Ellipse

Definition of the EllipseTo 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 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 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.