Because I started a small series about astronomical algorithms and the magic of math in space, I think we need to cover an important prerequisite. In the series, I will talk a lot about ellipses (duh), I will move from the semi-axis majors, to the periapsis, to eccentricity, to ellipse’s center and ellipse’s foci. I am concerned that things can get more complicated than expected if the readers does not know many of the geometric properties of the ellipse. For this reason, I put here this vade mecum on the ellipse geometry. A summary with all the basic points and lengths. A place that I can link everywhere I need to refresh a definition.
This only scratch the surface of ellipse properties. But I think it is enough for what we need now. So let’s start from the beginning.
You can click on the ellipse elements to jump to the corresponding definitions below.
The Ellipse Definition
We all probably know the classic ellipse definition: an ellipse is the set of points for each of which the sum of its distances to two foci is a fixed number. If we call this two special point \( F_1\) and \( F_2\), then, the ellipse is the set of points \( P \) such that
$$ E = \\{P \mid |PF_2| +|PF_1 | = 2a \\} $$In the formula, \( a \) is the semi-major axis.
However, you probably know better the analytic formula of an ellipse in the Cartesian plane:
$$ \frac{x^2}{a^2}+\frac{y^2}{b^2}= 1 $$Where \( a\) is, again, the semi-major axis and \( b\) is the semi-minor axis.
Important Points
Center
The ellipse center is usually defined by \( C\). This is the mid-point between the two foci \( F_1\) and \( F_2\). As a consequence, one formula involving the center is the obvious
$$ C = \frac{F_1 + F_2}{2} $$For what orbital mechanic is concerned, we can choose ellipse’s center as the origin of the reference frame. However, in general, it is preferred to put the origin in the focus in which the main body is.
Foci
As we have seen before, foci are two points along the ellipse major axis such that the sum of the distances between any point on the ellipse and the two foci is equal to the major axis itself. Foci are two of the most important points in the ellipse. They are the ellipse equivalent of the circle’s center. They are the locus of major ellipse’s properties and, most important, in astromechanics one of the focus is the place in which we can find the celestial body to which the other body is orbiting around.
In case we only have \( a\) and \( b\) as defining parameters, we can find the distance between one focus and the center with:
$$ c = a^2 - b^2 $$In this formula \( c\) is also called linear eccentricity.
As I said before, there are many properties involving the ellipse focus. One of my favorite is the fact that any ray passing through one focus and “bouncing” on the ellipse ends passing through the other focus.
Periapsis and Apoapsis
These two points represent respectively the closest and the farthest away point to the focus \( F_1\). Obviously we can choose \( F_2\) as reference focus and get completely symmetric results. Periapsis \( P^-\) and apoapsis \( P^+ \) are very important in astromechanics where they took different names depending on the center of mass of the orbiting system. For instance, when we consider the Sun we call them “perihelion” and “aphelion”; when we consider the Earth we call them “perigee” and “apogee”.
As you can imagine, they are very important when we are dealing with orbits and planets because they are much easier to measure empirically. Moreover, they are enough to compute all the other important points and measures. In fact, periapsis and apoapsis are probably at the center of one of my favorite set of relations ever.
Let’s call \( r_{min} \) the distance between the focus and \( P^- \) and \( r_{max} \) the distance between the focus and \( P^+\).
If we take the arithmetic mean of periapsis and apoapsis we get the semi-major axis \( a\).
$$ a = \frac{1}{2}(r\_{max} + r\_{min}) $$If we take the geometric mean of periapsis and apoapsis we get the semi-minor axis \( b\).
$$ b = \sqrt{r\_{max} r\_{min}} $$If we take the harmonic mean of periapsis and apoapsis we get the semi-latum rectus \( p\).
$$ p = \frac{2}{\frac{1}{r\_{max}} + \frac{1}{r\_{min}}} $$I find amazing how three different kinds of averages of the same two values can be connected to three such important measures.
Finally, because we can express \( a\) and \( b\) in function of \( r_{min}\) and \( r_{max}\), we can find a very useful formula for the ellipse eccentricity.
$$ e = \frac{r\_{max}-r\_{min}}{r\_{max}+r\_{min}} $$Important Measures
The Semi-Major Axis
The semi-major axis \( a\) is half of the major-axis, the line passing through the two foci from one end of the ellipse to the other. It is one of the main parameters of the ellipse and it is best known for being the parameter dividing \( x\) in the analytic formula.
The major-axis is also the constant to which distances from the two foci sum up in the ellipse definition.
The Semi-Minor Axis
Take the ellipse center and trace a line passing through it perpendicular to the major-axis. The size of this segment is the minor-axis. Take half its value, and you get the semi-minor axis \( b\). This value is better known for being the parameter dividing \( y\) in the analytic formula.
Linear Eccentricity
The linear eccentricity \( c\) is the distance between the center \( C\) and one of the foci. With a bit of algebraic juggling, it is easy to see that this value can be derived by \( a\) and \( b\).
$$ c = a^2 - b^2 $$The ratio of \( c\) and \( a\) defines the ellipse eccentricity \( e\). But we will see that there is an easier way to compute \( e\) than passing through \( c\).
Semi-Latus Rectum
Take the minor axis and slide it until it passes through one of the foci. The segment enclosed by the ellipse is called latus rectum. Take half of it and you get the semi-latus rectum. This value can be computed from \( a\) and \( b\).
$$ p = \frac{b^2}{a} $$It is probably the less interesting measures. But it may come handy, so it is important to know.
Eccentricity
Finally, the eccentricity \( e\) is another extremely important parameter. Poorly speaking, the eccentricity is a value between 0 and 1 (not included) that represents how much the ellipse is “stretched”. More formally, it measures how much the ellipse deviates from being a circle.
It is formally defined as the ratio between the linear eccentricity and the semi-major axis.
The reason this value is so important, is that it encodes the “asymmetry” of the ellipse in a dense and beautiful number between 0 and 1. As a result, it allows us to switch between the element of any pair of measures easily. We can use \( e\) from going from \( a\) to \( b\). From \( b\) to \( p\). From \( p\) to \( r_{min}\). And more.
As you can imagine, there are too many ways to compute eccentricity starting from any two parameters. Here they are the most useful ones!
$$ e = \frac{r\_{max}-r\_{min}}{r\_{max}+r\_{min}} $$$$ e = \frac{c}{a} $$$$ e = \sqrt{1 - \frac{b^2}{a^2}} $$Conclusions
As I said, this is just a fast introduction to some definition we need as a starting point for talking about orbits and stuff. In the next article we will continue our journey into space talking about seasons and their procedural generation.