Why slicing a cone gives an ellipse (beautiful proof)

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Strong rec Paul Sinclair · 2019-01 · Mathematics Stack Exchange · on “Why are two definitions of ellipses equivalent?”

3Blue1Brown recently posted a video giving a beautiful geometric proof of the relationship between the 3 most common descriptions of ellipses. [this video]

Recommends jonsno · 2017-04 · Mathematics Stack Exchange · on “Visualisation of ellipse as conic section”

I am sorry if this question sounds stupid, but I have this doubt in mind ever since I was a child. We know ellipses are symmetrical figures, having 2 axes of symmetry. We know that an ellipse is generated from a cone when a plane cuts it at some angle say $\theta$ ( see conic section [LINK] ) Now the cone is not same shape at both ends (my doubt), so a perfect ellipse shouldn't be formed from cone as it would result in a distorted ellipse with only one axis of symmetry (my thinking) I think that an ellipse that is a cylindric section [LINK] is seemingly more symmetrical than the ellipse that is a conic section How can I be satisfied that conic section ellipse is truly an ellipse? I think we

Recommends Paul Sinclair · 2019-01 · Mathematics Stack Exchange · on “How do we deduce that an ellipse, when defined as a "stretched circle", has foci?”

3Blue1Brown recently posted a video giving a beautiful geometric proof of the relationship between the 3 most common descriptions of ellipses. [this video]

Recommends AgentS · 2019-12 · Mathematics Stack Exchange · on “Why scaling a circle gives ellipse?”

There are at least $4$ definitions for ellipse: 1) Scaling a circle: $(x/a)^2+(y/b)^2=1$ 2) Sum of distances from two points is constant $PF_1+PF_2=\text{constant}$ 3) Focus, Directrix: $e=\dfrac{\text{distance between point and Focus}}{\text{distance between point and Directrix}}$ 4) Cutting a cone with a slanted plane. I managed to see the equivalence between #2 and #4 because of this [this video] awesome video. From then I've been trying to see a geometric connection between #1 and #2, but no luck yet. Using coordinate geometry & algebra easy to establish $ \#2\Rightarrow \#1$ . But how to see $\#1\Rightarrow \#2$ ? Any help?

Recommends Element118 · 2020-04 · Mathematics Stack Exchange · on “Why are two definitions of ellipses equivalent?”

From 3Blue1Brown [this video] , at about 9:40, expanding the comment [LINK] from @Paul Sinclair into an answer. A squashed circle with major axis $2a$ and minor axis $2b$ can sit inside tangent to a cylinder with radius $b$ . By constructing spheres tangent to the cylinder and the squashed circle, they would be tangent to the squashed circle at the foci, let them be $F_1$ and $F_2$ . The spheres are tangent to the cylinder, touching the cylinder at a circle each. These circles are perpendicular to the axis of the cylinder. Pick a point $P$ on the perimeter of the squashed circle. $F_1P$ is tangent to a sphere, and another line $l$ from $P$ tangent to the sphere at $Q_1$ runs along parallel to the

Recommends polfosol · 2020-09 · Mathematics Stack Exchange · on “Why ellipse is a "conic" section?”

3Blue1Brown has made a video about this question. Although their motivation came from reddit, and it seems they were not aware of this post. See: [this video]

Recommends D.R. · 2021-01 · Mathematics Stack Exchange · on “What do cones have to do with quadratics? Why is $2$ special?”

I've always been nagged about the two extremely non-obviously related definitions of conic sections (i.e. it seems so mysterious/magical that somehow slices of a cone are related to degree 2 equations in 2 variables). Recently I came across the following pages/videos: This 3B1B video about ellipses [this video] , which reignited my desire to understand conics Why are quadratic equations the same as right circular conic sections? [LINK] , which offers a very computational approach to trying to resolve this question Another 3B1B video on visualizing Pythagorean triples [LINK] (i.e. finding the rational points of a circle) and Manjul Bhargava's lecture on the Birch-Swinnerton-Dyer Conjecture [LINK]

Recommends John Alexiou · 2021-03 · Mathematics Stack Exchange · on “On the Equation of the Ellipse Formed by Intersecting a Plane and Cone”

Not a solution I remember seeing a YT video recently on this and it involved two spheres that are tangent to the inside of the cone and also the plane [LINK] Where these spheres touch the plane are the ellipse foci. From the two foci the equation of the ellipse can be found (or rather the parameters of the ellipse). Why slicing a cone gives an ellipse [3Blue1Brown] [this video]

Recommends omrir · 2022-06 · Mathematics Stack Exchange · on “Equivalence between ellipse definitions”

In a [LINK] by MathyJaphy, the creator uses a geometric\kinematic way to describe the focus and directrix definition of the ellipse (or, equivalently, the slice-of-a-cone definition). He even created a desmos file [LINK] if anyone wants to play with it a bit. When the moving line goes faster than the expanding circle, it draws an ellipse. I understand why it's equivalent to the conic section definition of the ellipse (or the focus and directrix one), but I can't see in a straightforward way why it's gives the same result as the two foci definition, or the squashed circle one (or for that matter - why the resulting shape is symmetric). Does anyone have a simple explanation why they're all the

Recommends Daniel · 2022-11 · Mathematics Stack Exchange · on “Is pure 3-d (solid) geometry basically 2d?”

I recently got interested in "pure" geometry (i.e. that doesn't use equations with x,y,z). I did some fun 2-d stuff like a circle inscribed in a square with a smaller circle in the corner, then find the relative radii. I then did similar 3-d stuff (a sphere in a cube, with a smaller sphere in the corner). In solving I noticed that I first converted to 2-d, by looking at a plane through the diagonal, then solved. So I got the impression than in pure solid geometry, you finally just end up resorting to 2-d geometry. But I just watched Why slicing a cone gives an ellipse [this video] by 3Blue1Brown and they did a pure approach to a conic section. I loved it but, is that approach unusual? Or are ther