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Some interesting ellipse stuff
What is an ellipse?
An ellipse is what you would see if you take a circular disk [flat plate] and rotate it around any diameter. The points on the circle, which mark the axis around which it is being rotated [solid line], remain the same distance apart, but the two points halfway between them appear to move closer together as the degree of rotation increases.
If you turn the disk until it is flat to your line of vision all you will see is the nearer edge. Try it with a CD. The further away you are from the disc the better, so if you can get someone to hold and rotate the disc, so much the better.
The ellipse is symmetrical around its axis of rotation, but also around another axis that goes vertically through its center [dashed line]. If a shape is symmetrical around an axis it simply means the part on one side of the axis is identical to the part on the other side except it is flipped over. In other words, the half of the shape on one side of the axis of symmetry is a mirror image of the half on the other side.
In the sketch you can see the part above the line called the axis of rotation is a mirror image of the part below in all three drawings. The same is true in all three cases for a vertical line going through the center. The shape on each side is a mirror image of the shape on the other side. It would not be true for any other line at any other angle except in the case of the circle where all lines going through the center are centerlines.
Drawing an ellipse
Drawing a circle is quite easy, but while drawing an ellipse is almost as easy, few people know how to do it. For a circle, all you need is a pencil, a bit of string or strong thread and a drawing pin or a small nail. Using a smooth board as an underlay, place some paper on the board and press the drawing pin firmly into the board. Make a loop with the string so when the loop is stretched from the pin it does not go beyond the edge of the paper. Place the point of the pencil into the loop, and, keeping a bit of tension in the loop, move the pencil around the pin so its point draws a circle. This needs to be done carefully because the point of the pencil can easily 'pop' over the thread or string.
Drawing an ellipse is almost as easy. All you need is one more drawing pin or nail. Push the pin firmly into the board some distance away from the first pin. This time, do not make a loop, but tie one end of the string to one pin and then tie the string to the other pin making sure the piece of string in the middle has a good amount of slack. Now you use the point of the pencil to pick up the string and pull it as far as it will go in line with the two pins to one side. There you put the point of the pencil onto the paper and move the pencil so the string moves away from the pins but remains taut all the time. When you get to the other side you already have half an ellipse. Now you move the string over the pins and continue on the other side in the same way, and when you reach the point where you started, you have an ellipse.
So, an ellipse is similar to a circle except it has two 'center' points but, because neither of them is in the center, they are called foci.
If you were to remove one of the drawing pins and move it a bit further away from the other one without changing the length of the string and repeat the procedure, you would get a flatter ellipse. If you move the pin closer you get a rounder one. In fact, if you slip the string off one pin and put it on the other pin and draw an 'ellipse' you will get a circle. So, ellipses and circles have a lot in common and belong to the same family of geometric shapes.
The sketch below shows a group of ellipses all drawn with the same length of string. They are drawn using the pin method described on the previous page (except they are done with a drawing program). The pin identified as '0' remains in the same place while the second pin is moved from 0 to 1 to 2 and so on. The first ellipse drawn is a circle. And, while the sketch shows the second point being moved to the left, it could just as well have been to the right or up or down.
There are two important common dimensions for an ellipse and they are the major and minor axes. They are commonly referred to as A and B respectively, and in mathematics, where it is common practice to start things off around the origin of the x-y co-ordinates of a graph, the half axes, a and b, are used.
There is more about the math of the ellipse and circle in the next section. However, there is one more thing you may have noticed while doing this little experiment. The length of the string between the knots is equal to the dimension of the major axis, and this also applies to the circle [which is why I have used a loop instead of a single length].
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