# Interaktive Ellipse

x-min= x-max=
y-min= y-max=
A =
B =
C =

## 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.