ellipse - Maple Help

geometry

 ellipse
 define an ellipse

 Calling Sequence ellipse(p, [A,B,C,E,F], n) ellipse(p, ['directrix'=dir, 'focus'=fou, 'eccentricity'=ecc], n) ellipse(p, ['foci'=foi, 'MajorAxis'=lma], n) ellipse(p, ['foci'=foi, 'MinorAxis'=lmi], n) ellipse(p, ['foci'=foi, 'distance'=dis], n) ellipse(p, ['MajorAxis'=ep1, 'MinorAxis'=ep2], n) ellipse(p, eqn, n )

Parameters

 p - the name of the ellipse A, B, C, E, F - five distinct points 'directrix'=dir - dir is the line which is the directrix of the ellipse 'focus'=fou - fou is a point which is the focus of the ellipse 'eccentricity'=ecc - ecc is a constant less than one denoting the eccentricity of the ellipse 'foci'=foi - foi is a list of two points which are the foci of the ellipse 'MajorAxis'=lma - where lma is the length of the major axis 'MinorAxis'=lmi - where lmi is the length of the minor axis 'distance'=dis - where dis is the sum of distance of any point on the ellipse to the foci 'MajorAxis'=ep1 - where ep1 is a list of two points which is the two endpoints of the major axis 'MinorAxis'=ep2 - where ep2 is a list of two points which is the two endpoints of the minor axis eqn - the algebraic representation of the ellipse (i.e., a polynomial or an equation) n - (optional) a list of two names representing the names of the horizontal-axis and vertical-axis

Description

 • 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:

 form(p) returns the form of the geometric object (i.e., ellipse2d if p is an ellipse). center(p) returns the name of the center of p. foci(p) returns a list of two foci of p. MajorAxis(p) returns the length of the major axis of p. MinorAxis(p) returns the length of the minor axis of p. Equation(p) returns the equation that represents the ellipse p. HorizontalName(p) returns the name of the horizontal-axis; or FAIL if the axis is not assigned a name. VerticalName(p) returns the name of the vertical-axis; or FAIL if the axis is not assigned a name. detail(p) returns a detailed description of the ellipse p.

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

Examples

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

define ellipse e1 from its algebraic representation:

 > $\mathrm{_EnvHorizontalName}≔'x':$$\mathrm{_EnvVerticalName}≔'y':$
 > $\mathrm{ellipse}\left(\mathrm{e1},2{x}^{2}+{y}^{2}-4x+4y=0\right):$
 > $\mathrm{center}\left(\mathrm{e1}\right),\mathrm{coordinates}\left(\mathrm{center}\left(\mathrm{e1}\right)\right)$
 ${\mathrm{center_e1}}{,}\left[{1}{,}{-2}\right]$ (1)
 > $\mathrm{foci}\left(\mathrm{e1}\right),\mathrm{map}\left(\mathrm{coordinates},\mathrm{foci}\left(\mathrm{e1}\right)\right)$
 $\left[{\mathrm{foci_1_e1}}{,}{\mathrm{foci_2_e1}}\right]{,}\left[\left[{1}{,}{-}{2}{-}\sqrt{{3}}\right]{,}\left[{1}{,}{-}{2}{+}\sqrt{{3}}\right]\right]$ (2)
 > $\mathrm{MajorAxis}\left(\mathrm{e1}\right),\mathrm{MinorAxis}\left(\mathrm{e1}\right)$
 ${2}{}\sqrt{{6}}{,}{2}{}\sqrt{{3}}$ (3)

define ellipse e2 from its foci and length of the major axis

 > $\mathrm{ellipse}\left(\mathrm{e2},\left['\mathrm{foci}'=\mathrm{foci}\left(\mathrm{e1}\right),'\mathrm{MajorAxis}'=\mathrm{MajorAxis}\left(\mathrm{e1}\right)\right]\right):$
 > $\mathrm{detail}\left(\mathrm{e2}\right)$
 $\begin{array}{ll}{\text{name of the object}}& {\mathrm{e2}}\\ {\text{form of the object}}& {\mathrm{ellipse2d}}\\ {\text{center}}& \left[{1}{,}{-2}\right]\\ {\text{foci}}& \left[\left[{1}{,}{-}{2}{-}\sqrt{{3}}\right]{,}\left[{1}{,}{-}{2}{+}\sqrt{{3}}\right]\right]\\ {\text{length of the major axis}}& {2}{}\sqrt{{6}}\\ {\text{length of the minor axis}}& {2}{}\sqrt{{3}}\\ {\text{equation of the ellipse}}& {96}{}{{x}}^{{2}}{+}{48}{}{{y}}^{{2}}{-}{192}{}{x}{+}{192}{}{y}{=}{0}\end{array}$ (4)

define ellipse e3 from its foci and length of the minor axis

 > $\mathrm{ellipse}\left(\mathrm{e3},\left['\mathrm{foci}'=\mathrm{foci}\left(\mathrm{e1}\right),'\mathrm{MinorAxis}'=\mathrm{MinorAxis}\left(\mathrm{e1}\right)\right]\right):$
 > $\mathrm{center}\left(\mathrm{e2}\right),\mathrm{coordinates}\left(\mathrm{center}\left(\mathrm{e2}\right)\right)$
 ${\mathrm{center_e2}}{,}\left[{1}{,}{-2}\right]$ (5)
 > $\mathrm{Equation}\left(\mathrm{e2}\right)$
 ${96}{}{{x}}^{{2}}{+}{48}{}{{y}}^{{2}}{-}{192}{}{x}{+}{192}{}{y}{=}{0}$ (6)

define ellipse e4 from its foci and the sum of distance of any point on the ellipse to the foci

 > $\mathrm{ellipse}\left(\mathrm{e4},\left['\mathrm{foci}'=\mathrm{foci}\left(\mathrm{e1}\right),'\mathrm{distance}'=2\sqrt{6}\right]\right):$
 > $\mathrm{Equation}\left(\mathrm{e2}\right)$
 ${96}{}{{x}}^{{2}}{+}{48}{}{{y}}^{{2}}{-}{192}{}{x}{+}{192}{}{y}{=}{0}$ (7)

define ellipse e5 from endpoints of the major and minor axis

 > $\mathrm{point}\left(A,4,0\right),\mathrm{point}\left(B,-4,0\right),\mathrm{point}\left(E,0,3\right),\mathrm{point}\left(F,0,-3\right):$
 > $\mathrm{ellipse}\left(\mathrm{e5},\left['\mathrm{MajorAxis}'=\left[A,B\right],'\mathrm{MinorAxis}'=\left[E,F\right]\right]\right):$
 > $\mathrm{Equation}\left(\mathrm{e5}\right)$
 ${144}{}{{x}}^{{2}}{+}{256}{}{{y}}^{{2}}{-}{2304}{=}{0}$ (8)

define ellipse e6 from its directrix, focus and eccentricity:

 > $\mathrm{line}\left(l,x=-2,\left[x,y\right]\right):$$\mathrm{point}\left(f,1,0\right):$$e≔\frac{1}{2}:$
 > $\mathrm{ellipse}\left(\mathrm{e6},\left['\mathrm{directrix}'=l,'\mathrm{focus}'=f,'\mathrm{eccentricity}'=e\right],\left[c,d\right]\right):$
 > $\mathrm{eq}≔\mathrm{Equation}\left(\mathrm{e6}\right)$
 ${\mathrm{eq}}{≔}\frac{{3}}{{4}}{}{{c}}^{{2}}{-}{3}{}{c}{+}{{d}}^{{2}}{=}{0}$ (9)