Construction and Analysis of a Hyperbola

Paper-folding construction

• As we did with the construction of the parabola and the ellipse, we're going to make a hyperbola using wax paper.
1.  On a piece of wax paper, draw a circle in the center of the paper.  You could use a compass or a circular object such as a lid of some kind.

2.  Next, mark a point outside the circle.  It really doesn't matter where this point is, but it might be easier if it's closer to the circle.  This point is the focus.

3.  Now, fold the paper so that the circle coincides (lies directly on top of) the focus.  Make a sharp crease of the fold.

4.  Do this numerous times.  Start at one point on the circle and work your way around the circle.

5.  What do you notice happening?  Is anything appearing on your paper.

• Let's try to repeat this process using Geometer's Sketchpad.
1.  Start Geometer's Sketchpad.  Then, under Display, select Preferences and mark Autoshow labels for points only. Check to make sure that precision for distances is hundredths.

2.  Near the center of the screen, construct a circle with a diameter of about 2-3 inches.

3.  Create a point (focus) outside the circle.

4.  Select the circle and construct a Point on Object.  (This point is movable along the circumference of the circle.  The other point that you see on the circle defines the circle, and if you try to move it, the entire circle will move.)

5.  When we created the parabola and ellipse, the next step was to create a perpendicular bisector of the segment connecting the focus and the sliding point.  Do the same for the construction of a hyperbola.  You can change to color of this line to aid in visualization.  This line is the fold line from the wax paper construction of an ellipse.

6.  Now, we want to simulate the process of folding.  Select the perpendicular bisector and under Display, choose the option Trace line.

7.  Next, select the sliding point and the circle.  Under Edit, choose Action Button and the type Animation.

8.  Once the animation button is created, double-click it and watch.

It looks like there is a hyperbola on the screen now, but how do we know that what we have constructed is really a hyperbola?

Let's try to prove that what we have constructed is a hyperbola:

Under Edit, select Undo to remove all of the trace lines from the animation.

What is the definition of a hyperbola?

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Let's go about this proof in a manner similar to what we did for the parabola and for the ellipse.  How do we determine the distance from a point to a circle?

(Hint: it's perpendicular distance)

9.  Construct a line that is perpendicular to the circle by constructing a line through the sliding point on the circle and the point that defines the center of the circle.  This line is perpendicular to the circle at this specific point.

10.  Notice that this line crosses the fold line.  Select both lines and construct the point of intersection.

11.  Next, construct the segment connecting the new point to the focus.

• To prove that point F in the diagram would actually "carve" out a hyperbola, we need to show that the difference of the measurements of FC and FA = a constant (i.e. |FC-FA| = a constant).
12.  Select points F and C and under Measure, select Distance.  Repeat for the segment FA, by selecting the points F and A.

13.  The measurements should be in the upper lefthand corner at this point.  Select both measurements and under Measure, select Calculate.  Under Functions, select abs[ to get the absolute value of the difference.  Then, under Values, you can select the distance of FC and FA.  Subtract one value from the other, select ), and click OK.  The difference of the two distances should now be displayed in the upper lefthand corner.

14.  Select the fold line and turn off the trace function under Display.  Select the point that defines the hyperbola (point F above) and under Display, choose Trace Point.

Select the segment CD, midpoint E, the perpendicular line, and the fold line and under Display, choose Hide objects.

Perform the animation.  What do you notice?  As the animation runs, keep watching the distance measures from #13.  What is significant about this calculation?  Does this help you prove that what you constructed is truly a hyperbola?