Figure 1
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 Geometer's Sketchpad, though no prior experience with the Sketchpad is assumed.
Let's begin with a focal point at the origin of our coordinate system and a vertical line k units from the origin, as shown in Figure 1.
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 closes to the point P. For example, in Figure 2, the ellipse is the set of all points P satisfying the relation FP = 0.53 PD.
Figure 2
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.
Figure 3
Let r and q represent the segment FP and the angle BFP, respectively. Because triangle BFP is a right triangle,
FB = r cos q.
Next, segment PD has length equal to the difference of segment FC and FB; i.e.,
PD = FC - FB.
Consequently,
PD = k - r cos q.
Thus, FP = e PD becomes
r = e (k - r cos q).
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 q).
When you start Sketchpad, you obtain a window similar to that in Figure 4.
Figure 4
The menus across the top of the Sketchpad window contain extensive commands for creating dynamic, geometrical objects. The toolbar to the left of the window contains some of the most commonly used construction tools. The red arrow is the selection tool, used for selecting objects in the window. Next comes a tool for constructing points, then a circle tool, and a line-making tool (actually several line-making tools). If you click and hold the mouse button over the line tool, you'll note three choices of line objects: lines, line segments, and rays. Last, the hand tool is used to place text on the Sketchpad window and is useful for labeling objects.
Before continuing with this activity, take some time to play with the tools and the menu items, enough to familiarize yourself with the Sketchpad environment.
Figure 5
We are now going to change some of the labels in Figure 5. Select the hand-tool from the toolbar at the left of the Sketchpad window. Click and drag each label in the sketch into a more favorable position. Next, double-click the label A with the hand-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 6.
Figure 6
Select the circle-tool from the toolbar. Click the circle cursor on the focus point at F and drag a circle about the size of that shown in Figure 7 (You can resize the circle later by dragging the point E). Select the point-tool from the toolbar and construct a point on the circle. Select the hand tool, double-click the label of the newly created point on the circle, and change the label to the letter Q, as shown in Figure 7.
Figure 7
Click and hold the mouse on the line tool, then slide and select the ray-drawing tool. Select the selection arrow-tool from the toolbar. Select and highlight the point F by clicking it with the selection tool. Depress and hold the shift key while clicking the point Q with the selection arrow-tool. This should result in both the points F and Q being selected and highlighted. Choose Construct-->Ray to construct a ray at F that emanates outward through the point Q, as shown in Figure 8. You may need to adjust the position of the label Q with the hand-tool.
Figure 8
Click and hold the mouse on the line tool, then slide and select the line-drawing tool. Select the selection arrow-tool from the toolbar. Select and highlight the point B by clicking it with the selection tool. Depress and hold the shift key while clicking the segment FB with the selection arrow-tool. This action should select and highlight both the point B and the segment FB. Select Construct-->Perpendicular Line. This should create a line through B, perpendicular to the segment FB, as shown in Figure 9.
Figure 9
Figure 10
Select Measure-->Calculate to open the calculator, as shown in Figure 11.
Figure 11
Single-click the measure FC in Figure 10, followed by the division symbol in the calculator, then single-click the measure CB in Figure 10. This should produce the calculation shown in Figure 12.
Figure 12
Clicking the OK button in the calculator places the result of this calculation in your Sketchpad window, as shown in Figure 13. The ratio FC/CB equals e, the eccentricity of our conic section. Note: Again, don't worry about the number you get for your ratio. Plunge ahead with the activity.
Figure 13
r = ek/(1 + e cos q).
Since e = FC/CB, k = FB, and q = angle BFQ, this polar equation for the conic section becomes
r = ( (FC/CB) FB )/(1 + (FC/CB) cos (angle BFQ).
Use the selection arrow-tool to select point F, then depress the shift key and select point B. Select Measure-->Distance to calculate the length of segment FB (note that this length equals k), as shown in Figure 14. Next, use the selection arrow-tool to select point B. Depress the shift key and select points F and Q, in that order. This should select and highlight the points B, F, and Q. Select Measure-->Angle to measure the angle BFQ, as shown in Figure 14. Note: Again, your numbers will differ. Don't worry about them and continue with the activity.
Figure 14
Select Measure-->Calculate to open the calculator. We must now calculate the radial length r. This is accomplished as follows:
Figure 15
Click OK to place the radial length computation in the Sketchpad window, as shown in Figure 16. Note: Again, don't worry about getting different numbers. Plunge ahead with the activity.
Figure 16
Note that we have enlarged the window and used the selection arrow-tool
to drag our calculations to the right-hand side of the window in Figure
16. This makes for a tidier sketch.
Figure 17
Click OK and note that the point F has been translated the correct distance along the ray FQ. Double-click the label of the translated point with the hand-tool, then rename the translated point as P. This point P satisfies the relation FP = e PD. The result is shown in Figure 18.
Figure 18
Next, use the selection arrow-tool to select the point P. Select Display-->Trace Point. Select Display-->Color and choose a color other than black (What's your favorite?). Now use the selection arrow-tool to again drag the point Q around the circumference of the circle. Note how the point P leaves a colored trace as it moves. In this case, the eccentricity is larger than one, so the trace of the point P is the hyperbola shown in Figure 19. Note: As you perform the trace, the point P leaves a trail of of circles representing points. When you finally release the mouse button, the points are replaced by a hyperbola, a curve passing through the trace points. Clicking anywhere in the window will erase the trace, allowing you to repeat the experiment.
Figure 19
Use the selection arrow-tool to select and highlight the point P. Select Display-->Trace Point. This toggles off the tracing of point P. Select the point Q, then depress the shift key and select the circle, in that order. Select Edit-->Action Button-->Animation. From the drop-down list boxes, select point Q moves one way, around Circle c1, normally. When you click the OK button, an animation button is created in your sketch, as shown in Figure 20.
Figure 20
Double-click the Animate button and watch the animation. Point Q travels
about the circle, while the point P travels along the hyperbola.
Figure 21
Pressing the OK button creates a Move button in your sketch, as shown in Figure 22.
Figure 22
Double-click the Move C->M button and watch the point C jump to the point M. Note that the locus is now a parabola because the eccentricity equals one.