Putting Together the Mathematics So Far

We’ve established the relationship:

We also saw earlier that eccentricity can be defined as:

This gives us enough to interrelate the four quantities:

• a — semi-major axis
• b — semi-minor axis
• c — focal distance
• e — eccentricity

and, given any two, calculate the other two.

In particular, we can square the second equation, rearrange it to get c² by itself and then substitute into the first equation to get:

This can be rearranged to get eccentricity as a function of the ratio of semi-minor axis to semi-major axis:

or the semi-minor axis as a function of the semi-major axis and the eccentricity:

These equations are used at various points in the code behind all these visualisations.

It is common for the semi-major axis and eccentricity to be given for an orbital body and the other two values calculated from them.