 Interaktive Ellipse

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Graphical ellipses calculator

Pulling the red dots A, B and C allows the ellipse to be varied. The currently calculated ellipse values are indicated in the box on the right.

Properties of the ellipse

The ellipse is the set of all geometric locations for which the sum of the distances of two fixed points is constant.

Circumference of the ellipse

With the semiaxis a = ME and b = MD is the circumference of the ellipse given by:

$U =πa+b 1+ ∑ n = 0 ∞ 2n n n+122n+1 22n+2$

$=\pi \left(a+b\right)\left(1+\frac{{\lambda }^{2}}{4}+\frac{{\lambda }^{4}}{64}+...\right)$

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$\lambda =\frac{\left(a-b\right)}{\left(a+b\right)}$

Approximation formula for the elliptical perimeter according to Ramanujan:

$U\approx \pi \left(a+b\right)\left(1+\frac{3{\lambda }^{2}}{10+\sqrt{4-3{\lambda }^{2}}}\right)$

Ellipse area

With the half-axes a and b the area of the ellipse is given by:

$F=\pi ab$

Focal distance

With the larger semi-axis a the distance of the focal points of the ellipse is given by:

$d=2\sqrt{{a}^{2}-{b}^{2}}$

Eccentricity

With the larger semi-axis a, eccentricity of the ellipse is given by:

$e=\frac{d}{2a}$

Tangent

The normal to the tangent of the ellipse halves the angle that the focal spot beams form at that point.