Conic sections

latexpage] A conic section is a plane curve that results from intersecting the surface of a double circular cone with a plane (cf. Figure 1).

The double circular cone is formed by rotating a straight line $g$ around an intersecting axis $a$. The surface of the cone then consists of the totality of all straight lines (the so-called surface lines), which result from the rotation of $g$ around $a$. The position of the intersection to the lateral surfaces determines which conic section is created.

Figure 1: The creation of a conic section

If the apex of the double cone does not lie in the respective section plane, the following curves can arise:

A parabola is created when the section plane is parallel to exactly one generatrix of the double cone. This means that the angle between the axis $a$ and the intersection plane is equal to half the opening angle of the double cone.

An ellipse occurs when the section plane is not parallel to any generatrix. This means that the angle between the $a$ axis and the section plane is greater than half the opening angle of the double cone. If this angle is a right angle, the circle appears as an intersection curve (as a special case of an ellipse).

A hyperbola occurs when the intersection plane is parallel to two generatrix lines of the double cone. This means that the angle between axis and plane is smaller than half the opening angle.

If you use a simple cone instead of a double cone and intersect it with a plane so that the plane does not pass through its apex, you get either a parabola, an ellipse or a branch of a hyperbola, analogous to the three cases just mentioned. Of course, the intersection of such a plane with the cone can also be empty.

In MATHEMATICS ADVENTURE LAND the conic sections just mentioned are generated by the following experiment:

In a transparent cone-shaped container, whose main axis can be tilted (by hand) up to 90°, there is a blue colored liquid. If one now sets an arbitrary angle, then the boundary line of the liquid in this container forms a conic section.

Figure 2: Apparatus in the Mathematics adventure land

And now … the mathematics of it:

We now consider the algebraic aspect of conic sections. In the plane Cartesian coordinate system, the general quadratic equations $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$

(with real coefficients $a,b,c,d,e$ and $f$) exactly describe the conic sections as the zero sets of such equations.

Such an equation can also be written in matrix notation as follows: $$\begin{pmatrix} x & y & 1\end{pmatrix}\begin{pmatrix} a & b & d\ b & c & e\ d & e & f\end{pmatrix}\begin{pmatrix} x\ y\ 1\end{pmatrix}=0\quad (\ast).$$

We write $B$ for the matrix $$\begin{pmatrix} a & b\ b & c\end{pmatrix}.$$

We now want to transform the system $(\ast)$ in such a way that one can immediately read the type of the described conic section. To do this, we allow ourselves to transform the coordinates $x,y$ of the plane by an orientation-preserving Euclidean motion $$\begin{pmatrix} x\ y\end{pmatrix}\mapsto O\begin{pmatrix} x\ y\end{pmatrix} + w$$

(here $O$ is a real orthogonal matrix with determinant 1 and $w\in\mathbb R^2$ is any vector). This rotates and shifts the conic section, but does not change its shape.

First, let’s consider the matrix $B$. Since it is a symmetric matrix ($B=B^\top$, i.e. $B$ merges into itself when mirrored on the main diagonal), there is — by a well-known theorem from linear algebra — a real orthogonal matrix $O$, so $O^\top BO$ is a diagonal matrix whose eigenvalues are on the main diagonal. By possibly swapping the columns of $O$, we can make it so that $\det(O)=1$. By the transition $$\begin{pmatrix} x\ y\end{pmatrix}\mapsto O\begin{pmatrix} x\ y\end{pmatrix}$$

we thus obtain a quadratic equation of the form $(\ast)$ where the mixed term $2bxy$ vanishes (since $b=0$).

Now we consider the following cases:

Case 1: $a\neq 0$ and $c\neq 0$: Then we can perform quadratic completion for both variables $x$ and $y$. This makes the terms $2dx$ and $2ey$ disappear. So we get a new equation of the form $ax^2+cy^2+f=0$. Now we have to distinguish two cases again:

(a) If $f\neq 0$ holds, we can transform the equation so that it is of the form $ax^2+cy^2=1$. Now, if $a,c\gt 0$ holds, then it is an ellipse with semi-axes $r_1=1/\sqrt{a}$ and $r_2=1/\sqrt{c}$. If exactly one of the parameters $a$ and $c$ is negative, we can assume by permutation that $a\gt 0$ and $c\lt 0$. Then it is a hyperbola with semiaxes $r_1=1/\sqrt{a}$ and $r_2=1/\sqrt{-c}$. But if $a,c\lt 0$ holds, we see immediately that the described conic section is the empty set.

(b) Now let $f=0$ hold (this corresponds to the case where the section plane intersects the double cone at its apex). If $a$ and $c$ have the same sign, it is easy to see that the conic section degenerates to exactly one point (namely $(0,0)$). If they have opposite sign then two straight lines of slope $\pm\sqrt{\lvert a/c\rvert}$ through the origin are generated.

Case 2: $ac=0$: If both parameters vanish, then we have an equation of at most 1st degree. So this describes either a straight line, the empty set, or the entire plane. Therefore we may assume that $a\neq 0$ and $c=0$ (except for permutation). By quadratic completion we may assume that $d=0$. Now we have to distinguish two cases again:

(a) If $e\neq 0$ holds, we can shift in $y$-direction so that we get an equation of the form $ax^2+ey=0$. This describes a parabola.

(b) But if $e=0$ is valid, the $y$ variable does not appear in our equation at all. We therefore obtain two (possibly identical) straight lines parallel to the $y$-axis as degenerate conic sections.

This finishes the classification of 2-dimensional conic sections.


Applications and examples

Figure 3: Conic sections (hyperbolas) as an architectural element: Brasilia Cathedral (Oscar Niemeyer, 1970)

One application of conic sections is in astronomy, since the orbits of celestial bodies are approximated conic sections. They are also used in optics — as a rotational ellipsoid for car headlights, as a paraboloid or hyperboloid for reflecting telescopes, etc.


Historical

The mathematician Menaichmos (c. 380–320 B.C.) studied conic sections at Plato’s Academy with the help of a cone model. He discovered that the so-called Delic problem (“cube doubling”) can be traced back to the determination of the intersection points of two conic sections. Euclid described in the 3rd century B.C. in four (so far not found again) volumes of his “Elements” the properties of conic sections. The entire knowledge of the mathematicians of antiquity about conic sections was summarized by Appolonios of Perge (around 262–190 B.C.) in his eight-volume work “Konika” (German: “Über Kegelschnitte”). The analytical description of the totality of conic sections by equations of the type $(\ast)$ was found by Pierre de Fermat (1607–1665) and René Descartes (1596–1650).


Literature

Koecher, M. u. Krieg, A.: Ebene Geometrie, 3. Auflage, Berlin, 2007.