conics
Conics are the curves or surfaces that arise from taking sections of a cone. Algebraically, they are second degree equations in two variables. Conics were studied and revered by the ancient Greeks, and were written about extensively by both Euclid and Appolonius. They remain important today, partly for their many and diverse applications.
Although to most people the word “cone” conjures up an image of a solid figure with a round base and a pointed top, to a mathematician a cone is a surface, one that is obtained in a particular way.
Imagine a vertical line, and a second line intersecting it at some angle \(\varphi\). We will call the vertical line the axis, and the second line the generator. The angle \(\varphi\) between them is called the vertex angle. Now imagine grasping the axis between thumb and forefinger on either side of its point of intersection with the generator, and twirling it. The generator will sweep out a surface, as shown in the diagram. It is this surface that we call a cone.
Notice that a cone has an upper half and a lower half (called the nappes), and that these are joined at a single point, called the vertex. Notice also that the nappes extend indefinitely far both upwards and downwards. A cone is thus completely determined by its vertex angle.
Now in intersecting a flat plane with a cone we have three choices, depending on the angle the plane makes to the vertical axis of the cone. First, we may choose our plane to have a greater angle to the vertical than does the generator of the cone, in which case the plane must cut right through one of the nappes. This results in a closed curve called an ellipse. Second, our plane may have exactly the same angle to the vertical axis as the generator of the cone, so that it is parallel to the side of the cone. The resulting open curve is called a parabola. Finally, the plane may have a smaller angle to the vertical axis (that is, the plane is steeper than the generator), in which case the plane will cut both nappes of the cone. The resulting curve is called a hyperbola, and has two disjoint “branches.”
Notice that if the plane is actually perpendicular to the axis (that is, it is horizontal) then we get a circle—showing that a circle is really a special kind of ellipse. Also, if the intersecting plane passes through the vertex then we get the so-called degenerate conics: a single point in the case of an ellipse, a line in the case of a parabola, and two intersecting lines in the case of a hyperbola.
Although intuitively and visually appealing, these definitions for the conic sections tell us little about their properties and uses. Consequently, one should master their “plane geometry” definitions as well. It is from these definitions that their algebraic representations may be derived, as well as their many important properties,such as the reflection properties. (That the definitions which follow are equivalent to those given above is not obvious – not at all! For an elegant proof, see the article on Dandelin’s Spheres.)
We will now look at each conic section in detail.
The Ellipse
An ellipse is the set of all points in the plane, the sum of whose distances from two fixed points, called the foci, is a constant. (“Foci” is the plural of “focus”, and is pronounced FOH-sigh.) Sometimes this definition is given in terms of “a locus of points” or even “the locus of a point” satisfying this condition—it all means the same thing.
For reasons that will become apparent, we will denote the sum of these distances by \(2a\).
We see from the definition that an ellipse has two axes of symmetry, the larger of which we call the major axis and the smaller the minor axis. The two points at the ends of the ellipse (on the major axis) are called the vertices. It happens that the length of the major axis is \(2a\), the sum of the distances from any point on the ellipse to its foci. If we call the length of the minor axis \(2b\) and the distance between the foci \(2c\), then the Pythagorean Theorem yields the relationship \(b^2 + c^2 = a^2\):
By imposing coordinate axes in this convenient manner, we see that the vertices are at the \(x\) intercepts, at \(a\) and \(-a\), and that the \(y\)-intercepts are at \(b\) and \(-b\). Let the variable point \(P\) on the ellipse be given the coordinates \((x, y)\). We may then apply the distance formula for the distances from \(P\) to \(F_1\) and from \(P\) to \(F_2\) to express our geometrical definition of the ellipse in the language of algebra:
\[\begin{eqnarray*} 2a & = & \overline{PF_1}+\overline{PF_2} \\ & & \\ & = & \sqrt{(x+c)^2+y^2}+\sqrt{(x-c)^2+y^2} \end{eqnarray*}\]
Substituting \(a^2-b^2\) for \(c^2\) and using a little algebra, we can then derive the standard equation for an ellipse centered at the origin,
\[\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\]
where \(a\) and \(b\) are the lengths of the semimajor and semiminor axes, respectively. (If the major axis of the ellipse is vertical, exchange \(a\) and \(b\) in the equation.) The points \((a, 0)\) and \((-a, 0)\) are called the vertices of the ellipse. If the ellipse is translated up/down or left/right, so that its center is at \((h, k)\), then the equation takes the form
\[\displaystyle\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\]
If \(a=b\) we have the special case of an ellipse whose foci coincide at the center—that is, a circle of radius \(a\).
The ellipse has the following remarkable reflection property. Let \(P\) be any point on the ellipse, and construct the line segments joining \(P\) to the foci. Then these lines make equal angles to the tangent line at \(P\).
Consequently, any ray emanating from one focus will always reflect off of the inside of the ellipse in such a way as to go straight to the other focus. Architects have exploited this property in many famous buildings. The “whisper chamber” in the United States Capitol is one; stand at one focus and whisper, and anyone at the other focus can hear you with perfect clarity, even though they are much too far away from you to hear a whisper normally. The Mormon Tabernacle in Salt Lake City was also designed as an ellipse (indeed, it is the top half of an ellipsoid), to provide a perfect acoustical environment for choral and organ music.
Ellipses occur in nature as well, and are critical to understanding the motion of planets and other bodies moving in space. See the article on Kepler’s Laws.
The Parabola
A parabola is the set of all points in the plane whose distances from a fixed point, called the focus, and a fixed line, called the directrix, are always equal.
The point directly between—and hence closest to—the focus and the directrix is called the vertex of the parabola.
To derive the equation of a parabola in rectangular coordinates, we again choose a convenient location for the axes, placing the origin at the vertex so that the \(y\)-axis is the axis of symmetry. We denote the distance from the vertex to the focus by \(p\), so that the directrix is then the line \(y = -p\).
Using the distance formula for the distance from the point \((x,y)\) to the focus at \((0,p)\), and noting that the distance from \((x,y)\) to the directrix is evidently \(y + p\), and setting these distances equal, we obtain,
\[\sqrt{x^2+(y-p)^2}=y+p\]
A direct application of ordinary algebra reduces this to,
\[x^2=4py\]
This then is the equation of a parabola opening upwards, with its vertex at the origin. If we introduce a negative sign, we get a parabola opening downwards. If we interchange the roles of \(x\) and \(y\), we get a parabola opening to the right (or to the left if there is a negative). We may translate the parabola up/down or back/forth, putting the vertex at the point \((h, k)\) if we write our equation as
\[(x-h)^2=4p(y-k)^2\]
The reflection property of parabolas is very important because it has so many practical uses. Construct a line segment joining any point on the parabola to the focus, and then also a ray emanating from the point that is parallel to the axis of symmetry (the \(y\) axis if its vertex is on the origin). The line segment and ray will always make equal angles to the tangent line to the parabola at their common point. Consequently, any ray emanating from the focus will reflect off of the parabola so as to point directly outwards, parallel to the axis. This property is made use of in the design of flashlights, headlights, and spotlights, for instance. Conversely, any ray entering the parabola that is parallel to the axis will be reflected to the focus. This property is exploited in the design of radio and satellite receiving dishes, and solar collectors.
The reflective property of parabolas can also be derived from the curious fact that the tangents to a parabola at the end-points of any chord that passes through the focus always meet on the directrix, and always at a right angle.
Parabolas also describe the movement of a body under constant accelaration, such as by the force of gravity. The common usage of the term ballistics derives from the fact that a bullet, once it leaves the gun, is acted on only by gravity.
The Hyperbola
A hyperbola is the set of points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.
Mimicking our procedure with ellipses, we will choose the constant \(2a\) to represent the difference of these distances, that is, \(\overline{PF}_1 – \overline{PF}_2 = 2a\). We will call the two points of the hyperbola which lie on the shortest line connecting the foci the vertices, and we then see that the distance between the vertices must be \(2a\). Also, we will call the distance between the foci \(2c\). Finally, we will define the constant \(b\) by \(b^2 = c^2 – a^2\). (We may do this since evidently \(c > a\).) Placing coordinate axes at the center as before, we obtain this picture:
Applying the distance formulas and substituting for \(c\) as we did in the previous cases, we can derive the standard formula of a hyperbola:
\[\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\]
We note that solving this equation for \(y\) yields
\[y=\pm \frac{b}{a}\sqrt{x^2-a^2}\]
and letting \(x\) become arbitrarily large causes this expression to become arbitrarily close to
\[y=\pm \frac{b}{a}x\]
Thus we see that the crisscrossing dotted lines in the diagram above are asymptotes for the hyperbola, that is, the curve becomes indefinitely close to these lines as the absolute value of \(x\) grows without bound.
As before, if the principal axis of the hyperbola is vertical instead of horizontal, we switch the roles of \(a\) and \(b\). We may also translate the hyperbola up/down and back/forth, placing the center at \((h, k)\) by modifying our equation thusly:
\[\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\]
The reflection property of the hyperbola is of great importance in optics. The line segments joining any point on the hyperbola to each of the foci form an angle which is bisected by the tangent line at that point.
Consequently, any ray approaching one of the foci from a convex side of the hyperbola is reflected to the opposite focus. An example of an application of this principle is the Cassegrain reflecting telescope:
A concave parabolic mirror forms the back of the telescope, and this shares a focus with a convex hyperbolic mirror, the other focus of which is at the eyepiece.
Eccentricity
The unifying idea among these curves is that they are all conics, that is, conic sections. We have seen the geometric realization of this unifying notion, but how can it be expressed algebraically? The key notion is that of eccentricity.
To define the eccentricity of a conic, we must first observe a feature of the ellipse and the hyperbola that we neglected before, namely, that each of these curves has a directrix, just as the parabola does. Indeed, the ellilpse and hyperbola each have two directrices. Now let \(P\) be a point on the conic curve, and consider its distance to a focus, and its distance to the corresponding directrix. The curve’s eccentricity is the ratio of these distances.
\[e=\frac{\overline{PF}_1}{\overline{PD}_1} = \frac{\overline{PF}_2}{\overline{PD}_2}\]
We will denote the eccentricity by the letter \(e\). It can be shown geometrically that \(e\) is always equal to the ratio of \(c\) and \(a\) as these constants were defined in each case. That is, we always have \(e = \displaystyle\frac{c}{a}\). It can also be shown that the directrices of an ellipse or hyperbola with principle axes horizontal are always the vertical lines given by
\[x=\pm\frac{a}{e}\]
Now recall that in a parabola the distance from a point to the focus, and from the same point to the directrix, are always the same. Consequently, a parabola always has eccentricity \(e = 1\). An ellipse, on the other hand, always has \(e 1\). (A circle is the special case of an ellipse with e = 0.) In summary, we have,
\[\begin{array}{rcl} e 1 & \longleftrightarrow & \mbox{hyperbola} \\ \end{array}\]
The names of these curves are related to their eccentricities. Ellipse comes from a Greek word meaning “deficiency” or “something left out,” and is related to the English words ellipsis and elliptical. The word hyperbola, on the other hand, comes from the Greek word for “excess,” and is related to the English word hyperbole. Finally, parabola means something like “just right,” and is related to the words compare and parable.
What this discussion shows is that we may consider that there is only one general kind of curve, called a conic, with special cases called ellipse, parabola, and hyperbola depending on the conic’s eccentricity. Algebraically, we may now consider conics in complete generality. To do so, consider a second degree polynomial in two variables, x and y.
\[p(x)=Ax^2+Bxy+Cy^2+Dx+Ey+F\]
The \(xy\) term can be eliminated by a rotation of axes, and the algebraic techniques for doing so can be found in any text on calculus with analytic geometry. By then completing the square with respect to both \(x\) and \(y\) one will obtain one of the standard equations given above, for either an ellipse or a hyperbola. If only one of \(x\) and \(y\) appears as a square in the original conic equation, then the standard equation of a parabola will be obtained.
The study of conic sections is one of the most beautiful topics in classical mathematics. Every student of mathematics should take the time to master conic sections thoroughly, not only for the esthetic appeal of the subject, and not only because their applications are so varied and important, but also because they show—in a deep and clear way—the fundamental unification of geometry and algebra in the field of analytic geometry.
