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Ellipse
Definition of the Ellipse
To read how the ellipse got its name, and what it means, see Parabola. That page also contains some background information on
conic sections and other topics that also applies to ellipses, that won't be repeated here. Finding the arc length of an ellipse, which
introduces elliptic integrals, and Jacobian elliptic functions, are treated in their own articles.
An oval is generally regarded as any ovum (egg)-shaped smooth, convex closed curve. Convex means that any chord connecting
two points of the curve lies completely within the curve, and smooth means that the curvature does not change rapidly at any point.
The ellipse is a typical oval, but a very particular one with a shape that is regular and can be exactly specified. It has two diameters
at right angles that are lines of symmetry. It's best to reserve the word ellipse for real ellipses, and to call others ovals. A diameter
is any chord through the center of the ellipse. The diameters that are lines of symmetry are called the major axis 2a, and the minor
axis 2b, where a > b. If a = b, we have the very special ellipse, the circle, which has enough special properties that it should be
distinguished from an ellipse, though, of course, it has all the properties of an ellipse in addition to its own remarkable properties.
A vertex of a curve is a point of maximum or minimum radius of curvature. An ellipse has vertices at the ends of the major axis
(minimum) and at the ends of the minor axis (maximum).
An ellipse can be represented parametrically by the equations x = a cos θ and y =
b sin θ, where x and y are the rectangular coordinates of any point on the ellipse,
and the parameter θ is the angle at the center measured from the x-axis
anticlockwise. This is one method of drawing an ellipse, called the concentric
circle method, shown at the right. The larger circle, especially, is often of help in
working with ellipses. The ellipse is just this auxiliary circle with its ordinates
shrunk in the ratio b/a = √(1 - e2). The constant e in this expression is the
eccentricity of the ellipse (not the base of natural logs!), which we shall soon
define.
An ellipse is the curve described implicitly by an equation of the second degree
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 when the discriminant B2 - 4AC is less than
zero. The standard form of the equation of an ellipse is (x/a)2 + (y/b)2 = 1, where
a and b are the lengths of the axes.
The polar equation of an ellipse is shown
at the left. The θ in this equation should not be confused with the parameter θ in the
parametric equation. In celestial mechanics, the θ in the polar equation is called the true
anomaly (sometimes denoted by w), while the parameter is called the eccentric anomaly
(sometimes denoted by E). The two constants in the polar equation are the semi-latus rectum
p and the eccentricity e. The origin is a focus F of the ellipse. There is a second focus F'
symmetrically located on the axis. The point P at which r is a minimum is called perihelion
in an orbit about the sun, while A is the aphelion. Hence, 2a = p/(1 + e) + p/(1 - e) = 2p/(1 e2), relating p to the semi-major axis a. p is, of course, the radius when θ = 90°.
Now let c be the distance from the center to either
of the foci. Then c = a - p/(1 + e) = ep/(1 - e2) = ea. This is the clearest definition of the
eccentricity: e = c/a. We see that e < 1, and that e = 0 gives a circle. These things are
illustrated in the diagram at the right. Note especially the right triangle with legs b and c,
and hypotenuse a. From this triangle, we can prove that b = a √(1 - e2), which we claimed
above. Below the ellipse is shown the canonical equation of an ellipse, which includes the
lengths a and b. Two parameters are necessary to specify an ellipse, either a, b or p, e for
example.
Yet another way to specify an ellipse is that it is
the locus of points the sum of whose distances
from two given points (the foci) is constant. The two parameters in this case are the
distance between the foci, 2c, and the sum of the radii, 2a. Then a > c, and e = c/a is less
than unity. We'll show below that each of the definitions we have proposed gives the same
curve. Since the sum of the distances to the foci is constant, with waves the phase
difference will be the same along any path, so that waves emitted from one focus will be
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focussed at the other. In ray optics, this is the same as the law of reflection, as shown in
the sketch. Therefore, a normal to the ellipse is found by bisecting the angle between the radii to F and F', and a tangent is
perpendicular to the normal. The radius of curvature of the ellipse at any point is R = (rr')3/2/ab, where r and r' are the distances to
the two foci. At the end of a major axis, R = b2/a, and at the end of a minor axis, R = a2/b.
Any chord through the center, such as AB in the diagram at the right, specifies a family of
parallel chords that includes the tangents of the same slope. In fact, the mid-points of all
these chords define another chord CD, that is the diameter conjugate to AB. This gives
another method of drawing tangents to the ellipse. The tangents corresponding to
conjugate diameters "box" the ellipse in a parallelogram, which becomes a rectangle when
the conjugate diameters are the major and minor axes. Properties that hold relative to the
perpendicular axes also hold with respect to the conjugate diameters.
Suppose an ellipse is given, and the major and minor
axes are to be found. They can be measured
accurately if their directions are determined. The
figure at the left shows one way of doing this. The first thing is to find the center O of the
ellipse. This can be done by bisecting parallel chords, such as AB at J, and CD at K, and
joining the points J and K. Another set of parallel chords, such as EF and GH are also
bisected at L and M, and the points joined. The centre O is at the intersection of JK and LM.
With the centre known, draw a circle of any radius that cuts the ellipse at P, Q and R. The
lines PQ and QR are the directions of the major and minor axes. They are perpendicular
because they are drawn in a semicircle. If a diameter of the ellipse is known, the midpoint of
the diameter is at the centre O of the ellipse, and a circle of diameter equal to the length of
this diameter determines the points P, Q and R.
The ellipse is one of the conic sections, the intersection of a right circular cone with a cutting
plane, as shown in the diagram at the right. A plane perpendicular to the axis gives a circle, while
a plane parallel to a generator of the cone gives a parabola. Apollonius used this property to study
the ellipse, but it is very difficult to analyze in this way, so that the conic section property is of
little use to us, although it is remarkable. A point on the ellipse is determined as the intersection
of the generator o'd' with the cutting plane in the front view. Point d in the top view was projected
to point d' in the front view to specify the generator, and then point c' was projected back to point
c. Since this point is known in two views, it may be projected to point c" in an auxiliary view that
shows the ellipse in true size. Although the top view in this case looks like an eccentric circle, the
auxiliary view gives a good ellipse. The reader may want to make a drawing and determine
enough points on the ellipse to draw it.
The focal definition of an ellipse is that an ellipse is the
locus of points the ratio of whose distances from a fixed
line, the directrix, to a fixed point, the focus, is a constant
e, called the eccentricity. The property that the sum of the
radii from the foci to a point on the ellipse is constant is easily derived from the focal
definition, as shown in the figure. F and F' are the foci, and the directrices pass through D
and D'. From the focal definition, r'/h' = r" / h" = e, so that r' + r" = e(h' + h") = 2eh, which
is a constant. By considering a vertex V, we find that r' + r" = 2a, so a = eh. The directrices
of a circle are at infinity, since e = 0.
The parameters of an ellipse are shown at the right. If we start with the focal distance c,
then we may choose either the semimajor axis a, the semiminor axis b, or the eccentricity e.
If we choose a, then the ellipse can be drawn using a loop of length 2(a + c). If b is chosen
instead, then a = √(b2 + c2). Finally, if e is chosen, a = c/e. It is usually not as convenient to
use the directrix to construct the ellipse, but points on the ellipse are easily found by
intersecting a vertical line a distance d from the directrix and a radius of ed from the focus.
The area of an ellipse is πab = πa2 √(1 - e2), which we easily see is true because of the
vertical compression of the ellipse relative to the auxiliary circle. The length L of the
circumference of an ellipse is more difficult to determine. Calculus gives L = 4aE(e), where a is the semimajor axis and E(e) is the
complete elliptic integral of the second kind for the eccentricity e = sin φ (tables are usually given in terms of φ instead of e). E(0)
= π/2, and E(1) = 1, corresponding to the limits of a circle, b = a, and a straight line, b = 0. An approximate expression, for e not
too close to 1, is L = π[(3/2)(a + b) + √(ab)].
The curves known as Cassini's Ovals may be ovals, but are not ellipses. These are the loci of points the product of whose distances
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from two foci a distance 2c apart is a constant, or r1r2 = a2. When a > c√2, the curves resemble ellipses, but have none of its useful
properties. As a becomes smaller, a waist forms and becomes more and more pronounced, until when a = c, we have the lemniscate
of Bernoulli, which looks like the symbol for infinity. When a is still smaller, we get two ovals, one around each focus. These ovals
are fourth-order curves, in contrast to ellipses, which are second-order. They are curiosities more than useful tools.
Ellipses and Drawing
Ellipses are useful in drawing because they are perspective views of circles. However, the perspective projections may be difficult
to draw, because major and minor axes do not project as major and minor axes, centres do not project to centres (in general),
concentric circles do not project as concentric ellipses, and so on. Ellipses may be drawn from the definitions above by calculating
and plotting points, or directly by precise or approximate constructions. The concentric-circle method introduced above is one
practial method for constructing an accurate ellipse when the axes are known.
A construction based on the focal property may be used for laying out small ellipses in the field, called gardeners' ellipses. Place
two stakes in the ground with tacks in their tops marking the desired foci. Then make a loop of stout cord of length 2a + 2c, and
loop it around the tacks. Taking something like a chaining pin or pointed metal rod, put it in the loop and draw the loop taut.
Everywhere the pin may be will be a point on the ellipse, and as many points as desired may be marked. This method works quite
well and is rapid. Stakes and string make it possible to lay out many decorative curves: circles, ellipses, parabolas and spirals, as
the constructors of corn circles know.
The simplest way to draw an ellipse on the drawing board is by using a trammel, as shown
at the right. This is simply a straight strip of cardboard, with distances o-a = a and o-b = b
marked on it. The perpendicular axes are drawn where the ellipse is required, and the
position of point "o" when the other points are carefully held on the axes is a point on the
ellipse. There are machines for drawing ellipses based on this principle, but they are
expensive and probably no longer manufactured. The trammel requires care, but is dead
easy to use.
A standard method is called the parallelogram
method, and is applicable to conjugate diameters as well as to the usual perpendicular axes.
OE and EA are divided into N equal intervals. In the figure, N = 2 for simplicity. Then draw
CB and intersect with DG extended to determine a point P on the ellipse. A similar method
can be used to draw a parabola--it is all in how the intersecting lines are drawn. Note that we
could draw circles with this method or the trammel, but there are easier ways!
The theory behind the parallelogram method can be found from
the diagram at the right. The point P on the ellipse is the
intersection of lines AB and CD. s is a parameter that runs from 0 to 1 as P goes from the end of the
minor axis to the end of the major axis. Line AB has the equation y = b - sbx/a, while line CD has the
equation x = sa(b + y)/b. Eliminating s between these two equations, we find quite readily that x2/a2 +
y2/b2 = 1, which is the equation of an ellipse with semimajor axis a and semiminor axis b.
If you have the major minor axes of an ellipse, it is easy to find the foci by swinging an arc of radius
equal to the semimajor axis from the end of a minor axis. This arc will cut the major axis at the focal
points. Conversely, if you know the foci and the major axis, the intersection of arcs drawn from the
foci with radius equal to the semimajor axis will determine the ends of the minor axis.
The above methods all make accurate ellipses. It is much more convenient to represent an
ellipse by circular arcs that can be drawn with a compass than to connect points
laboriously determined. The simplest case is shown in the figure at the left, called a threecenter arch. To a draftsman, it is a "four-center ellipse." The centers are the symmetricallyplaced D, C, D' and C'. On the line AB joining the ends of the major and minor axes, lay
off distance BM = a - b, the difference in the semiaxes. Now draw the perpendicular
bisector of the remainder AM, the line L, which determines the centers D and C, as well as
the point G where the two arcs meet. Now the symmetrical points D' and C' can be laid
out. Arcs of radii DG and CG are then drawn to complete the arch or ellipse. This method
is excellent for representing ellipses on drawings, but may be a little crude for an actual
arch, where the approximation may give an uneasy feeling. It would probably be better in
any case to calculate an exact ellipse and lay it out by coordinates for the actual structure,
while the three-centered approximate ellipse will always do for a drawing.
Stevens' Method gives a somewhat better three-centred ellipse useful in
isometric projection. In the figure at the right, an ellipse is to be drawn in the
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rhombus ABCD, which is the isometric projection of a square. Using two
triangles, first find the tangent point E, and draw an arc of radius R equal to ED, cutting the minor axis at Q. Do the same thing
with C as centre. With radius OQ, draw an arc locating points P and P' on the major axis. Now draw PD and extend it to the arc of
radius R. This is not shown to avoid confusing the diagram. Now, draw arcs of radius r equal to the distance from P to the curve at
each end. They will be tangent to the arcs of radius R, so the approximate ellipse has been constructed.
A five-centred arch gives a better approximation by using three different radii, but is a little
complicated to lay out. Here is the method: refer to the figure at the left. Start with the box
AFDO with width equal to the half-span and height equal to the rise. Draw AD, and then
from F draw a perpendicular to this line, which intersects the minor axis at H. Make OK
equal to OD, and then draw a circle on AK as a diameter (Q is not the center of the line
AK). Now lay off OM equal to LD, and draw an arc with center at H and radius HM. You
will not have the point N yet, but be patient. From A, lay off AQ equal to OL, and AP equal
to half of AQ. P should be on the line FH. Draw an arc with center P and radius PQ,
intersecting the arc through M at point N. Now, P, N and H are the centers of the arcs
defining one-quarter of the approximate ellipse. The sectors are shaded to make this clear,
and to show that the arcs are tangent to each other. Find the centers of the remaining arcs
by symmetry.
For accurate drawing, it is useful to be able to find the directions of
tangents to an ellipse, and points of tangency. Tangents to a circle
are perpendicular to the radius at the point of tangency, so it is easy
to draw a tangent at a given point on a circle, or to construct the
tangent from an external point. The principles of two methods for
drawing tangents to an ellipse are shown in the figure at the right.
The method at (a) uses the auxiliary circle. First, the point S on the
auxilary circle corresponding to the point of tangency P is found,
and a tangent to the auxiliary circle is constructed, cutting the axis
at point T. The desired tangent is the line TP. This method uses the
vertical compression that turns the auxiliary circle into the ellipse.
Method (b) uses the focal properties of the ellipse. The focal radii from P to the foci F and F' are drawn, and the external angle is
bisected. The bisector is the desired tangent. This direction is perpendicular to the bisector of the internal angle between the focal
radii.
The fact that the tangent is perpendicular to the bisector of the angle between the focal radii at P is
not difficult to prove. In the figure at the right, the neighborhood of a point A on the ellipse is shown.
Suppose B is a neighboring point, so close that the directions to F and F' are not sensibly changed.
For B to be a neighboring point on the ellipse, the sum of the focal radii must remain unchanged (and
equal to the major axis). In the direction shown, it is clear that the lengthening BC of the focal radius
to F' is equal to the shortening AD of the focal radius to F, so that their sum remains constant, and B
is also a point on the ellipse, or, in this approximation, on the tangent from A. The tangent, of course,
is the limit of the chord through BA as B approaches A.
The method for drawing a tangent from an external point P to an ellipse is shown at the
right. First, draw a circle with centre at P and radius PF. The intersection of this circle
with an arc of a radius equal to the major axis (2a) is point E. Point R is the intersection of
radius F'E and the ellipse. In the triangles PER and PFR, corresponding sides are equal,
since FR + RF' = 2a = ER + RF', or FR = ER. PE and PF are equal by construction, while
the side PR is common. Therefore, PR bisects the external angle and is thus the tangent. A
similar construction gives the other tangent from point P.
The line PR also bisects the line FE. In the diagram, the triangles do not appear exactly congruent because the foci are not
accurately located. If you make a careful drawing, the triangles will be congruent, and PR will be an accurate perpendicular
bisector of FE. Since FE is perpendicular to the tangent, if we want a tangent with a certain direction (and there is no point P
given), a line is drawn in the perpendicular direction from F, and E is the intersection with an arc of length 2a from F. Now EF is
bisected, and the perpendicular bisector is the desired tangent. We can now find tangents at a point on the ellipse, from an external
point, and in a certain direction.
If the focal points F and F' approach one another, direction of the tangent is more and more restricted. In the limit as the ellipse
becomes a circle, F = F', and the tangent will be perpendicular to the radius, as we know from Euclid.
The drawing utilities in Windows draw an ellipse in a circumscribed rectangle, which made preparing the graphics for this page
quite easy, compared with the effort required for parabolas and hyperbolas, which are not easy to draw with the Windows routines.
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It is annoying not to be able to draw a circle from center and radius, as in real drafting programs, but this can be worked around by
putting the center of a square of side equal to the diameter at the desired center point. If the Shift key is held down in a graphics
program, it is only necessary to move the cursor along one diameter to get a circle, which makes it a little easier. A template of
three small crosses can be constructed and copied to wherever a circle is desired if you are drawing a number of circles of the same
diameter. One cross is the center, the other two the ends of a horizontal line equal to the diameter.
Ellipses and Orbits
The bound orbits under an inverse-square central force are ellipses, with the
attracting center in one focus. This fact alone would make ellipses important and
worthy of study. This is, in fact, Kepler's First Law, which he deduced for the
planet Mars, using Tycho Brahe's naked-eye observations. The orbital parameters
of the major planets are given in the table at the right. In this table, a is the semimajor axis in units of the earth's distance from the sun (the astronomical unit, A.U.),
e its eccentricity, and i the inclination of the plane of the orbit to the average plane
of the earth's orbit. It is remarkable that the planets revolve in the same direction,
and almost in the same plane. Only Pluto is errant, with its large inclination. All the
eccentricities are small, except for Mercury and Pluto. To the eye, all the orbits
would appear to be circles, though the sun would not be accurately at the centers of
the circles, quite obviously eccentric in the cases of Mercury, Mars and Pluto. In
the case of the solar system, the mass of the sun is so predominant, and the planets
so far apart, that to a first approximation they all revolve independently about a
fixed center. The sun is about 1000 times as massive as Jupiter, which is 4 times as
massive as Saturn. The sun is 328,900 times as as massive as the earth plus moon.
Planetary Orbits
Planet
a
e
i
Mercury 0.387
0.206 7° 00'
Venus
0.723
0.007 5° 24'
Earth
1.000
0.017 0° 00'
Mars
1.881
0.093 1° 51'
Jupiter
11.857 0.048 1° 18'
Saturn
29.42
0.056 2° 29'
Uranus
83.75
0.046 7° 00'
Neptune 163.72 0.009 1° 46'
Kepler's Second Law was that the radius vector of a planet swept out equal areas in equal
Pluto
248.0 0.249 17° 09'
times, or that the areal velocity was constant. Newton showed that this was a conseqence of
the constancy of the angular momentum of the planet, since the force of attraction had no sideways component, or was central. The
angular momentum is the product of the mass of the planet, the distance from the sun and the velocity normal to the radius. A
quantity proportional to the angular momentum is h = r(dθ/dt)2, which is twice the areal velocity. When the equation of the orbit is
determined from Newton's Laws, the parameter p = (h/k)2, where k is the Gaussian gravitational constant, equal to 0.01720209895
when the length unit is the A.U. and the time unit is the mean solar day. From the polar equation of the ellipse, we also find that p
= a (1 - e2), so we can express h in terms of the orbital constants.
Since the total area of the orbit is A = πab = πa2(1 - e2)1/2, the time for one revolution, the period P = A/(h/2) = 2πa3/2/k. Check
this result by applying it to the earth, where a = 1 and P = 365.25 days. You should get the value of k given above. In fact, this is
how k can be determined accurately, since its value depends only on the length of the year. This result expresses Kepler's Third
Law, that the square of the period is proportional to the cube of the semimajor axis. Check this for Jupiter, whose period is 11.86
years. The mean daily motion of a planet is n = 2π/P radians per day, or 360/P degrees per day. The problem of finding the true
anomaly (the angle of the radius vector) as a function of time is called Kepler's Problem. Its solution is not only practical, but quite
interesting.
To start, the areal velocity is constant, so the area that has been described by the radius from the moment of perihelion is
proportional to time. The ratio of this area to the total area of the orbit is equal to the ratio of the mean motion M in angle in the
same time to 2π, and M = n(t - T), where n is the mean daily motion, T the time of perihelion passage, and t the time. This is the
fundamental relation between uniform time t and the nonuniform dθ/dt.
In the diagram, the shaded area AFQ is the area swept out at some time t. As we
have explained above, M/2π = AFQ / πa2(1 - e2)1/2 = AFQ' / πa2, since the
ellipse is the squeezed auxiliary circle. Now area AFQ' is equal to the sector
ACQ' less the triangle CQ'F, with base ae and altitude a sin E, or Ea2/2 - (1/2)
ea2 sin E. When we divide by πa2 and multiply by 2π, we find that M = E - e
sin E, Kepler's Equation. This is a relation between the mean anomaly M, that
increases uniformly with time, and the eccentric anomaly E, which does not.
Unfortunately, it gives M directly in terms of E, not vice-versa, which would
have been more convenient.
From the figure, we can work out how r and w depend on E. r2 = FN2 + QN2.
FN = a cos E -ae and QN = a(1 - e2)1/2 sin E. Working through the algebra, we
find r = a (1 - e cos E) and tan(w/2) = [(1 + e)/(1 - e)]1/2 tan (E/2). This is
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simply another way to parameterize the equation of an ellipse. Our procedure is
to find M = n(t - T), then to find E from Kepler's Equation, and finally to compute r and w. With computers, solving Kepler's
Equation is easy. For small e, there are approximations that work well for the solar system.
Miscellaneous Ellipses
If you look at the bright patch cast by some small aperture in a tree or a hole in the window blind by the sun, you will see that it is
an ellipse. In this case, the aperture is the apex of a slender cone of light of about 32' of arc angle, and the patch is a section of that
cone. This is an unusual example of an ellipse that is really a conic section. In a partial eclipse, the patch assumes the crescent
shape of the eclipsed sun, as noted by Minnaert in his famous book. By measuring the elliptical patch, and knowing the cone angle,
the distance and height of the aperture can be determined.
There will be two points in an elliptical room where whispers at one point can be heard distinctly at the other. These are, of course,
the foci. Somewhat more common is the whispering gallery where sound is propagated along the walls of the gallery. A famous
one exists in St. Paul's Cathedral, London, that was explained by Lord Rayleigh in 1904 as not due to reflection from the dome
overhead, but to a waveguide effect.
An elliptical cylinder has two focal lines. A discharge tube with its axis along one of them can "pump" a laser along the other.
If an ellipse is rotated about one of its principal axes, a spheroid is the result. If it is rotated about the major axis, the spheroid is
prolate, while rotation about the minor axis makes it oblate. A more general figure has three orthogonal axes of different lengths a,
b and c, and can be represented by the equation x2/a2 + y2/b2 + z2/c2 = 1. This figure is called an ellipsoid, of which the spheroids
and the sphere are special cases. The volume is 4πabc/3, which can be specialized as required. These quadric surfaces have many
uses, but here we only consider their relations to the ellipse.
Suppose you see the reflection of a street lamp in a wire strung above the street. What determines the location of this bright patch?
Imagine the lamp and your eye as the foci of a prolate spheroid. Then, a ray from the lamp to any point on the spheroid will be
reflected into your eye. Therefore, the wire is tangent to the spheroid, which determines the location of the reflection. This
observation is mentioned in Minnaert. His shiny wires were the polished contact wires of streetcars.
The Roman arch was usually a circular arch. These arches were finely made without mortar, so they would last, and last they have.
The circular form is an esthetic choice, but it also gives a reliable and strong arch. Segmental arches were found pleasing to the
eye, and could be used where the increased lateral thrust was properly resisted by the abutments or neighboring arches. They also
gave a larger opening under the bridge without excessive height. Unlike Roman circular arches, damage to one span could cause
adjacent spans to collapse. A particularly elegant choice is the elliptical arch, which gives a feeling of lightness and grace to the
bridge, plus the practicality of a larger opening. I. K. Brunel built a particularly nice two-span elliptical brick arch over the Thames
at Maidenhead for the Great Western Railway around 1840, and it is still in service. It has a particularly small rise, and his critics
expected it to collapse as soon as the centering was removed. Telford built a beautiful elliptical arch over the Severn at Over, west
of Gloucester, that is also still standing. Modern arches will not last as long as the Roman ones, because mortar is more perishable
than stone or brick, and reinforced concrete weathers readily. Medieval builders had troubles with the crowns of their arches
collapsing because of indifferent workmanship, and invented the Gothic arch that made this less of a hazard. Three- and fivecentred arches were often used in bridges and buildings, but now there is little excuse for not using a precise ellipse.
Return to Mathematics Index
Composed by J. B. Calvert
Created 6 May 2002
Last revised 6 September 2005
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