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According to Wikipedia,

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.

However most of the images actually show a double cone instead of a cone and it makes sense to me since a hyperbola has two components.

So is it true that despite its name and definition, conic section is actually an intersection of a plane with a double cone?

MCL
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    If you really want to make a hard distinction between "cone" and "double cone", then, yes: a conic section requires the latter. However, it's quite common to refer to a double cone simply as a "cone". (Context should make the usage clear.) Consider, for instance, the characterization of quadric surfaces. – Blue Sep 01 '19 at 08:27
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    You only need double cone for a hyperbola with two branches. You can form a circle, ellipse, parabola with single side of a cone. – Sonal_sqrt Sep 01 '19 at 08:28

2 Answers2

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There's a very general definition of "cone" which is the following.

Let $\cal C$ be a (closed, regular) curve in $\Bbb R^3$ and let $P$ a point not in $\cal C$. Then the cone through $\cal C$ with vertex $P$ is the ruled surface made of all lines $QP$ as $Q$ varies in $\cal C$.

When $\cal C$ is a circle with center $C$ and $P$ is chosen so that $PC$ is orthogonal to the plane containing $\cal C$ you get what you call "double cone".

You really need double cones if you want your section to be an hyperbola, which is made of two disconnected parts, lying on different "parts" of the double cone.

General cones are a basic example of flat surfaces (they can be flattened out isometrically on a plane).

Also note that if you let $P$ move far away from $\cal C$ the lines $QP$ tend to become parallel and will if you let $P$ "go to infinity". In this sense cylinders are degenerate cones (the vertex of a cylinder being "at infinity")

Andrea Mori
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Cone/Plane Intersection scenario

A full cone consists of two nappes/sheets.

Only when the inclinations of cone generator semi-vertical angle $ \alpha$ is more than sectioning plane angle $\beta$ to the cone axis we get the hyperbola as a real disjuncted double curve.

A parabola or ellipse is produced by intersection when $ \alpha < \beta$ on a single nappe.

In this case we are not concerned with generality when only one cone is partaking and resulting in an intersection.

Narasimham
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