HOW TO MAKE AN ELLIPSE:
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First put two pins into some surface such as cardboard or a bulletin board as shown below.
Then tie a piece of string or thread in a loop, and put the loop around the two pins:
Take a pencil and use it to.
draw the loop tight
Keep
the string tight while drawing all the way around the two pins. The result will be an ellipse. The two spots where the pins entered the
cardboard or bulletin board are the foci.
Each of these points is a focus.
If the ellipse is filled with a medium that can support waves, such as air (sound waves) or water (water surface waves), for example, then the foci really can focus waves. A whispering gallery can be built this way. If someone stands at one focus and talks (or whispers) the sound waves will be focused on the other focus and can be easily heard there.
If the two pins are far apart, the ellipse comes out long and skinny. In mathematical terms, this is called an ellipse with a high eccentricity.
If the two pins are close together, the ellipse turns out to be close to a circle, which is called low eccentricity.
The ellipse can even turn into a circle, if the two points overlap one another exactly.
Kepler found that the planets move around the sun in ellipses with the sun at one focus. The other focus is empty. As it turns out, the orbits of the planets are nearly circles, but not quite. This is a description of Kepler’s First law of Planetary Motion. Sometimes it happens that, if you are tying to solve a problem and the solution just eludes you no matter what you do, there is one simple thing that you are missing. Everyone from Hipparchus to Ptolemy to Copernicus had been going crazy trying to fit the motion of the planets to circles. They should have been using this curve called an ellipse that can be almost a circle. Then the theory fits the observed motion.
As it happens, any object that revolves around any other object is moving in an ellipse and is following Kepler’s three laws. Comets revolve around the sun in long, thin ellipses with the sun at one focus. Satellites revolve around the earth in ellipses with the center of the earth at one focus.
Kepler was able to verify this to an accuracy of about two minutes of arc (There are 60 minutes of arc to a degree and 360 degrees to a complete circle.) So Kepler was pretty accurate, but we can be much more accurate now. It turns out that there are a few very small deviations from ellipses that Kepler could not have observed. For example, planets pull on one another with a gravitational attraction that is much smaller than the sun’s gravitational pull. This perturbs the orbits a bit. In fact, the small changes from an ellipse produced by this are often called perturbations. However, the planets follow Kepler’s Laws to an accuracy greater than anyone had observed up to the time of Kepler.