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David Terr
Ph.D. Math, UC Berkeley

9.1. The Ellipse

In this chapter, we describe three very important types of curves, known as conic sections. These include circles, ellipses, parabolas, and hyperbolas. (Circles are sometimes classified as ellipses.) Conic sections are aptly named, because they are the intersections of a cone and a plane. Actually, seven types of intersections of a cone and a plane are possible, as shown below. Besides the four curves mentioned above, there are three degenrate cases, which include a single point (the intersection of the plane with the vertex of the cone), a line (the intersection of the plane with the cone in the case in which the plane is tangent to the cone), and two intersecting lines (the intersection of the plane with the cone in the case in which the plane passes through the vertex of the cone and also slices the cone. Note that by "cone", we really mean both halves of the cone, which meet at the vertex, which otherwise might be regarded as two cones.

Figure 9.1.1: The Four Conic Sections

(Images taken from www.math2.org)

Figure 9.1.2: The Three Degenerate Conic Sections

(Images taken from www.math2.org)

In this section, we discuss the ellipse, corresponding to the second diagram in Figure 9.1.1. The ellipse is the only one of the three conic sections (not including the circle) which is closed. An ellipse is usually defined as a curve in the plane which is the locus of all points P for which the sum of the distances from P to two fixed points F1 and F2 in the plane, is constant. Mathematically, this amounts to the following formula:

• (9.1.3) d(P, F1) + d(P, F2) = 2a

for some positive constant a. The fixed points F1 and F2 are known as foci (singular focus) of the ellipse. A diagram of an ellipse, along with its foci F1 and F2, is shown below.

Figure 9.1.4: An Ellipse

Also shown in the diargam are the major and minor axes of the ellipse. The major axis is the line segment going between points A and B shown above. Specifically, it is the line segment passing through each focus and starting and ending at the intersection points of the ellipse with the line passing through the foci. The length of the major axis, usually also just called the major axis, is clearly 2a, as can be seen if we let P coincide with A or B. The minor axis of the ellipse is the line segment perpendicular to the major axis, passing through the center of the ellipse, and starting and ending at the points C and D of the ellipse. The length of the minor axis, usually just called the minor axis, is denoted above as 2b. The lengths a and b are called the semi-major axis and semi-minor axis respectively.

There is an important dimensionless quantity associated with an ellipse, known as its eccentricity, and denoted above as e. The eccentricity of an ellipse is defined as the ratio of the distance between the foci and the length of the major axis. Thus, the distance between the foci is 2ea and the distance from a focus to the center of the ellipse is ea. Note that by this definition, the eccentricity is a positive number less than 1. (A circle may be thought of as an ellipse with eccentricity 0 with both its foci at the center.)

One can calculate e in terms of a and b as follows: Consider the point C in the diagram above. By the Pythagorean Theorem, we see that the distance from C to either focus is given by

d(C, F1) = d(C, F2) = √(ea)2 + b2.

But we also know that this distance is equal to a, since the sum of the distances is 2a. Therefore, we have the equation

e2a2 + b2 = a2,

whence

• (9.1.5) b = a√(1-e2).

Thus, if we know a and e, we can solve for b. We can also solve for e if we know a and b as follows:

• (9.1.6) e = √a2 - b2 / a.

Equation (9.1.6) easily follows from (9.1.5), or even more easily from the equation above it.

Example 1: Compute the semi-minor axis of an ellipse with semi-major axis 5 and eccentricity 0.6.

Solution: From (9.1.5) we find b = 5√(1-0.62) = 5√(1-0.36) = 5√0.64 = (5)(0.8) = 4.

Example 2: Compute the eccentricity of an ellipse with semi-major axis 5 and semi-minor axis 3.

Solution: From (9.1.6) we find e = √52-32/5 = √25-9/5 = √16/5 = 4/5 = 0.8.

Example 3: It is well-known that the orbit of a planet around the sun is an ellipse with the sun at one of the foci. The maximum distance from the earth to the sun along its orbit is 1.521 * 1011 m and the minimum distance is 1.469 * 1011 m. Compute the eccentricity of the orbit of the earth.

Solution: From Figure 9.1.4, we see that the maximum distance is given by dmax = a + ea = (1+e)a and the minimum distance is given by dmin = a - ea = (1-e)a. The sum of these distances dmax+ dmin = 2a and the difference is given by dmax- dmin = 2ea. Dividing the second equation by the first, we find

e = (dmax- dmin) / (dmax+ dmin) = (1.521 - 1.469) / (1.521 + 1.469) = 0.052 / 2.990 = 0.017.

There is a nice equation for a graph of an ellipse in the plane. The equation for an ellipse with semi-major axis a along the x-axis, semi-minor axis b along the y-axis, and centered at the origin, is as follows:

• (9.1.7) x2 / a2 + y2 / b2= 1.

Example 4: Graph the ellipse with semi-major axis 5 along the x-axis, semi-minor axis 4 along the y-axis, and centered at the origin.

Solution: Note that we can use (9.1.7) to solve for y in terms of x, obtaining

y(x) = ±b√1-(x2/a2) = ±(b/a)√(a2-x2).

Plugging in a = 5 and b = 4, this equation becomes

y(x) = ±0.8√(25-x2).

Below we tabulate values of y(x) corresponding to simple values of x:

x y
-5.00
0.00
-4.00
±2.40
-3.00
±3.20
-2.00
±3.67
-1.00
±3.92
0.00
±4.00
1.00
±3.92
2.00
±3.67
3.00
±3.20
4.00
±2.40
5.00
0.00

From this table, we can plot a smooth graph, such as the one shown below:

Like the circle, it is easy to generalize Equation (9.1.7) to the case of ellipses not centered at the origin, but still with horizontal major axis and vertical minor axis. To derive the formula for such an ellipse centered at the point (x0, y0), we merely need to replace x with x - x0 and y with y - y0 in (9.1.7). The resulting formula is as follows:

• (9.1.8) (x - x0)2 / a2 + (y - y0)2 / b2 = 1.

Note that in the cases in which a = b = R, formulas (9.1.7) and (9.1.8) reduce to equations (1.4.1) and (1.4.2) respectively, which are the equations for the graph of a circle or radius R.

We should point out one more fascinating property of ellipses, namely their reflective property. Imagine a room with floor plan in the shape of an ellipse. If one person stands at one focus of the ellipse and another person stands at the other focus, the two people can whisper to each other and hear each other as clearly as if they were standing right next to one-another! (For this reason, such elliptical rooms are often called whispering galleries.) How is this possible? The secret comes from considering the two blue line segments in Figure 9.1.4. One property of the ellipse is that these line segments both make the same angle with the ellipse at the point X. Since it is a well-known property of waves, such as light or sound waves, that the angle of incidence is equal to the angle of reflection, this means that sound waves eminating from one focus will bounce off the wall and converge on the other focus.