Understanding Conic Sections: Exploring the Plane-Cone Intersection

[Greg Bernhardt]

Definition/Summary

A conic section (or conic) is any curve which results from a plane slicing through an upright circular cone.

If the slope of the plane is zero, it cuts only one half of the cone, and the conic is a circle (or a point, if the plane goes through the apex of the cone).

If the slope of the plane is less than the slope of a generator of the cone (the tangent of the half-angle of the cone), it cuts only one half of the cone, and the conic is an ellipse (or, again, a point).

If the slope of the plane is equal to the slope of a generator, it cuts only one half of the cone, and the conic is a parabola (or a straight line, if the plane goes through the apex of the cone).

If the slope of the plane is greater than the slope of a generator, it cuts both halves of the cone, and the conic is a hyperbola (or a pair of crossed straight lines, if the plane goes through the apex of the cone).

Equations

A circle of radius a:

$$x^2\,+\,y^2\,=\,a^2$$

An ellipse of major axis 2a minor axis 2b and eccentricity $\sqrt{1\,-\,b^2/a^2}$:

$$\frac{x^2}{a^2}\,+\,\frac{y^2}{b^2}\,=\,1$$

A parabola whose focus is 2a from its directrix:

$$y^2\,=\,4ax$$

A hyperbola of major axis 2a and eccentricity $\sqrt{1\,+\,b^2/a^2}$:

$$\frac{x^2}{a^2}\,-\,\frac{y^2}{b^2}\,=\,1$$

Extended explanation

Dandelin spheres:

A Dandelin sphere of a conic section is a sphere of maximum size which fits between the cone and the plane.

So it touches the original plane in a point, known as a focus (plural: foci).

And it touches the cone in a "horizontal" circle, and the plane of that circle intersects the original plane (unless the original plane is also "horizontal") in a line, known as a directrix.

An ellipse (other than a circle) or hyperbola has two Dandelin spheres, and therefore two foci, each with its own directrix.

A parabola has only one Dandelin sphere, and therefore only one focus and one directrix.

A circle has two Dandelin spheres, but they touch the plane (and each other) at the same point, and therefore a circle also has only one focus (or: a circle's two foci are coincident).

Moreover, that plane is "horizontal", and therefore cannot meet the "horizontal" planes through the centre of the spheres, and therefore a circle has no directrix.

Alternatively: the planes meet "at infinity", and therefore a circle has infinitely many directrices, all "at infinity", or one circular directrix (or two coincident circular directrices) "at infinity".

Cylindrical sections:

Using Dandelin spheres, it is easy to prove that a curve which results from a plane slicing through an upright circular cylinder is an ellipse or a pair of parallel lines.

Eccentricity:

Using the theorem that the length of two tangents to the same sphere from the same point are equal, it is easy to prove that the following ratio is a constant for all points on the conic:

the distance from that point to a focus divided by the distance from that point to the accompanying directrix.​

This ratio is the eccentricity of the conic.

"String definition":

Using the same theorem, it is easy to prove that the sum of the distances from any point on an ellipse to both foci is a constant.

This justifies the "string construction" of an ellipse: pin two ends of a string to two points on a flat surface, and with a pencil held tight against the string, draw a curve round the two points: the result is an ellipse whose foci are those two points.

Similary, it is easy to prove that the difference of the distances from any point on a hyperbola to both foci is a constant.

Focussing:

Light from one focus will be reflected (if the conic is considered to be a curved mirror), from anywhere on the conic, so as to converge on the other focus, forming a real image there (for a circle ellipse or parabola), or to diverge as if it had come from the other focus, forming a virtual image there (for a hyperbola).

Both foci of a circle are in the same place.

One focus of a parabola is "at infinity" (so light "from infinity" is focussed onto the other focus, and vice versa).

For this reason, most reflecting telescopes have mirrors which are paraboloids (the rotation of a parabola about its axis).

The word "focus" comes from the Latin for "hearth".

Coordinate geometry:

In Cartesian x and y coordinates, the equation of a conic is a quadratic in x and y.

Orbits and trajectories:

The trajectory of a small body moving near a large gravitational body (in Newtonian mechanics) is a conic.

This is because [can someone else please provide a simple explanation, not involving coordinates? ]

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[fresh_42]

The Greek mathematician Menaechmus studied the conic sections at Plato's academy with the help of a cone model. He found out that the Delian problem can be traced back to determining the point of intersection of two conic sections. Euclid wrote four books on conic sections, but they are not preserved. The complete knowledge of the conic sections of the ancient mathematicians was summarized by Apollonius von Perge in his eight-volume work Konika. The description of conic sections through coordinate equations was introduced by Fermat and Descartes.