Conic sections are curves that are described by equations that are special cases of the general quadratic equation:

Ax² + Bxy + Cy² + Dx + Ey + F = 0

Among other things, these curves describe the motions of bodies in gravitational fields.

A circle describes an object moving about a point from which a central force is being exerted on it, such as a ball being swung on a string, or a satellite orbiting the earth. A satellite given greater energy than that required for a circular orbit will take the path of an ellipse, which also describes the shape of the orbits of planets about the sun (which, of course, are also subject to a central gravitational force from the sun). A parabola describes the motion of a falling body (or one that has been shot upwards) that has some horizontal velocity (not sufficient horizontal (or upwards) velocity, though, to send it into orbit). This kind of motion is referred to as projectile motion. The circular and elliptical orbits described above are bound orbits. That is, the satellite, for example, cannot escape the earth, and it keeps to its orbit. Given sufficient energy to escape the earth, the satellite would follow the path of a hyperbola.

The parabola also has the interesting property that rays entering parallel to its central axis all intersect it at such an angle to the tangent at the intersection that all the rays leaving the parabola from these intersections at the same angles to the tangents meet at a common point (the focus; vide infra). Put another way, if one has a surface whose shape corresponds to a rotated parabola – a paraboloid – rays entering parallel to its central axis will all be reflected to a common point (the focus). Conversely, rays emitted from a source placed at the focus are reflected parallel to the axis, across the entire opening of the paraboloid. Parabolic mirrors are widely used in various types of telescopes, and in satellite television dishes and parabolic microphones.

The hyperbola also describes the lines of constructive interference and destructive interference set up by two fixed point sources emitting waves of the same frequency and amplitude.

These curves are known as conic sections, because they constitute the intersections of a right circular cone and a plane that cuts through the cone. We can define the cone by its central axis and the angle, β, that the surface of the cone makes with the axis. We must keep in mind that this cone also has above it an identical cone in mirror image, which the plane may also intersect if it cuts the cone at an appropriate angle. If we call the angle that the plane makes with respect to the central axis of the cone α, we obtain the following sections:

1) If α = 90°, the curve is a circle, shown second from right in the right-hand photograph.

A circle is the locus of all points in a plane that are equidistant from a given point. Its equation is: (x - h)² + (y - k)² = r², where h and k are the x,y coordinates of the center of the circle, and r is the radius.

2) If β < α < 90°, which is the case for the sample intersection shown in the left-hand photograph, then we obtain an ellipse, second from left in the right-hand photograph. (Note that the major axis of the ellipse points from front to back, and the minor axis goes from left to right; the ellipse is tilted toward the back by the angle shown in the photographs.)

An ellipse is the locus of all points in a plane, the sum of whose distances from two fixed points is constant. Its equation is: x²/a² + y²/b² = 1, where a and b are constants. The two fixed points are called foci, and they always lie on the major (long) axis of the ellipse. The larger of a and b gives the length of the semimajor axis, and the smaller gives the length of the semiminor axis. Taking a as the semimajor axis, the sum of the distances from any point on the ellipse to the two foci is 2a. If we take the intercept of the ellipse at the semiminor axis, we then have a right triangle with the semiminor axis, b, the hypotenuse from the intercept to the focus, and the base from the focus to the center of the ellipse. The length of the hypotenuse is merely a (the same as the distance to the other focus), so the square of the distance, c, from the center to the focus is c² = a² - b², or c = √(a² - b²).

The eccentricity of the ellipse, that is, how flat it is, or how much it deviates from being a circle, is defined as e = c/a. If c = 0, the two foci coincide, and the ellipse is a circle. At the other extreme, c = a, and the ellipse flattens to a straight line segment between the foci.

All of the planets in the solar system follow elliptical orbits about the sun, with the sun at one focus and nothing at the other focus. With the exceptions of the orbits of Mercury and Pluto (which is no longer classified as a planet), whose eccentricities are 0.206 and 0.249, respectively, the orbits the planets are nearly circular. The one with the greatest eccentricity, 0.093, is that of Mars, and the least eccentric is that of Venus, with an eccentricity of 0.007. Earth’s orbit has an eccentricity of 0.017. Because the orbits of the planets are elliptical, as a planet travels around the sun its distance from the sun varies. The point at which the planet is closest to the sun is called its perihelion, and when the planet is farthest from the sun it is at its aphelion. In terms of the semimajor axis, a, and eccentricity, e, (and c, the distance from a focus to the center of the ellipse) at perihelion, the planet is at a distance a (1 - e) (or a - c) from the sun, and at aphelion, it is at a distance of a (1 + e) (or a + c) from the sun.

Johannes Kepler determined that orbits of the planets were as described above, by carefully studying the copious data compiled by Tycho Brahe from decades of observation and very accurate measurements of the positions of the planets and various stars. This constituted a refinement of the heliocentric model of the solar system proposed earlier by Copernicus, in which the planets traveled in circular orbits about the sun. Both models provided much simpler planetary orbits than those of the geocentric model proposed even earlier by Ptolemy, in which the sun and all the planets revolved around the earth, and which necessitated the use of epicycles – motion of the planets in small circles superimposed on their larger orbital motion. Because Copernicus’s model used circular orbits, it also required some use of epicycles, but the orbits were far simpler than those of the geocentric model. Kepler’s model completely eliminated the need for epicycles. From his analysis of Tycho Brahe’s data, Kepler arrived at three laws. The first law is that the orbits of the planets are ellipses, with the sun at one focus. The second law is that the line that connects the sun to any planet sweeps out equal areas of the ellipse in equal time intervals. Planets move faster when they are closer to the sun, and more slowly when they are farther from the sun. The third law is that the square of a planet’s orbital period is proportional to the cube of the semimajor axis of its orbit, essentially its average distance from the sun. An astronomical unit (A.U.) is the semimajor axis of Earth’s orbit around the sun. It turns out that if we use Earth years and A.U., the proportionality constant equals one, and if we call the orbital period p and the semimajor axis of the orbit a, we can write Kepler’s third law as p² = a³.

3) If α = β, we obtain a parabola, shown at right in the right-hand photograph.

A parabola is the locus of all points in a plane that are equidistant from a given point and a given line. The point is called the focus, and the line is called the directrix (so called because it determines the direction in which the parabola opens). If we choose a point, (0,p) on the y-axis as the focus, and the directrix as a line parallel to the x-axis going through the point (0,-p), then the equation that describes the parabola is x² = 4py. This parabola is centered about the y-axis and opens upwards. If we change this to x² = -4py, the parabola opens downwards. If we exchange x and y, that is, y² = 4px, we obtain a parabola centered about the x-axis and opening rightward, and with y² = -4px, the parabola opens leftward. The vertex of the parabola is the intersection of the parabola with its axis of symmetry, or just the axis of the parabola. This point is, of course, equidistant from the focus and the directrix. In the preceding examples, the vertex is at the origin. If we move the vertex to a point (h,k), the preceding equations take the forms (x - h)² = 4p(y - k), (x - h)² = -4p(y - k), (y - k )² = 4p(x - h) and (y -k)² = -4p(x - h). For the first two of these, the axis is the line x = h. For the first, the focus is the point on the axis p units above the vertex, and the directrix is the line p units below the vertex and perpendicular to the axis. For the second, the focus is below the vertex and the directrix is above it, both p units away from it. For the second two, the axis is the line y = k. For the first of these, the focus is the point on the axis p units to the right of the vertex, and the directrix is the vertical line p units to the left of the vertex. For the second, the focus is to the left of the vertex, and the directrix is to the right of it, both p units away from it. Any equation that is quadratic in one coordinate and linear in the other describes a parabola. If such an equation has a cross term, it indicates that the axis and directrix of the parabola are tilted with respect to the coordinate axes.

4) If 0 ≤ α < β, the plane intersects both cones, and we have a hyperbola, one branch of which is shown at left in the right-hand photograph.

A hyperbola is the locus of all points in a plane, the difference of whose distances from a fixed point is constant. Its equation is x²/a² - y²/b² = 1, where a and b are constants.

The equations given above for the ellipse, parabola and hyperbola do not include the cross term (i.e., B = 0). If B is nonzero, the resulting conic section is tilted with respect to the x- and y-axes. One can always find an angle through which to rotate the coordinate axes to eliminate this cross term.

One can characterize all conic sections in terms of a focus (fixed point) and a directrix (fixed line), such that the ratio of the distance of a point on the curve from the focus, call it PF, to its distance from the line, call it PD, is a constant. This distance ratio is e, the eccentricity. If e = 0, the conic section is a circle. If 0 < e < 1, it is an ellipse. If e = 1, it is a parabola, and if e > 1, it is a hyperbola.

You can find good discussions of conic sections here and here. You can also find similar material in any good calculus text that includes analytic geometry.

References:

1. Chaisson, Eric and McMillan, Steve. Astronomy Today (Upper Saddle River, New Jersey: Prentice-Hall, 1999)
2. Resnick, Robert and Halliday, David. Physics, Part One (New York: John Wiley and Sons, 1977)
3. Thomas, George B., Jr. Calculus and Analytic Geometry (Reading, Massachusetts: Addison-Wesley Publishing Company, Inc., 1972)