Conic Sections

A conic section is the plane curve formed by the intersection of a plane and a right-circular, two-napped cone. Such a cone is shown in Figure 1.

The cone is the surface formed by all the lines passing through a circle and a point. The point must lie on a line, called the "axis," which is perpendicular to the plane of the circle at the circle's center. The point is called the "vertex," and each line on the cone is called a "generatrix." The two parts of the cone lying on either side of the vertex are called "nappes." When the intersecting plane is perpendicular to the axis, the conic section is a circle (Figure 2).

Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Figure 3. Illustration by Hans & Cassidy. Courtesy of Gale Group.

When the intersecting plane is tilted and cuts completely across one of the nappes, the section is an oval called an ellipse (Figure 3).

When the intersecting plane is parallel to one of the generatrices, it cuts only one nappe. The section is an open curve called a parabola (Figure 4).

When the intersecting plane cuts both nappes, the section is a hyperbola, a curve with two parts, called "branches" (Figure 5).

All these sections are curved. If the intersecting plane passes through the vertex, however, the section will be a single point, a single line, of a pair of crossed lines. Such sections are of minor importance and are known as "degenerate" conic sections.

Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Figure 4. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Figure 5. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Figure 6. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Figure 8. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Since ancient times, mathematicians have known that conic sections can be defined in ways that have no obvious connection with conic sections. One set of ways is the following:

Ellipse: The set of points P such that PF₁ + PF₂ equals a constant and F₁ and F₂ are fixed points called the "foci" (Figure 6).

Parabola: The set of points P such that PD = PF, where F is a fixed point called the "focus" and D is the foot of the perpendicular from P to a fixed line called the "directrix" (Figure 7).

Hyperbola: The set of points P such that PF₁ -PF₂ equals a constant and F₁ and F₂ are fixed points called the "foci" (Figure 8).

If P, F, and D are shown as in Figure 7, then the set of points P satisfying the equation PF/PD = e where e is a constant, is a conic section. If 0 < e < 1, then the section is an ellipse. If e = 1, then the section is a parabola. If e > 1, then the section is a hyperbola. The constant e is called the "eccentricity" of the conic section.

Because the ratio PF/PD is not changed by a change in the scale used to measure PF and PD, all conic sections having the same eccentricity are geometrically similar.

Figure 7. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Figure 9. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Conic sections can also be defined analytically, that is, as points (x,y) which satisfy a suitable equation.

An interesting way to accomplish this is to start with a suitably placed cone in coordinate space. A cone with its vertex at the origin and with its axis coinciding with the z-axis has the equation x² + y² -kz² = 0. The equation of a plane in space is ax + by + cz + d = 0. If one uses substitution to eliminate z from these equations, and combines like terms, the result is an equation of the form Ax²+ Bxy + Cy² + Dx + Ey + F = 0 where at least one of the coefficients A, B, and C will be different from zero.

For example if the cone x² + y² -z² = 0 is cut by the plane y + z - 2 = 0, the points common to both must satisfy the equation x² + 4y - 4 = 0, which can be simplified by a translation of axes to x² + 4y = 0. Because, in this example, the plane is parallel to one of the generatrices of the cone, the section is a parabola (Figure 9).

Figure 11. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Figure 10. Illustration by Hans & Cassidy. Courtesy of Gale Group.

One can follow this procedure with other intersecting planes. The plane z - 5 = 0 produces the circle x² + y² - 25 = 0. The planes y + 2z - 2 = 0 and 2y + z - 2 = 0 produce the ellipse 12x² + 9y² - 16 = 0 and the hyperbola 3x² - 9y² + 4 = 0 respectively (after a simplifying translation of the axes). These planes, looking down the x-axis are shown in Figure 10.

As these examples illustrate, suitably placed conic sections have equations which can be put into the following forms:

The equations above are "suitably placed." When the equation is not in one of the forms above, it can be hard to tell exactly what kind of conic section the equation represents. There is a simple test, however, which can do this. With the equation written Ax² + Bxy + Cy² + Dx + Ey + F = 0, the discriminant B² - 4AC will identify which conic section it is. If the discriminant is positive, the section is a hyperbola; if it is negative, the section is an ellipse; if it is zero, the section is a parabola. The discriminant will not distinguish between a proper conic section and a degenerate one such as x² - y² = 0; it will not distinguish between an equation that has real roots and one, such as x² + y² + 1 = 0, that does not.

Students who are familiar with the quadratic formula

will recognize the discriminant, and with good reason. It has to do with finding the points where the conic section crosses the line at infinity. If the discriminant is negative, there will be no solution, which is consistent with the fact that both circles and ellipses lie entirely within the finite part of the plane. Parabolas lead to a single root and are tangent to the line at infinity. Hyperbolas lead to two roots and cross it in two places.

Conic sections can also be described with polar coordinates. To do this most easily, one uses the focus-directrix definitions, placing the focus at the origin and the directrix at x = -k (in rectangular coordinates). Then the polar equation is r = Ke/(1 - e cos θ) where e is the eccentricity (Figure 11).

The eccentricity in this equation is numerically equal to the eccentricity given by another ratio: the ratio CF/CV, where CF represents the distance from the geometric center of the conic section to the focus and CV the distance from the center to the vertex. In the case of a circle, the center and the foci are one and the same point; so CF and the eccentricity are both zero. In the case of the ellipse, the vertices are end points of the major axis, hence are farther from the center than the foci. CV is therefore bigger than CF, and the eccentricity is less than 1. In the case of the hyperbola, the vertices lie on the transverse axis, between the foci, hence the eccentricity is greater than 1. In the case of the parabola, the "center" is infinitely far from both the focus and the vertex; so (for those who have a good imagination) the ratio CF/CV is 1.