Orbital geometry

Figure 1 shows the geometry and characteristics of an elliptical orbit. F₁ and F₂ are called the focal points of the ellipse, a is called the semi-major axis, b is the semi-minor axis, and e is the eccentricity. For every orbit, the body being orbited is at one of the focal points of that orbit. For example, the sun is at a focal point of the Earth's orbit, and the Earth is at a focal point of the International Space Station's orbit. The distance between the center of the ellipse and either focal point is equal to the product of the semi-major axis and the eccentricity. For an ellipse, the eccentricity is between 0 and 1. For the special case where e is zero, the orbit geometry is a circle. A circle is an ellipse with a = b and both focal points in the same location.

Figure 1: Geometry of an Ellipse

Figure 2:

Figure 3:

For every ellipse, the sum of the distance between every point on the ellipse and each focal point is a constant value. This is illustrated in Figure 2 for a particular point. The following is true for an ellipse:
c₁+c₂=2a

Figure 3 introduces many important parameters for elliptical orbits. The perifocal frame is defined by i_e, i_p, and with the third axis, i_h, pointing out of the plane of the figure. Periapsis, designated by r_p, is defined as the point in the orbit closest to the orbited body. Apoapsis, designated by r_a, is defined as the point in the orbit farthest from from orbited body. There are specific terms that may be used in place of periapsis and apoapsis that are dependent on the body being orbited. For example, for orbits about the sun periapsis is called perihelion and apoapsis is called aphelion; for orbits about the earth periapsis is called perigee and apoapsis is called apogee.