The simple pair of formulas derived in problem (8)
is very powerful and useful. For instance, it can reproduce rotations in 3dimensional space by first rotating (x,y) by some angle α around the z axis,, giving the above result and producing (x',y'), while z remains the same, so that in the new system z' = z. Then we rotate (x',y',z') by an angle β around the x' axis, using the same formulas but with (y',z', β) in place of (x,y, α). Let the new variables be (x",y",z"). Then x' remains the same, i.e. x"=x', but (y',z') are replaced by
Suppose a satellite's position is given by its orbital elements. Knowing its semimajor axis a, eccentricity e and mean anomaly M, one can derive its true anomaly f (using appropriate formulas, not given in #12a) and from it the radial distance r
In cartesian coordinates (x',y',z'), with z' perpendicular to the orbital plane and x' along the major axis, the satellite's position is

But what are its actual celestial coordinates (x,y,z)? From the illustration shown here (taken from section #12a), (x,y,z) can be reached by conducting two simple rotations by angles (ω, i, Ω). To connect the two systems, the simple transformation derived above is applied, consecutivelyfirst a rotation around line N by an angle i giving an intermediate system (x",y",z"), after which z' = z" = z. And then another rotation by an angle ω+Ω around the shared z axis, bringing the x' axis, which originally passed the apogee P, into the celestial x axis. Special notations exist which simplify the job. Problems involving the orientation of satellites in 3dimensional space (the "attitude" of those satellites) are handled in a similar way. 
Back to #M11a Trigonometry Proficiency Drill
Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: stargaze("at" symbol)phy6.org .
Last updated (formatting) 9 January 2005