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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 end points of the major axis

'MinorAxis'=ep2

-

where ep2 is a list of two points which is the two end points 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 'directrix'=dir,'focus'=fou,'eccentricity'=ecc 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 'foci'=foi,'MajorAxis'=lma 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 'foci'=foi,'MinorAxis'=lmi 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 'foci'=foi,'distance'=dis where foi and dis are explained above.

– 

from the end points of the major and minor axis. The input is a list of the form 'MajorAxis'=ep1,'MinorAxis'=ep2 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

withgeometry:

define ellipse e1 from its algebraic representation:

_EnvHorizontalName:='x':_EnvVerticalName:='y':

ellipsee1,2x2+y24x+4y=0:

centere1,coordinatescentere1

center_e1,1,2

(1)

focie1,mapcoordinates,focie1

foci_1_e1,foci_2_e1,1,23,1,2+3

(2)

MajorAxise1,MinorAxise1

26,23

(3)

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

ellipsee2,'foci'=focie1,'MajorAxis'=MajorAxise1:

detaile2

name of the objecte2form of the objectellipse2dcenter1,−2foci1,23,1,2+3length of the major axis26length of the minor axis23equation of the ellipse96x2+48y2192x+192y=0

(4)

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

ellipsee3,'foci'=focie1,'MinorAxis'=MinorAxise1:

centere2,coordinatescentere2

center_e2,1,2

(5)

Equatione2

96x2+48y2192x+192y=0

(6)

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

ellipsee4,'foci'=focie1,'distance'=26:

Equatione2

96x2+48y2192x+192y=0

(7)

define ellipse e5 from end points of the major and minor axis

pointA,4,0,pointB,4,0,pointE,0,3,pointF,0,3:

ellipsee5,'MajorAxis'=A,B,'MinorAxis'=E,F:

Equatione5

144x2+256y22304=0

(8)

define ellipse e6 from its directrix, focus and eccentricity:

linel,x=2,x,y:pointf,1,0:e:=12:

ellipsee6,'directrix'=l,'focus'=f,'eccentricity'=e,c,d:

eq:=Equatione6

eq:=34c23c+d2=0

(9)

See Also

geometry[conic], geometry[draw], geometry[HorizontalName], geometry[objects], geometry[VerticalName]


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