (x=0,y=0,z=0) and is intersected by the plane
z=m for some m.

Here's a proof someone (I forget who) wrote earlier in response to AP,
that tilts the cone and the cone is intersected by the plane
z=<constant>. It may be unclear in the last line why that is the
equation of an ellipse if C>0, but the left side is a constant, and the
right side is C*(x-K)^2 + y^2, which is the formula of an ellipse. K=k*S/C.

I'll start with the cone z^2 = x^2 + y^2, and rotate it through an angle
'theta' around the 'y' axis, and consider the intersection of that
rotated cone with the plane z = <constant>

To simplify things, let c = cos(theta) and s = sin(theta). Then the
rotation is defined by

z --> cz + sx
x --> -sz + cx
y --> y

So the equation of the rotated cone is

(cz+sx)^2 = (-sz+cx)^2 + y^2

and now let C = c^2-s^2 and S = 2sc (again, just to simplify the look
of things)

so we get

Cz^2 = Cx^2 - 2Szx + y^2

and letting 'z' equal the constant 'k' gives

Ck^2 + k^2*S^2/C = C(x - k*S/C)^2 + y^2

which is the equation of an ellipse if C > 0.

On 10/23/2022 1:51 AM, Michael Moroney wrote:
> It may be unclear in the last line why that is the
> equation of an ellipse if C>0, but the left side is a constant, and the
> right side is C*(x-K)^2 + y^2, which is the formula of an ellipse. K=k*S/C.

This proof skips several steps before the last line, so it's far from
obvious. I will have to make it clearer, and add the skipped steps back in.

On Sunday, October 23, 2022 at 12:51:25 AM UTC-5, Michael Moroney wrote:
> On 10/23/2022 1:06 AM, Earle Jones wrote:
> > *
> > Anyone who has taught mathematics at the college freshman level (as I have at Georgia Tech as a TA) went through the fairly simple process, using analytic geometry to define what is a conic section. If you can follow this, just perform these steps: First, write the definition of a cone (in x, y, z space). It is not difficult. Then, write the equation of an inclined plane in three-space. Then, if you have done the work accurately, you can solve these two equations simultaneously. That gives the locus of all points common to the cone and the inclined plane. (This is the definition of a section.) The resulting equation will be an ellipse, a circle, a parabola, or a hyperbola, depending on the exact inclined plane you have chosen. By the way, this was first demonstrated in about 300 BC, even before the original Archmedes (not the Plutonium version.)
> >
> > earle
> > *
> Plutonium's argument is based on axes of symmetry. While his so-called
> "proof" is rambling and in no way a valid math proof, its basic argument
> is a tilted plane intersecting a cone will have the side nearest the
> apex of the cone will be smaller than the side tilting away from the
> apex, simply because the cone itself gets smaller near the apex and
> larger away from it. Thus his "proof" is that the cone isn't symmetric
> _around the axis of the cone_. However the cone's formula will have
> something like (x-k)^2 in it, which is obviously symmetric around the
> x=k plane.
>
> I know if you look at a drawing, it doesn't look like it could be an
> ellipse. I think of this as like a "mathematical optical illusion".
>
> This is easier to visualize if the cone is tilted around the y axis,
> with its apex at the point (x=0,y=0,z=0) and is intersected by the plane
> z=m for some m.
>
>
> Here's a proof someone (I forget who) wrote earlier in response to AP,
> that tilts the cone and the cone is intersected by the plane
> z=<constant>. It may be unclear in the last line why that is the
> equation of an ellipse if C>0, but the left side is a constant, and the
> right side is C*(x-K)^2 + y^2, which is the formula of an ellipse. K=k*S/C.
>
>
>
> I'll start with the cone z^2 = x^2 + y^2, and rotate it through an angle
> 'theta' around the 'y' axis, and consider the intersection of that
> rotated cone with the plane z = <constant>
>
> To simplify things, let c = cos(theta) and s = sin(theta). Then the
> rotation is defined by
>
> z --> cz + sx
> x --> -sz + cx
> y --> y
>
> So the equation of the rotated cone is
>
> (cz+sx)^2 = (-sz+cx)^2 + y^2
>
> and now let C = c^2-s^2 and S = 2sc (again, just to simplify the look
> of things)
>
> so we get
>
> Cz^2 = Cx^2 - 2Szx + y^2
>
> and letting 'z' equal the constant 'k' gives
>
> Ck^2 + k^2*S^2/C = C(x - k*S/C)^2 + y^2
>
> which is the equation of an ellipse if C > 0.

Fascinating how a failure of science Kibo Parry M. who even failed arithmetic with his 938 is 12% short of 945, has the mindless balls and mindless skull to tackle a geometry proof.

Why the Rensselaer Polytech failure Kibo Parry M. cannot even understand that Geothermal is caused from radioactive decay inside of Earth, but then the failure Kibo still testifies that geothermal is recycled Solar.

There should be a award for the loud mouth idiot of sci.math and Kibo surely would win it year after year.