An ellipse is the set of all points in the plane, the sum of whose distances from two fixed points is a given positive constant that is greater than the distance between the fixed points.

The two fixed points are called the foci.

An ellipse p can be defined as follows:

from five distinct points. The input is a list of five points. Note that a set of five distinct points does not necessarily define an ellipse.

from the directrix, focus, and eccentricity. The input is a list of the form $\left['\mathrm{directrix}'=\mathrm{dir}\,'\mathrm{focus}'=\mathrm{fou}\,'\mathrm{eccentricity}'=\mathrm{ecc}\right]$ where dir, fou, and ecc are explained above.

from the foci, and the length of the major axis. The input is a list of the form $\left['\mathrm{foci}'=\mathrm{foi}\,'\mathrm{MajorAxis}'=\mathrm{lma}\right]$ where foi and lma are explained above.

from the foci, and the length of the minor axis. The input is a list of the form $\left['\mathrm{foci}'=\mathrm{foi}\,'\mathrm{MinorAxis}'=\mathrm{lmi}\right]$ where foi and lmi are explained above.

from the foci, and the sum of distance of any point on the ellipse to the foci. The input is a list of the form $\left['\mathrm{foci}'=\mathrm{foi}\,'\mathrm{distance}'=\mathrm{dis}\right]$ where foi and dis are explained above.

from the endpoints of the major and minor axis. The input is a list of the form $\left['\mathrm{MajorAxis}'=\mathrm{ep1}\,'\mathrm{MinorAxis}'=\mathrm{ep2}\right]$ where ep1 and ep2 are explained above.

from its algebraic representation eqn. The input is an equation or a polynomial. If the optional argument n is not given, then:

if the two environment variables _EnvHorizontalName and _EnvVerticalName are assigned two names, these two names will be used as the names of the horizontal-axis and vertical-axis respectively.

if not, Maple will prompt for input of the names of the axes.

To access the information relating to an ellipse p, use the following function calls:

The command with(geometry,ellipse) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{geometry}\right)\:$

$\mathrm{\_EnvHorizontalName}≔'x'\:$$\mathrm{\_EnvVerticalName}≔'y'\:$

$\mathrm{ellipse}\left(\mathrm{e1}\,2x^2+y^2-4x+4y=0\right)\:$

$\mathrm{center}\left(\mathrm{e1}\right),\mathrm{coordinates}\left(\mathrm{center}\left(\mathrm{e1}\right)\right)$

$\mathrm{center\_e1},\left[1\,\mathrm{-2}\right]$

$\mathrm{foci}\left(\mathrm{e1}\right),\mathrm{map}\left(\mathrm{coordinates}\,\mathrm{foci}\left(\mathrm{e1}\right)\right)$

$\left[\mathrm{foci\_1\_e1}\,\mathrm{foci\_2\_e1}\right],\left[\left[1\,-2-\sqrt{3}\right]\,\left[1\,-2+\sqrt{3}\right]\right]$

$\mathrm{MajorAxis}\left(\mathrm{e1}\right),\mathrm{MinorAxis}\left(\mathrm{e1}\right)$

$2\sqrt{6},2\sqrt{3}$

$\mathrm{ellipse}\left(\mathrm{e2}\,\left['\mathrm{foci}'=\mathrm{foci}\left(\mathrm{e1}\right)\,'\mathrm{MajorAxis}'=\mathrm{MajorAxis}\left(\mathrm{e1}\right)\right]\right)\:$

$\mathrm{detail}\left(\mathrm{e2}\right)$

$\begin{array}{ll}\text{name of the object}& \mathrm{e2}\\ \text{form of the object}& \mathrm{ellipse2d}\\ \text{center}& \left[1\,\mathrm{-2}\right]\\ \text{foci}& \left[\left[1\,-2-\sqrt{3}\right]\,\left[1\,-2+\sqrt{3}\right]\right]\\ \text{length of the major axis}& 2\sqrt{6}\\ \text{length of the minor axis}& 2\sqrt{3}\\ \text{equation of the ellipse}& 96x^2+48y^2-192x+192y=0\end{array}$

$\mathrm{ellipse}\left(\mathrm{e3}\,\left['\mathrm{foci}'=\mathrm{foci}\left(\mathrm{e1}\right)\,'\mathrm{MinorAxis}'=\mathrm{MinorAxis}\left(\mathrm{e1}\right)\right]\right)\:$

$\mathrm{center}\left(\mathrm{e2}\right),\mathrm{coordinates}\left(\mathrm{center}\left(\mathrm{e2}\right)\right)$

$\mathrm{center\_e2},\left[1\,\mathrm{-2}\right]$

$\mathrm{Equation}\left(\mathrm{e2}\right)$

$96x^2+48y^2-192x+192y=0$

$\mathrm{ellipse}\left(\mathrm{e4}\,\left['\mathrm{foci}'=\mathrm{foci}\left(\mathrm{e1}\right)\,'\mathrm{distance}'=2\mathrm{sqrt}\left(6\right)\right]\right)\:$

$\mathrm{Equation}\left(\mathrm{e2}\right)$

$96x^2+48y^2-192x+192y=0$

$\mathrm{point}\left(A\,4\,0\right),\mathrm{point}\left(B\,-4\,0\right),\mathrm{point}\left(E\,0\,3\right),\mathrm{point}\left(F\,0\,-3\right)\:$

$\mathrm{ellipse}\left(\mathrm{e5}\,\left['\mathrm{MajorAxis}'=\left[A\,B\right]\,'\mathrm{MinorAxis}'=\left[E\,F\right]\right]\right)\:$

$\mathrm{Equation}\left(\mathrm{e5}\right)$

$144x^2+256y^2-2304=0$

$\mathrm{line}\left(l\,x=-2\,\left[x\,y\right]\right)\:$$\mathrm{point}\left(f\,1\,0\right)\:$$e≔\frac{1}{2}\:$

$\mathrm{ellipse}\left(\mathrm{e6}\,\left['\mathrm{directrix}'=l\,'\mathrm{focus}'=f\,'\mathrm{eccentricity}'=e\right]\,\left[c\,d\right]\right)\:$

$\mathrm{eq}≔\mathrm{Equation}\left(\mathrm{e6}\right)$

$\mathrm{eq}≔\frac{3}{4}{c}^2-3c+d^2=0$