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Ellipse circumference
#1
Hi all! Just a quick question...

Recently I tried to look up the formula for the circumference of an ellipse, and found that it is highly non-trivial (involving elliptical integrals). However, I wonder if the following special case may be easier to calculate: given a circle of fixed circumference C and radius r, how easy is it to find an ellipse E with one of the axes already given, such that E has circumference C? E.g., if I deform a circle into an ellipse with major radius R, what must the minor radius be in order for its circumference to be constant? Thanks.
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#2
Hmm, OK, I just found an approximation of elliptic circumference with radii and :

(Ramanujan's approximation). Does anyone know how accurate this approximation is?
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#3
Hm in my formula reference the formula
is given ... *headscratch*

Ever thought about the volume of a polyhedron given by its vertices?
Seams a similar but completely different difficulty.
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#4
bo198214 Wrote:Hm in my formula reference the formula
is given ... *headscratch*

Ever thought about the volume of a polyhedron given by its vertices?
Seams a similar but completely different difficulty.

Well, Ramanujan's formula is supposed to be "very accurate", so I don't know. But I like your version better, it's definitely a lot easier to remember!

But regarding polyhedral volumes... isn't it just a matter of dissecting the polyhedron into right pyramids and summing them up? This isn't too hard to do from the vertex coordinates.
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#5
Ellipses seem to be a really hard topic.
There is not even a closed formula for the location of a planet revolving around the sun as a function of time.
Instead we have all those Kepler's laws.
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