I ran across this wonderful reference when searching for a solution to your question:
http://forum.swarthmore.edu/dr.math/faq/formulas/faq.ellipse.circumference.html which gives a complete explanation of the "calculus version" of the problem. It also details several of the approximation formulas, which are based on infinitely repeating series that approximate the correct result. However, for a non-calculus explanation of the reason the formula doesn't settle down easily, consider this one: The circumference of a circle is easily calculated because the rate at which the radius changes is constant (i.e., it is zero.). That means, from one degree to the next, the radius value--which is related to the arc length subtended-- doesn't change, so the arc length for a small angle (which equals the radius times the angle as a good approximation if the angle is measured in radians), when added together for an entire circle is 2 pi radians x the radius, and simplifies readily to the familiar C = 2 pi r. This works because a) arc length (a tiny chunk of circumference) is accurately estimated by using the small angle approximation tan (theta) = theta for small theta in radians, and because b) r doesn't change at all. However, if r DOES change, as it does for an ellipse, you have to account for the RATE at which the r changes in your estimation. Now, if r changed at a constant, steady, fixed rate, the expression would still simplify to a simple formula. But r is (drum roll) continuously variable (cymbal crash) which means that to account for the changing r, the rate of which r is changing is itself not a constant, but a formula (a sine and cosine function). So to simpifly THAT, you find out how fast the rate changes-- getting a rate of a rate-- and .... that is also continuously variable too, so you try a rate of a rate of a rate and find .... that is also continuously variable too... so in fact it NEVER settles down to a nice, simple, constant number or even a standard polynomial algebra expression like x squared or something. So the best you can do--since even calculus won't crack that nut apparently-- is to use an infinitely repeating series to approximate it or to evaluate the integral numerically, perhaps with a computer. Now, at a step even further removed from calculus, you could say to your nonmathematical friends the following statement: To get the circumference of a circle, you look at 2 pi r, where r is the radius. Well, ellipses have radii too, but they don't stay constant. As you measure the radius, it keeps getting smaller and larger by uneven amounts, and the closer you look at it precisely, the more variation you see. So it doesn't settle into a simple formula because it doesn't settle down and behave like a nice constant as it does in a circle. I wonder though, if you used the definition of Kepler's Second Law for an orbit computed on a focus instead of the center, if it would be possible to compute the circumference that way. Something worth investigating... Jeff Adkins "Frans W. MAES" wrote: > Hi all, > > Just to relieve the recent boredom of this list, how about this one: > > As some of you may know, I have an analemmatic sundial in my > garden (story on my homepage). The person who did the actual > work had to know how much material (tiles, bricks etc.) he would > need. Then I found out that there is no 'simple' formula for the > circumference of an ellipse. (I also found out that this led to a lot of > interesting mathematics, called 'elliptic integrals'. I knew the term, > but never realized where it came from.) > > The formula for the area of a circle generalizes simply to the area of > an ellipse (pi x r x r -> pi x a x b ; a and b being half the major and > minor axes, resp.). What puzzles me since is, why the > circumference of a circle does NOT generalize simply (actually, not > at all) to an ellipse. > > My question thus is: does anyone of you happen to know of a NON- > mathematical, intuitively convincing explanation for the fact that > there is no 'simple' formula for the circumference of an ellipse? > > Regards, > Frans Maes > > ===================================== > Frans W. Maes > Peize, The Netherlands > 53.1 N, 6.5 E > www.biol.rug.nl/maes/ > ===================================== Content-Type: text/x-vcard; charset=us-ascii; name="Astronomer.vcf" Content-Transfer-Encoding: 7bit Content-Description: Card for Jeff Adkins Content-Disposition: attachment; filename="Astronomer.vcf" Attachment converted: Macintosh HD:Astronomer.vcf 1 (TEXT/ttxt) (00025A60)
