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perimeter of ellipse
The usual parametric presentation of the ellipse (blue in the diagram)
using as parameter the eccentric anomaly , is
| (1) |
If, according to the diagram, the point on the circumscribed circle corresponds the point of the ellipse, the geometrical meaning of is the polar angle of .
From (1) we get
Substituting to the arc length formula
the limits and and the derivatives we can express the perimeter length of the ellipse as
The expression may be converted by taking the factor out and by using the formula :
Here we have used the eccentricity (). Because of the symmetry of the ellipse in regard to the coordinate axes, we get
and since on the interval the sine function attains the same values as the cosine function (but in reverse order), the length of the perimeter of ellipse may be written also as
| (2) |
This is a complete elliptic integral of second kind. The name “elliptic integral” is due to this task to determine the length of the perimeter of an ellipse ().
The integral (2) is not expressible in closed form with elementary functions, but can be expanded e.g. to the series
One quite accurate approximative formula, due to S. Ramanujan, is
with and the relative error about .
Mathematics Subject Classification
53A04 Curves in Euclidean space51N20 Euclidean analytic geometry
51-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
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