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where P is the perimeter of the ellipse
given by:
and e is the eccentricity of the ellipse:
There is no simple exact formula for the solution of the
integral defining the perimeter of an ellipse. ETB uses the
following approximate solution proposed by David W. Cantrell in 2001:
where:
The value of the exponent p may be
optimized for different criteria. In the ETB implementation, p is
given by:
Using this value of p, Cantrell's equation
for the perimeter of an ellipse gives exact results for both flat ellipses
(b=0) and circles (a=b). The worst relative error is about 0.022%,
obtained for an ellipse of eccentricity slightly below 0.99742841112. See Reference
1 for more details.
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