Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: How we get a Ellipse from a Conic, and how we get a Oval from
Cylinder Sections-- knifes that are V and asymmetrical V shaped

Replies: 27   Last Post: Oct 8, 2017 12:41 AM

 Messages: [ Previous | Next ]
 Me Posts: 1,716 Registered: 1/23/16
Re: How we get a Ellipse from a Conic, and how we get a Oval from
Cylinder Sections-- knifes that are V and asymmetrical V shaped

Posted: Oct 5, 2017 5:18 PM

On Thursday, October 5, 2017 at 6:38:55 PM UTC+2, Archimedes Plutonium wrote:

> the Conic section is always a Oval

No, it isn't.

> never an Ellipse.

Nope, *some* cone sections *are* ellipses (see proof below).

> So, that is true and I proved it.

No, it's NOT true, and you didn't prove it.

You will find a simple proof below that shows that certain cone sections as well as certain cylinder sections are ellipses.

It turns out that a cylinder can be considered as a special case of a cone in this context. Actually, the presented proof works for both cases, cone and cylinder.

Some preliminaries:

Top view of the cone/cylinder section and depiction of the coordinate system used in the proof:

^ x
|
-+- <= x=h
.' | `.
/ | \
. | .
| | |
. | .
\ | /
`. | .´
y <----------+ <= x=0

Cone/Cylinder (side view):

/ | \
/b | \
/---+---´ <= x = h
/ |´ \
/ ´ | \
/ ´ | \
x = 0 => ´-------+-------\
/ a | \

2 cases: 1.) cone: b < a, 2.) cylinder: b = a = r.

Proof:

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)^2/h^2

=> (1/ab)y(x)^2 + (4/h^2)(x - h/2)^2 = 1 ...equation of an ellipse

qed