Ellipses have multiple definitions, how are these the same?

A link to the full video is at the bottom of the screen.

The full video this comes from proves why slicing a cone gives the same shape as the two-thumbtacks-and-string construction, which is beautiful.

Editing from long-form to short by Dawid Kołodziej

[DonOtto]

You know for the longest time I couldn’t tell the difference between a cone sliced ellipsis and a cylinder sliced ellipsis. It turns out that the cone ellipsis foci rides on the center radius of the cone, and the cylinder ellipsis center rides along the the radius line of the cone. Oddly enough this concept really bothered me until i found this out.

[nocapstoast]

Here are 2 other ways to define an ellipse that ive learnt

1st Way
Take a variable point M on a fixed line

Take a fixed point S (not on the fixed line)

Draw perpendicular from point M to a P

Now the ellipse will be formed by locus of point P only when
PS/PM = constant ratio
this ratio is eccentricity
(locus is a path that a point follows)
(basically a locus of P, PS=ePM)

2nd Way
Take a circle and draw a diameter

From each point on circle draw a perpendicular to the diameter

Now take a point P on these perpendiculars so that it divides all those perpendiculars in a fixed ratio
Now the locus of P is ellipse
i learnt this yesterday in math class :]

[mikeholt]

The answer is that eclipse exists independently of the methods used generate them.

[Achill]

Construction methods 2 and 3 can be shown to be equivalent. And if you allow cones that come closer and closer to a cylinder, you get method 1, too.

[ynes]

Dandelin? Looking at the full video: yes, Dandelin.

[deltavega]

Creating 2 foci or localized center where there would have been one in a circle?

Just guessing

[happmacdonald]

I wanna know why it's hard to calculate their circumference

[peachycoffee]

ah yes thanks for the mathgasem needed that now i cant think stright

[Shooo]

How about looking at a circle from an angle, that's also ellipse

[Vecto]

Nice I am from india __/\__ is there anyone

[angelorf]

Mathematicians have a low bar for what counts as a definition.