The Conic Sections in Polar Coordinates
This activity is an interactive study of the polar form of the equation for a conic section. Readers should be familiar with polar coordinates and triangle trigonometry. This activity is also a vehicle for the introduction of the Geogebra, though no prior experience with the Geogebra is assumed.
You can also download a Sketchpad Version of this activity.
We begin by introducing the focus-directrix definition of the conic section. The directrix is a line. The focus is a point, usually chosen so that it does not lie on the directrix.
Let's begin with a focal point `F` at the origin of our coordinate system and a vertical line (called the directrix) k units from the origin, as shown in Figure 1.
Our task is to locate all points `P` such that the distance from `F` to `P` is a constant multiple of the distance from `P` to the directrix, the line `x=k`. In symbols, we want to locate all points `P` satisfying the relation
`FP = e PD`,
where `e` is a proportionality constant called the eccentricity of the conic section, and `D` is the point on the directrix closest to the point `P`. For example, in Figure 2, the ellipse shown is the set of all points `P` satisfying the relation `FP = 0.53 PD`.
In Figure 2, the eccentricity of the ellipse is `e = 0.53`, a number less than unity (`e<1`).; All ellipses have eccentricity less than one.
For purposes of discussion, we need to draw the auxiliary line segment `PB` and label some existing line segments, angles, and points. This we do in Figure 3.
Let `r` and `theta` represent the segment `FP` and the angle `BFP`, respectively. Because triangle `BFP` is a right triangle,
`FB = r cos theta`.
Next, segment `PD` has length equal to the difference of segment `FC` and `FB`; i.e.,
`PD = FC - FB`.
`PD = k - r cos theta`.
Thus, `FP = e PD` becomes
`r = e (k - r cos theta)`.
Solving this last equation for `r`, we arrive at the equation of the conic section in polar form; i.e.,
`r = (ek)/(1 + e cos theta)`.
Geogebra is an open source clone (plus some of its own fine bells and whistles) of the Geometer's Sketchpad. Geogebra can be downloaded at the following URL:
Alternately, providing you have the proper Java Runtime installed, you can start up Geogebra directly from the above website by clicking on the "Start Geogebra" link. In a sense, this is a better starting technique as you will always have the latest update of the software.
When you start Geogebra, you obtain a window similar to that in Figure 4.
The menus across the top of the Geogebra window contain extensive commands for creating dynamic, geometrical objects. We begin by selecting View->Axes from the menu system, which toggles on and off the axes (Note: We use the notation View->Axes to indicate to our readers to first select the main menu item View, followed by the sub-menu item Axes.) Secondly, we use the mouse to drag the separation bar between the algebra window and sketch window a bit to the left. The result is shown in Figure 5.
Geogebra's toolbar contains some of the most commonly used construction tools. Each has a little down-arrow in the lower-right corner of the tool icon. When you click this down-arrow, a drop-down list expands, showing a variety of extra tools. Before continuing with this activity, take some time to play with the tools and the menu items, enough to familiarize yourself with the Geogebr environment.
The Basic Sketch
Select the Segment between two points tool and create a line segment, as shown in Figure 6. Use the New Point tool to construct a point on the segment, near the position `C` indicated in Figure 6. Use the Midpoint or Center tool, then select the segment to create the midpoint of the segment at `D`. At this point, your sketch should resemble that in Figure 6.
We are now going to change some of the labels in Figure 6. Select the Move-tool from Geogebra's toolbar. Click and drag each label in the sketch into a more favorable position. Next, double-click the label `A` with the Move tool. Change the label in the resulting dialog box to the letter `F`, because this is the point we will use as the focus of our conic section. Similarly, change the letter `D` to the letter `M`, a much more satisfactory letter to represent the midpoint of the segment `FB`. Your sketch should closely resemble that of Figure 7.
Select the Circle with center through point tool from the toolbar. Single-click the focus at `F` (you must let up the mouse after the single-click --- don't hold it down), then move the mouse and click in the vicinity of point `A` in Figure 8. You can resize the circle later by dragging the point `A`. Select the New Point tool from the toolbar and construct a point on the circle. Select the Move tool (the selection arrow), double-click the label of the newly created point on the circle, and change the label to the letter Q, as shown in Figure 8.
Select the Ray through two points tool. Click the point `F`, then the point `Q` to construct a ray at F that emanates outward through the point Q, as shown in Figure 9. You may need to adjust the position of the label Q with the Move tool.
Select the Perpendicular line tool, click the point `B`, then click the segment `FB`. This should create a line through `B`, perpendicular to the segment `FB`, as shown in Figure 10.
Finding the Eccentricity
The point `C` in Figure 10 is a point on the conic section. At least, that's our plan. Consequently, `FC = e CB`; i.e., the ratio `FC//CB` equals `e`, the eccentricity of the conic section. We need to calculate this ratio, but first we must measure the segments `FC` and `CB`. Select the Distance or length tool (on the fourth icon from the right end of the toolbar). Click the points `F` and `C`, which will calculate the distance between the points `F` and `C`. In a similar manner, calculate the distance between the points `C` and `B`. When this is accomplished, the measures of the segments `FC` and `CB` will appear in the Algebra window as distanceFC and distanceCB, respectively, similar to that shown in Figure 11. Note: The actual numbers may differ, due to your personal configuration of Geogebra. Don't worry about numbers; everyone will have different numbers at this point. Continue to the next step.
Computing the Eccentricity
We need to computer the eccentricity, in this case defined as `e=FC//CB`. In the Input window, as shown in Figure 12, enter the expression e=distanceFC/distanceCB.
When you press Enter, the result of this calculation in the Input window is placed in the algebra window, in our case as `e=3.79`, as shown in Figure 13. Note: Again, don't worry about the number you get for your ratio. Plunge ahead with the activity.
Computing the Radial Length
Recall that the polar form of the conic section is
`r = (ek)/(1 + e cos theta)`.
Recall that `k` represents the distance of the directrix from the origin (See Figure 3), so we'll have to find the distance `k=FB`. Similarly, we'll need to find the measure of angle `theta` (again, see Figure 3).
Thus, we need two additional measurements, the length of segment `FB` and the measure of angle `\angle BFQ`. Use the Distance or length tool to measure `FB`. Then set k=distanceFB in the Input box as shown in Figure 14. (Turns out this is a bug/feature in Geogebra and `k` winds up being a "free object", which is not what we want. A better move here is to type the following in the Input box, then hit Enter: k=Distance[F,B]. This makes `k` a dependent object, which is what we need.)
Pressing Enter executes the calculation in the Input window and the result shows up in the Algebra window, in our case `k=5.56` as seen in Figure 15. Next, use the Angle measurement tool (found in the fourth icon from the right on the Geogebra toolbar), then click `B`, `F`, and `Q`, in that order. The result shows up in the Algebra window, in our case as `alpha=41.65^\circ` as seen in Figure 15. Note: Again, your numbers will differ. Don't worry about them and continue with the activity.
Calculating the Radial Length
We're now ready to calculate the radial length using our formula
`r = (ek)/(1 + e cos theta)`.
In the Input window, enter the expression r=e*k/(1+e*cos(α)), as shown in Figure 16.
Some comments are in order.
- Simply type cos( to get the cosine shown in Figure 16.
- Getting the `alpha` is a bit trickier, unless you know what to look for. Note that to the right of the Input window, there are three drop-down lists. The second one contains the Greek characters. Select `alpha` from this drop down list.
Pressing Enter executes the calculation in the Input window and places the result in the Algebra window, in our case `r=5.5`, as shown in Figure 17. Note: Again, don't worry about getting different numbers. Plunge ahead with the activity.