Finding the perimeter of an ellipse
A sequence of approximations, and learning math in reverse order
The perimeter of a circle is of simply P = 2πr, but there is no such simple equation for the perimeter of an ellipse.
A simple idea, going back at least four centuries to Kepler, is to estimate the perimeter of an ellipse by find the perimeter of a circle whose radius is the average of the largest and smallest radius of the ellipse. This post explores how well that works.
If you need more accuracy, or if your ellipse is far from being a circle, the approximation above will not work. But there is a series of improvements, discovered by James Ivory two centuries ago, that can give you all the accuracy you need.
I’ve been learning and applying math for a long time now, and I’m still learning things that were known hundreds of years ago. One reason is simply that a lot of math has been developed over the last two millenia. Another reason is that I learned math in something like reverse-chronological order.
You have to do original research to get a PhD, so the aim grad school is to get you to the frontier of some narrow bit of math as quickly as possible, rushing past everything along the way that you won’t need in order to write a dissertation. I learned 20th century math in college, and I’ve been learning 18th and 19th century math ever since. You’ll never see things like Ivory’s series in grad school, not because they’re not useful, but because they’re not active areas of research.
Learning in reverse -- in other words, bringing much-needed context to "the tip of the spear".
This is using the tall 'tree' you just climbed up...to see "the forest". Absolutely critical to knowledge generation...