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Ellipse

An ellipse is a curve that is the locus of all
points in the plane the sum
of whose distances and from two fixed points and (the foci) separated
by a distance of is a given positive
constant (Hilbert and Cohn-Vossen 1999, p. 2). This
results in the two-center bipolar
coordinate equation

(1)

where is the semimajor
axis and the origin of the
coordinate system is at one of the foci. The ellipse
is a conic
section and a Lissajous
curve. The corresponding parameter is known as the semiminor
axis.

An ellipse can be specified in Mathematica
using Circle[x, y, a, b].

If the endpoints of a segment are moved along two
intersecting lines, a fixed point on the segment (or on the line
that prolongs it) describes an arc of an ellipse. This is known as
the trammel construction of an ellipse (Eves 1969, p. 177).

The ellipse was first studied by Menaechmus,
investigated by Euclid , and named by Apollonius . The focus and conic
section directrix of an ellipse were considered by Pappus. In
1602, Kepler  believed that the orbit of Mars  was oval; he later
discovered that it was an ellipse with the Sun  at one focus. In fact,
Kepler introduced the word "focus" and
published his discovery in 1609. In 1705 Halley  showed that the comet  now named after him moved in an elliptical orbit around
the Sun  (MacTutor Archive). An ellipse rotated about its minor
axis gives an oblate
spheroid, while an ellipse rotated about its major axis gives a
prolate
spheroid.

A ray of light passing through a focus will pass
through the other focus after a single bounce (Hilbert and
Cohn-Vossen 1999, p. 3). Reflections not passing through a focus will be
tangent to a confocal hyperbola or
ellipse, depending on whether the ray passes between the foci or not.

Let an ellipse lie along the x-axis
and find the equation of the figure (◇) where and are at and . In Cartesian
coordinates,

(2)

Bring the second term to the right side and square
both sides,

(3)

Now solve for the square root
term and simplify

(4)

(5)

(6)

Square one final time to clear the remaining square root,

(7)

Grouping the terms then gives

(8)

which can be written in the simple form

(9)

Defining a new constant

(10)

puts the equation in the particularly simple form

(11)

The parameter is called the semiminor
axis by analogy with the parameter , which is called the semimajor
axis (assuming ). The fact that as defined above is actually the semiminor
axis is easily shown by letting and be equal. Then two right
triangles are produced, each with hypotenuse
, base , and height . Since the largest distance along the minor axis
will be achieved at this point, is indeed the semiminor
axis.

If, instead of being centered at (0, 0), the center of the
ellipse is at (, ), equation (◇) becomes

(12)

The ellipse can also be defined as the locus of points
whose distance from the focus is
proportional to the horizontal distance from a vertical line known
as the conic
section directrix, where the ratio is . Letting be the ratio and the distance from the center at which the
directrix lies, then in order for this to be true, it must hold at
the extremes of the major and minor axes, so

(13)

Solving gives

(14)

(15)

The focal
parameter of the ellipse is

(16)

(17)

(18)

where is a characteristic of the ellipse known as the eccentricity,
to be defined shortly.

An ellipse whose axes are parallel to the coordinate
axes is uniquely determined by any four non-concyclic points on it,
and the ellipse passing through the four points , , , and has equation

(19)

Like hyperbolas,
noncircular ellipses have two distinct foci and two
associated directrices,
each conic
section directrix being perpendicular
to the line joining the two foci (Eves 1965, p. 275).

In polar
coordinates, the angle measured from the center of the ellipse is
called the eccentric
angle. Writing for the distance of a point from the ellipse
center, the equation in polar
coordinates is just given by the usual

(20)

(21)

Here, the coordinates and are written with primes to distinguish them from
the more common polar coordinates for an ellipse, which are centered
on a focus. Plugging the polar equations into the Cartesian
equation (◇) and solving for gives

(22)

Define a new constant called the eccentricity
(where is the case of a circle) to
replace

(23)

from which it also follows from (◇) that

(24)

(25)

(26)

Therefore (◇) can be written as

(27)

(28)

If , then

(29)

so

(30)

Summarizing relationships among the parameters , , , and characterizing an ellipse,

(31)

(32)

(33)

The eccentricity
can therefore be interpreted as the position of the focus as a
fraction of the semimajor
axis.

If and are measured from a focus instead of from the center (as they commonly are in orbital mechanics ) then the equations of the ellipse are

(34)

(35)

and (◇) becomes

(36)

Clearing the denominators
gives

(37)

Substituting in gives

(38)

Plugging in (◇) and (◇) to re-express and in terms of and ,

(39)

Dividing by and simplifying gives

(40)

which can be solved for to obtain

(41)

The sign can be determined by requiring that must be positive. When
, (◇) becomes , but since is always positive, we
must take the negative sign,
so (◇) becomes

(42)

(43)

(44)

The distance from a focus to a point
with horizontal coordinate (where the origin is taken to lie at the center of
the ellipse) is found from

(45)

Plugging this into (◇) yields

(46)

(47)

In pedal
coordinates with the pedal point
at the focus, the
equation of the ellipse is

(48)

As can be seen from the Cartesian
equation for the ellipse, the curve can also be given by a
simple parametric form analogous to that of a circle, but with
the and coordinates having different scalings,

(49)

(50)

The unit tangent
vector of the ellipse so parameterized is

(51)

(52)

A sequence of normal and
tangent
vectors are plotted above for the ellipse.

The arc length of
the ellipse is

(53)

(54)

(55)

where is an incomplete elliptic
integral of the second kind with elliptic
modulus (the eccentricity). Again, note that is a parameter which does not have a direct
interpretation in terms of an angle. However,
the relationship between the polar angle from the ellipse center
and the parameter follows from

(56)

This function is illustrated above with shown as the solid curve and as the dashed, with . Care must be taken to make sure that the correct
branch of the inverse
tangent function is used. As can be seen, weaves back and forth around , with crossings occurring at multiples of . The curvature and
tangential
angle of the ellipse are given by

(57)

(58)

The entire perimeter
of the ellipse is given by setting (corresponding to ), which is equivalent to four times the length of
one of the ellipse's quadrants,

(59)

(60)

(61)

where is a complete
elliptic integral of the second kind with elliptic
modulus (the eccentricity). The perimeter can
be computed using the rapidly converging Gauss-Kummer
series as

(62)

(63)

(Sloane's A056981
and A056982),
where is a binomial
coefficient and

(64)

This can also be written analytically as

(65)

(66)

where is a hypergeometric
function, is a complete elliptic
integral of the first kind.

Approximations to the perimeter
include

(67)

(68)

(69)

where the last two are due to Ramanujan (1913-14), and
(◇) has a relative error of for small values of . The error surfaces are illustrated above for
these functions.

The maximum and minimum distances from the focus are called
the apoapsis and
periapsis,
and are given by

(70)

(71)

The area of an ellipse
may be found by direct integration

(72)

(73)

(74)

(75)

(76)

(77)

The area can also be
computed more simply by making the change of coordinates and from the elliptical region to the new region . Then the equation becomes

(78)

or , so is a circle of radius . Since

(79)

the Jacobian is

(80)

(81)

The area is therefore

(82)

(83)

as before. The area of an
arbitrary ellipse given by the quadratic
equation

(84)

is

(85)

The area of an ellipse
with semiaxes and with respect to a pedal point
is

(86)

The ellipse inscribed in
a given triangle and
tangent at its midpoints is
called the midpoint
ellipse. The locus of the
centers of the ellipses inscribed in
a triangle is
the interior of the medial
triangle. Newton  gave the solution to inscribing an ellipse in a convex quadrilateral
(Dörrie 1965, p. 217). The centers of the ellipses inscribed in
a quadrilateral
all lie on the straight line segment joining the midpoints of
the polygon
diagonals (Chakerian 1979, pp. 136-139).

The area of an ellipse
with barycentric
coordinates inscribed in
a triangle of
unit area is

(87)

(Chakerian 1979, pp. 142-145).

The locus of the apex
of a variable cone containing an
ellipse fixed in three-space is a hyperbola
through the foci of the
ellipse. In addition, the locus of the apex
of a cone containing
that hyperbola is
the original ellipse. Furthermore, the eccentricities
of the ellipse and hyperbola are
reciprocals. The locus of centers
of a Pappus
chain of circles is an
ellipse. Surprisingly, the locus of the end of a garage door mounted
on rollers along a vertical track but extending beyond the track is
a quadrant of an ellipse (Wells 1991, p. 66). (The envelope of
the door's positions is an astroid.)

SEE ALSO: Circle, Conic
Section, Eccentric
Anomaly, Eccentricity,
Elliptic
Cone, Ellipse
Tangent, Elliptic
Curve, Elliptic
Cylinder, Hyperbola, Lissajous
Curve, Midpoint
Ellipse, Parabola, Paraboloid,
Quadratic
Curve, Reflection
Property, Salmon's
Theorem, Steiner
Circumellipse. [Pages Linking Here]

REFERENCES:

Beyer, W. H. CRC
Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC
Press, pp. 126, 198-199, and 217, 1987.

Casey, J. "The Ellipse." Ch. 6 in A
Treatise on the Analytical Geometry of the Point, Line, Circle, and
Conic Sections, Containing an Account of Its Most Recent Extensions,
with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges,
Figgis, & Co., pp. 201-249, 1893.

Chakerian, G. D. "A Distorted View of Geometry."
Ch. 7 in Mathematical
Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc.
Amer., 1979.

Courant, R. and Robbins, H. What
Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd
ed. Oxford, England: Oxford University Press, p. 75, 1996.

Coxeter, H. S. M. "Conics" §8.4 in Introduction
to Geometry, 2nd ed. New York: Wiley, pp. 115-119, 1969.

Dörrie, H. 100
Great Problems of Elementary Mathematics: Their History and
Solutions. New York: Dover, 1965.

Eves, H. A
Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon,
1965.

Fukagawa, H. and Pedoe, D. "Ellipses," "Ellipses
and One Circle," "Ellipses and Two Circles," "Ellipses and Three
Circles," "Ellipses and Many Circles," "Ellipses and Triangles,"
"Ellipses and Quadrilaterals," "Ellipses, Circles, and Rectangles,"
and "Ellipses, Circles and Rhombuses." §5.1, 6.1-8.2 in Japanese
Temple Geometry Problems. Winnipeg, Manitoba, Canada:
Charles Babbage Research Foundation, pp. 50-68, 135-160, 1989.

Harris, J. W. and Stocker, H. "Ellipse." §3.8.7
in Handbook
of Mathematics and Computational Science. New York:
Springer-Verlag, p. 93, 1998.

Hilbert, D. and Cohn-Vossen, S. Geometry
and the Imagination. New York: Chelsea, pp. 2-3, 1999.

Kern, W. F. and Bland, J. R. Solid
Mensuration with Proofs, 2nd ed. New York: Wiley, p. 4,
1948.

Lawrence, J. D. A
Catalog of Special Plane Curves. New York: Dover, pp. 72-78,
1972.

Lockwood, E. H. "The Ellipse." Ch. 2 in A
Book of Curves. Cambridge, England: Cambridge University
Press, pp. 13-24, 1967.

MacTutor History of Mathematics Archive.
"Ellipse." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Ellipse.html.

Ramanujan, S. "Modular Equations and
Approximations to ." Quart. J. Pure. Appl. Math. 45,
350-372, 1913-1914.

Sloane, N. J. A. Sequences A056981
and A056982
in "The On-Line Encyclopedia of Integer Sequences."

Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry.
London: Penguin, pp. 63-67, 1991.

Yates, R. C. "Conics." A
Handbook on Curves and Their Properties. Ann Arbor, MI: J.
W. Edwards, pp. 36-56, 1952.

CITE THIS
AS:

Eric W. Weisstein. "Ellipse." From MathWorld--A Wolfram
Web Resource. http://mathworld.wolfram.com/Ellipse.html

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