A cone is defined as a surface generated by lines passing through a fixed point and interesecting a given conic, or touching a given surface.
A pair of non-parallel planes seems to loosely satisfy these properties. We can loosely call a pair of planes as a cone with an infinite number of vertices (any point on the line of intersection of the planes being a vertex) and the guiding curve being two non-intersecting lines meeting at infinity. The pair of planes satisfies the property of cone which says that any line joining the vertex with a point of the cone has to lie on the cone. (being a generating line)
I understand that most of my claims are hand wavy but I doubt that this is just a coincidence. So the question is mathematically speaking, what is the relationship between cones and pairs of planes?
Kindly explain in simple language. I am familiar with only undergraduate level mathematics.
Geometry – Nature of Relationship Between a Cone and a Pair of Planes
analytic geometrygeometryquadrics
Related Solutions
I wish to express my sincere thanks to @Blue and @amd for clearing my doubt in the comments.
The location of the focus, vertex, and directrix of a conic can be easily determined by knowing some basics about Dandelin Spheres. Wikipedia gives the following explanation:
In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called focal spheres.
In the above diagram, the yellow plane cuts the blue cone forming an ellipse. Next, imagine inserting two spheres of maximum volume in the upper and lower parts of the cone demarcated by the slicing plane, such that they just touch the surfaces (curved surface of the cone and slicing plane) but don't peep out. These two are called as Dandelin spheres. The points at which these spheres touch the yellow plane (slicing plane), $F_1$ and $F_2$ are the foci of the ellipse. So now, we have located the foci of the ellipse.
The same Dandelin Spheres are helpful in determining the directrix and the vertex. The two spheres in the above diagram, touch the curved surface of the cone in a circle (represented by white circles $k_1$ and $k_2$. Let us consider, two planes passing through these two circles separately. These planes meet the yellow plane (slicing plane) in straight lines (unless and until the yellow plane cuts out a circle in the cone, when it's parallel to the base of cone). The two lines formed by the intersection of the three planes (slicing plane, and the two planes through the two circles) are parallel to each other. These lines are the directrices of the ellipse.
The above [.gif] explains the same visually. In this, the light blue plane is the slicing plane, orange sphere is one of the Dandelin spheres, transparent planes are those which pass through the circular region formed by the intersection of the spheres and the curved surface of the cone. Here the slicing plane, forms an ellipse which is shown by blue. The two parallel blue lines are the directrices.
Even though I've explained using ellipses, the same concept can be extended to other ellipses. For example, parabola since it has only one focus, has only one Dandelin sphere. Hyperbola has two foci in opposite nappes, so it has two such spheres in the two nappes. Using this we can determine the directrix and the focus of the conic.
Now, coming to the last part of the answer, finding the vertex. This is simple once we've found the directrix and the focus. Just draw a line perpendicular to the directrix passing though the focus. This line is the axis of the conic (and not that of the cone!). The point where axis meets the curve is the vertex.
I am belatedly offering these remarks as an answer rather than just a comment.
In projective geometry, a point at infinity is just as good as one at a finite distance. Take a circle; call this the base circle. Construct a perpendicular axis through the center of the base circle and go infinitely far to get to a point (in either direction; it gets to the same point either way). Using that point at infinity as the vertex, construct a cone consisting of all lines that pass through the vertex and the circle.
The object constructed this way is a cone in projective space, but the finite part of it (excluding the vertex) is a cylinder, and if the base circle is perpendicular to a plane and intersects the plane in two points, the conic generated by the plane is a pair of parallel lines.
Alternatively, without explicitly invoking projective geometry, we could consider a cylinder to be a degenerate cone which is the limiting case as the vertex of the cone goes off to infinity. The parallel lines then are produced by intersection with a suitable plane parallel to the axis of the cylinder.
Best Answer
Actually, cones enjoy different definitions.
1 - In linear algebra, a cone is a subset of a real vector space that is closed under multiplication by a positive scalar, cf. Wikipedia: https://en.m.wikipedia.org/wiki/Convex_cone
So yes, with this definition two non-parallel planes make a cone.
Apart from being one of Wikipedia's definitions for a cone (and they add "sometimes called a linear cone for distinguishing it from other sorts of cones"), this is also the definition used in conic optimization, cf. https://en.m.wikipedia.org/wiki/Conic_optimization
This closedness by positive multiplication is actually usefull in proofs related to optimization algorithms, i.e. it is a well-thought concept; although usually coupled with convexity: cones considered in optimization are convex cones.
2 - That being said, two non-parallel planes are also a cone according to the definition you adopt, which is also widely used, i.e. "lines passing through a fixed point and intersecting a given conic": two non-parallel planes are equal to lines passing through a fixed point and two intersecting lines, and two intersecting lines are a (degenerate) conic: https://en.m.wikipedia.org/wiki/Degenerate_conic
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EDIT
Following comments, here is a construction to relate a circular cone and two planes.
The idea is: as two intersecting lines are a degenerate conic, they can be parameterized in the same way as other conics.
We'll use as two lines in the projective plane $\mathbb P^2$ the principal axes: $x=0$ and $y=0$.
These are parameterized with $t \in [0, 2)$:
$\begin{cases} t \in [0, 1): x=\tan(\pi t), y = 0\\ t \in [1, 2): x=0, y = \tan(\pi(1-t)) \end{cases}$
For $t=\frac 1 2$ the point $(\pm \infty, 0)$ is the point at infinity that closes line $y=0$ in $\mathbb P^2$. And similarly for $t=\frac 3 2$ and line $x=0$.
Hence this parameterization of the two lines makes them into a closed curve, with one double point obtained for $t=0$ and $t=1$.
The right-angle turn at $(0,0)$ is because we want to connect the infinity branches that are on the same line, i.e. $(0, +\infty)$ with $(0, -\infty)$, and $(+\infty, 0)$ with $(-\infty, 0)$. For $2$ reasons:
Another option would be to connect $(+\infty, 0)$ to $(0, -\infty)$, and $(0, +\infty)$ to $(-\infty, 0)$. That would be possible if we close the plan $\mathbb R^2$ not with one point at infinity for each parallel set of lines, like in $\mathbb P^2$, but by only one point at infinity, which would topologically be a sphere $S^2$. The two lines would be a closed curve, but with two crossings, and conics do not have crossings, so this looks less natural.
Then we can take a circle $x^2+y^2=1$ centered on the origin, and parameterize it on $t \in [0,2)$ with $x = \cos (\pi t), y = \sin (\pi t)$.
This allows a bijection between the parameterization of the two lines, and the parameterization of the circle. This is not a bijection between the two lines and the circle, as the two lines have a double point at the origin, which correspond to two opposite points $(0, 1)$ and $(0, -1)$ on the circle.
This relation can now be made a relation between a cone and two planes, by adding a third dimension $z \in (-\infty, +\infty)$ which acts as a second parameter:
Two planes $y=0$ and $x=0$:
$\begin{cases} t \in [0, 1): x=z\tan(\pi t), y = 0\\ t \in [1, 2): x=0, y = z\tan(\pi(1-t)) \end{cases}$
The cone $x^2+y^2=z^2$:
$x = z\cos(\pi t), y = z\sin(\pi t)$
However it would be more fun to have a continuous mapping between the two lines and the circle, using a third parameter, with intermediate values being ellipses and hyperbolas, so I am now working on that.