Here's one way to draw a perfect ellipse. You'll need:

• A flat board, made of any material into which pins or nails can easily be pushed
• Two pins or nails
• A loop of thread or string
• A pen or pencil
• Paper

Place a piece of paper on the board, and stick in the two pins (not too close together). Loop the the thread around the pins and pull taut with the tip of the pen. Now move the pen around, always keeping the loop of thread taut. As the pen rotates around the two pins it will trace out an ellipse!

Here's a Java simulation of such an ellipse drawer. The two dots represent the pins. The positions of the dots are called the foci of the ellipse. You can adjust the position of the foci (the pins) by clicking on one of them and dragging it left or right. The numbers in the top left corner show you the lengths of the string from the pen to each pin. Notice that the sum of the two numbers is always the same!

If Java worked here, you'd see something like this.

The two lines with the cross marks correspond to the lengths of string from the pen (which sits at the point where the lines meet) to each of the pins. In a real pin-and-string device, when the string has not been pulled taut, the sum of the distances from the pen to each of the two pins is too small. You can understand how the computer indicates this if you imagine that the string is lying on a table and that the (green) focus is actually a hole in the table through which the string passes. If the the string has been pulled taut, some of it still hangs down underneath the table, waiting to be pulled up through the hole at the focus. The string below the table is shown in grey, and without cross marks. As the amount of string above the table increases, the amount below the table must decrease. When the grey string has all disappeared, the string above the table is at its full length -- and the pen then lies on the curve we want to draw. If you try to drag the pen further than the fixed length of string will allow, nothing happens -- the computer won't allow it!

This construction can be used to derive equations for an ellipse.