(11)  Graphs and Ellipses

The laws of orbital motion are mathematical, and one cannot explore them without some mathematics. The math used here is rather elementary; if you need a refresher click here. Otherwise, you can just skip the equations and follow the narrative.

The Mathematical Description of a Curve

As already noted, the cartesian system labels any point in a plane (e. g. on a flat sheet of paper) by a pair of numbers (x,y), its distance from two perpendicular axes. These numbers are known as the "coordinates" of the point.

A line in the plane--straight or curved--contains many points, each with its own (x,y) coordinates. Often there exists a formula ("equation") which connects x and y: for instance, straight lines have a relationship

y = ax + b

where any pair of numbers (a,b), positive, negative or zero, gives some straight line. The plot of a line given by one of such relationship (or indeed by any relationship--even pure observation, e.g. temperature against time--is known as a graph. More complicated relationships give graphs that are curves: for instance

y = ax²

gives a parabola, with a any number. Usually (though not always) y is isolated, so that the formula has the form

y = f(x)

where f(x) stands for "any expression involving x" or in mathematical terms, a "function of x." The curves drawn here are the straight line y = -(2/3)x + 2 and the parabola y = x². A list of some of their points follows.

The Equation of a Circle

Such a form makes it very easy to find points of the graph. All you have to do is choose x, calculate f(x) (= some given expression involving x) and out comes the corresponding value of y.

However, any equation involving x and y can be used as the property shared by all points of the graph. The main difference is that with more complicated equations, after x is chosen, finding the corresponding y requires extra work, (and sometimes it is easier to choose y and find x).

Perhaps the best-known graph of this kind is a circle of radius R, whose equation is

Draw a circle of radius R centered at the origin O of a system of (x,y) axes . Given any point P on the circle with specified values of (x,y), draw a perpendicular line from P to point A on the x-axis. Then

Here x and/or y may be negative, if they are to the left of the y-axis or below the x-axis, but regardless of sign, x² and y² are both always positive. Since the triangle OAP has a 90° angle in it, by the theorem of Pythagoras, for any choice of P, the relation below always holds:

Since this can also be written

The equation of the circle is satisfied by any point located on it. For instance, if the graph is defined by the equation:

this equation is satisfied by all the points listed below:

The Equation of an Ellipse

The equation of an ellipse is a small modification of this:

where (a,b) are two given numbers, for example (8,4). What would such a graph look like? Near the x axis, y is very small and the equation comes close to

From which

also comes close to x² = a² in this region. In exactly the same way you can show that near the y-axis, where x is small, the graph cuts the axis at y=±b and its shape there resembles that of a circle of radius b.

An example

We already know that it cuts the axes at x=±8 and at y=±4. Let us now add a few points:

Substract 1/4 from both sides

Take roots (marked here by √) and retain only 3-4 figures:

from which  x = 6.93 within resonable accuracy.

Substract 9/16 from both sides

Take square roots (to an accuracy of 3-4 figures):

from which, approximately,  x = 5.29

Again, either sign can be attached to x and y. We get 12 points, enough for a crude graph:

(If you first wish to learn or review your trigonometry, go to
Trigonometry--what is it good for? and the sections that follow.)