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Ellipses

Definitions
General Equation of an Ellipse
Standard Forms of an Ellipse
Reduction to Standard Form
Ellipse from Vertices and Eccentricity
Ellipse from Foci and Eccentricity
Ellipse from Focus and Directrix
Parametric Equations
Explorations

The visible universe is filled with ellipses, or near ellipses, traced by celestial bodies revolving around each other, such as planets and the sun. The fact that the angle formed by two focal radii through a point on an ellipse is bisected by the normal to the curve may be used in a device for re-concentrating sound waves, at illustrated in the acoustics of the Mormon Tabernacle in Salt Lake City, Utah. Various types of rotating machinery use elliptical components to generate special types of linear and rotational motions. This chapter develops the mathematics of an ellipse.

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Definitions [Top]

An ellipse is the locus of a point that moves so that the ratio of its distance from a fixed point and from a fixed line is a positive constant less than one. As with the parabola, a focus, directrix and eccentricity are associated with the curve as described in Table [ell:tbl00].

"ell_1.gif"

Definition of an ellipse.

"ell_2.gif"

Ellipse definition.

Consider the line through the focus perpendicular to the directrix. From the definition PF/PD=e/1 there are obviously two points V and "ell_3.gif" which divide the (undirected) segment FD, internally and externally respectively, into the ratio of e/1. Therefore V and "ell_4.gif" are points (on the same side of D) on the ellipse; they are called the vertices. The segment "ell_5.gif" is called the major axis. By symmetry there is another point "ell_6.gif" and another line "ell_7.gif" such that "ell_8.gif" and "ell_9.gif" would serve as the definition of this curve. Thus an ellipse has two foci and two directrices associated in pairs F, D and "ell_10.gif", "ell_11.gif". The midpoint of "ell_12.gif", which is also the midpoint of "ell_13.gif", is called the center C. It is evident that the locus is contained between the vertices, that it is bounded in all directions and that it is symmetric both with respect to the major axis and to a line perpendicular to it through C.

The focal chord perpendicular to the major axis is called the latus rectum. The length of the central chord perpendicular to the major axis is called the minor axis.

Example: Plot the ellipse with center at coordinates (2,1), major axis length of 6, minor axis length of 2, and rotated "ell_14.gif" (π/6 radians) about the center point.

Solution: Ellipse2D[{h,k},a,b,θ] is the standard representation of an ellipse in Descarta2D. The ellipse is centered at coordinates {h,k}, has semi-major axis of a, semi-minor axis of b and is rotated about the center point by an angle θ (the semi-major axis is half the length of the major axis; the semi-minor axis is half the length of the minor axis).

Sketch2D[{Ellipse2D[{2,1},3,1,Pi/6]}]

"ell_15.gif"

General Equation of an Ellipse [Top]

Take any point "ell_16.gif" as focus and any line, "ell_17.gif" as directrix, where "ell_18.gif". The normalized form of the line is used to simplify the derivation. By definition the equation of the ellipse is

"ell_19.gif"

which may be expanded to

"ell_20.gif"
"ell_21.gif"

This is of the form "ell_22.gif", an equation of the second degree. Moreover, it can be verified that "ell_23.gif" (when e<1).

Therefore, a necessary condition that "ell_24.gif" represent an ellipse is that "ell_25.gif". The general equation reveals that if the defining directrix line is parallel to one of the coordinate axes then B=0, since either "ell_26.gif" or "ell_27.gif" will be zero. The equation of an ellipse in this position will have no xy term.

Standard Forms of an Ellipse [Top]

By an appropriate choice of coordinate axes the general equation of an ellipse can be reduced to one of the following standard forms.

"ell_28.gif"

Ellipse in standard position (x-axis).

"ell_29.gif"

Ellipse in standard position (y-axis).

Major Axis Parallel to the x-Axis

The equation of an ellipse in standard position whose major axis is parallel to the x-axis and whose center is at the origin is

"ell_30.gif"

where a and b are the lengths of the semi-major and semi-minor axes, respectively. If the ellipse is centered at (h,k), then the equation is

"ell_31.gif"

as shown in Figure [ell:fig4]. When an ellipse is in this special position, the formulas for the important points, lines and constants associated with the ellipse are simply determined and are summarized in Table [ell:tbl01].

"ell_32.gif"

Ellipse equations (x- and y-axis).

Major Axis Parallel to the y-axis

The equation of an ellipse in standard position whose major axis is parallel to the y-axis and whose center is at the origin is

"ell_33.gif"

where a and b are the lengths of the semi-major and semi-minor axes, respectively. If the ellipse is centered at (h,k), then the equation is

"ell_34.gif"

as shown in Figure [ell:fig6]. When an ellipse is in this special position, the formulas for the important points, lines and constants associated with the ellipse are simply determined and are summarized in Table [ell:tbl01].

Reduction to Standard Form [Top]

The most general equation of an ellipse with no xy term (and hence one whose axes are parallel to the coordinate axes) is of the form

"ell_35.gif"

The condition "ell_36.gif" reduces to AC>0 which implies that A and C are of like sign. This equation can be reduced to one of the standard forms by completing the square.

Example: Reduce "ell_37.gif" to standard form and plot.

Solution: The Descarta2D function Loci2D[quad] reduces a quadratic equation to a standard form.

crv1=Loci2D[Quadratic2D[1,0,4,4,0,0]]

"ell_38.gif"

The equation in standard form is "ell_39.gif".

Sketch2D[{crv1}]

"ell_40.gif"

Example: Reduce "ell_41.gif" to standard form. Find the center, foci, vertices, directrices, the lengths of the semi-major and semi-minor axes and the eccentricity. Plot the geometric objects.

Solution: The function Loci2D[quad] reduces a quadratic equation to a standard form. The function Point2D[ellipse] returns the center point of an ellipse; the function Foci2D[ellipse] returns a list of the two foci of an ellipse; the function Vertices2D[ellipse] returns a list of the two vertex points of an ellipse; the function Directrices2D[ellipse] returns a list of the two directrix lines of an ellipse; SemiMajorAxis2D[ellipse] and SemiMinorAxis2D[ellipse] return the lengths of the semi-major and semi-minor axes of an ellipse, respectively.

crv1=Loci2D[Quadratic2D[5,0,9,-10,-54,41]]

"ell_42.gif"

The standard form of the equation is "ell_43.gif".

objs=Map[(#[ crv1[[1]] ])&,
         {Point2D, Foci2D, Vertices2D, Directrices2D,
          SemiMajorAxis2D, SemiMinorAxis2D,
          Eccentricity2D}]

"ell_44.gif"

Sketch2D[{crv1,Drop[objs,-3]},
         PlotRange->{{-5,7},{-1,6}},
         CurveLength2D->15]

"ell_45.gif"

Ellipse from Vertices and Eccentricity [Top]

Suppose we are given the two vertices, "ell_46.gif" and "ell_47.gif", and the eccentricity, e, of an ellipse and we wish to find the standard equation of the ellipse. The center point (h,k) of the ellipse is clearly the midpoint between the vertices and is given by

"ell_48.gif"

The length of the semi-major axis, a, is one-half the distance between the vertices, yielding "ell_49.gif". The eccentricity is given by

"ell_50.gif"

so, solving for b gives the length of the semi-minor axis as

"ell_51.gif"

The line through the two vertex points determines the rotation angle of the ellipse as

"ell_52.gif"

Example: Find the ellipse whose vertices are (4,2) and (-2,1), and whose eccentricity is 7/8.

Solution: The Descarta2D function Ellipse2D[{point,point},e] returns the ellipse whose vertices are the given points with the specified eccentricity.

p1=Point2D[{4,2}];
p2=Point2D[{-2,1}];
e1=Ellipse2D[{p1,p2},7/8] //N

"ell_53.gif"

Sketch2D[{p1,p2,e1}]

"ell_54.gif"

Ellipse from Foci and Eccentricity [Top]

It is evident from Table [ell:tbl01] that the distance between the foci of an ellipse is "ell_55.gif" and that the distance between the vertices is "ell_56.gif". Therefore, the eccentricity, e, given by

"ell_57.gif"

is the ratio of the distance between the foci to the distance between the vertices. This relationship allows us to construct an ellipse by specifying the two foci and the eccentricity. The semi-major axis length, a, is given by "ell_58.gif" and the semi-minor axis length is "ell_59.gif". The center point of the ellipse is clearly the midpoint of the two foci and the angle of rotation is

"ell_60.gif"

where "ell_61.gif" and "ell_62.gif" are the coordinates of the foci.

Example: Find the ellipse whose foci are (-1,-1) and (1,1) and whose eccentricity is 1/2.

Solution: The Descarta2D function Ellipse2D[point,point,e] constructs an ellipse given the two foci points and the eccentricity.

e1=Ellipse2D[Point2D[{-1,-1}],Point2D[{1,1}],1/2]

"ell_63.gif"

{Foci2D[e1], Eccentricity2D[e1]}

"ell_64.gif"

Ellipse from Focus and Directrix [Top]

Given the focus point "ell_65.gif", the directrix line L≡px+qy+r=0, and the eccentricity, 0<e<1, of an ellipse we wish to determine the standard equation of the ellipse. The rotation angle of the ellipse is the angle the line perpendicular to L makes with the +x-axis and is given by "ell_66.gif". The distance, d, from F to L is given by

"ell_67.gif"

It is clear from Table [ell:tbl01] that the distance from F to L is also given by d=a/e-ae. Solving for a (the length of the semi-major axis) yields

"ell_68.gif"

Table [ell:tbl01] shows that the eccentricity, e, is related to the lengths of the semi-major and semi-minor axes, a and b, respectively, by

"ell_69.gif"

Solving this equation for b yields

"ell_70.gif"

Table [ell:tbl01] reveals that the distance from the focus F to the center C(h,k) is given by ae. If F' is the projection of F onto L, then we can find the center point C of the ellipse by offsetting F in the direction from F to F' a distance -ae. This computation is easily accomplished using Descarta2D and is provided in the exploration ellfd.html. The resulting defining constants of the ellipse are given by

"ell_71.gif"

"ell_72.gif"

where

"ell_73.gif"

Example: Find the ellipse whose focus point is (3,2), directrix line x-y+2=0 and eccentricity is 1/4.

Solution: The Descarta2D function Ellipse2D[point,line,e] constructs an ellipse for the focus, directrix and eccentricity.

e1=Ellipse2D[p1=Point2D[{3,2}],l2=Line2D[1,-1,2],1/4]

"ell_74.gif"

{Foci2D[e1],
Directrices2D[e1],
Eccentricity2D[e1]} //Simplify

"ell_75.gif"

Parametric Equations [Top]

The parametric equations for a standard ellipse

"ell_76.gif"

are very similar to those of a circle, with the exception that the radius is replaced by either the length of the semi-major axis, a, or the semi-minor axis, b. The appropriate equations are

x=h+acosθ   and   y=k+bsinθ

where (h,k) is the center of the ellipse, a and b are the lengths of the semi-major and semi-minor axes, respectively, and parameter values in the range 0≤t<2π generate a complete curve. The validity of these equations can be verified by direct substitution.

Example: Plot 16 points on the ellipse

"ell_77.gif"

at equally spaced parameter values.

Solution: The Descarta2D function Ellipse2D[{h,k},a,b,θ][t] returns the coordinates of a point at parameter value t on the ellipse.

e1=Ellipse2D[{0,0},4,2,0];
pts=Map[Point2D[e1[2*Pi*#/16]]&,Range[0,15]];
Sketch2D[{e1,pts}]

"ell_78.gif"

As with the circle, a pair of rational equations may be used as the parametric equations for an ellipse. The ellipse

"ell_79.gif"

has the parametric equations

"ell_80.gif"

Values of t in the range 0≤t≤1 generate coordinates on the ellipse in the first quadrant. The point (-a,0), which is on the ellipse, cannot be generated using these equations.

Example: Plot the ellipse "ell_81.gif" using the rational parametric equations in the parameter range -10≤t≤10.

Solution: The Mathematica function ParametricPlot plots curves defined by parametric equations.

Clear[t];
ParametricPlot[{2*(1-t^2)/(1+t^2),2*1*t/(1+t^2)},
               {t,-10,10},AspectRatio->Automatic]

"ell_82.gif"

Explorations [Top]

Length of Ellipse Focal Chord. [elllen.html]

Prove that the length of the focal chord of an ellipse is "ell_83.gif", where a is the length of the semi-major axis and b is the length of the semi-minor axis.

Sum of Focal Distances of an Ellipse. [ellips2a.html]

Show that the sum of the distances from the two foci to any point on an ellipse is 2a, where a is the length of the semi-major axis.

Ellipse from Focus and Directrix. [ellfd.html]

Show that the ellipse with focus "ell_84.gif", directrix line L≡px+qy+r=0 and eccentricity, 0<e<1, is defined by the constants

"ell_85.gif"

"ell_86.gif"

where

"ell_87.gif"

Focus of Ellipse is Pole of Directrix. [elfocdir.html]

Show that the focus of an ellipse is the pole of the corresponding directrix.

Ellipse Locus, Distance from Two Lines. [elldist.html]

A point moves so that the sum of the squares of its distances from two intersecting straight lines is a constant. Prove that its locus is an ellipse.

Similar Ellipses. [ellsim.html]

All ellipses of equal eccentricity are essentially similar in that by a proper choice of scales (and axes) they can be made to coincide. Show this property is true for two ellipses of equal eccentricity centered at the origin.

Polar Equation of an Ellipse [polarell.html]

Show that the polar equation of an ellipse with a horizontal major axis and centered at (0,0) is given by

"ell_88.gif"

where a and b are the lengths of the semi-major and semi-minor axes, respectively.

Apoapsis and Periapsis of an Ellipse. [ellrad.html]

Show that the greatest (apoapsis) and least (periapsis) radial distance of a point on an ellipse as measured from a focus point is given by r=a(1+e) and r=a(1-e), respectively, where e is the eccentricity and a is the length of the semi-major axis of the ellipse.


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