Move the points in the grafic or define the point coordinates in the numeric field. Pulling the focal points F_{1}, F_{2} and the peripherie point P allows the ellipse to be varied. The center of the ellipse is marked by the center point C The currently calculated ellipse values are shown below.

Scale:

Number of digits =

Ellipse:

Eccentricity *e*=

Circumference *U*≈

Area *F*=

semi-axis:

|C V_{1}|=

|C V_{2}|=

semi-axis sum=

focal lines:

|p F_{1}|=

|p F_{2}|=

focal lines sum=

foacl distance:

|F_{1} F_{2}|=

α=

Slope at p=

The ellipse is the set of all geometric locations for which the sum of the distances of two fixed points (A and B) is constant.

The ellipse equation in cartesian co-ordinates is given by:

$\frac{{\left(x-{m}_{x}\right)}^{2}}{{a}^{2}}+\frac{{\left(y-{m}_{y}\right)}^{2}}{{b}^{2}}=1$

with the ellipse center M at m_{x} and m_{y}.

In parameter representation the ellipse equation is given as follows

$\left(\begin{array}{c}{m}_{x}+a\mathrm{cos}t\\ {m}_{y}+b\mathrm{sin}t\end{array}\right)$

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

$$U=\pi \left(a+b\right)\left(1+\sum _{n=0}^{\infty}\frac{\left(\begin{array}{c}\mathrm{2n}\\ n\end{array}\right)}{\left(n+1\right){2}^{2n+1}}{2}^{2n+2}\right)$$

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

with

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

Approximation formula for the elliptical perimeter according to Ramanujan:

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

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

$F=\pi ab$

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

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

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

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

The perpendicular to the tangent of the ellipse halves the angle that the focal spot beams (AF and FB) form at the tangent point F.

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Here is a list of of further useful sites:

Index Trigonometric calculations Ellipse Triangle Parallelogram Rectangle