Drawing/ Creating an Ellipse

          There are several different ways to create an ellipse. We can use any of the technology applications to help us draw an ellipse, we can use everyday materials such as a piece of string or paper folding, and lastly we can use our minds and our math abilities to draw an ellipse.

          We start with the equation and we substitute different values for x,y with constants at a and b. This method, as I said before, is sloppy at best when drawing an ellipse. So for this reason we’ll stay away from this method and go on to using the string and technology to draw ellipses.

          The easiest and most available method for drawing an ellipse is with two push pins, a pencil, and a piece of string. We begin by trying the string at the two ends to make a circle. Next we put the push pins into a wall or a desk and lay the string around the pins. Lastly we put the pencil inside the string and pull it tight around the pins. The trace that we produce will be an ellipse. Here’s a picture.

Several interesting questions arise from this picture such as Why does the trace create an ellipse? Can you measure the distance from one foci to another foci? And since the string is at a fixed ratio or length then where does that fixed length come into play? These and other questions will be addressed as we move forward in our exploration.

          The second method of drawing an ellipse is with some type of graphing software whether it be Graphing Calculator or some type of TI-83 calculator. Either one will suffice, but for this method I will use Graphing Calculator. If we set the equation  with a = 1.5 and b = 1 then we get a picture that looks like this.

If we just let a and b equal the same value then we’ll get a circle, which is an ellipse of a different form. A circle is an ellipse with eccentricity equal to 1, but that will be later in the discussion of an ellipse if we even get to it. Here’s an example of a movable ellipse on Graphing Calculator.  Just click the play button to watch it rise and fall depending on the values of n. Can you make any conjectures or assumptions that may be involved with this graph?

          The last way to create an ellipse is with Geometer’s Sketchpad. This, in my opinion, turns out to be the most useful and the most interesting drawing of the ellipse. With GSP we can create the ellipse and then move it and watch the changes occur in a dynamic way. Here’s a picture of the creation as well as a detailed explanation of how to make it on GSP.

          1. Start by making a circle, C, with center F1.

          2. Create a point on the outside of circle C and label it B. Also place a point, F2 anywhere inside of the circle.

          3. Make a segment, l, from F2 to B and find the midpoint and call it M.

          4. Construct a perpendicular bisector with l, or a perpendicular line with l through M, call this line q.

          5. Now create a line through M and F1 and call it x.

          6. Find the intersection of q and x and make a line through it and F2, call this intersection E.

          7. Lastly, select E and B in that order, and go to Construct and Locus to create the ellipse.

Now we have an ellipse that we can interact with. Click here to play around with it. Ellipse