Technical Notes Relevant to Astrodynamics
A Fine Way to Kiss the Hyperellipsoid
A closed form solution determining the intersection of a pointing vector
from a reference point to an arbitrarily oriented hyperellipsoid is
presented. The tangent to the hyperellipsoid from a line originating at
the reference point, in the direction of the pointing vector, is also
derived. The solutions are obtained by transforming the hyperellipsoid
into a unit n-sphere, followed by a second transformation converting
the geometry to a unit circle.
Representation of a hyperellipsoid as a unit n-sphere subject to an
affine transformation is first illustrated. Next, the process of
mapping the three dimensional intersection and tangent point problems to
two dimensions is described. This method is then extended to
n-dimensional space through the generation of an orthonormal
transformation via the tensor calculus definition of the cross product.
The two dimensional solutions are derived, concluding with justifying
why the intersection and tangent points remain valid through the
application of an affine transformation.
Astrodynamic Pedantry—Earth’s Gravitational
Parameter, Equatorial Radius, and Angular Velocity
A great deal of confusion revolves around the earth's gravitational
parameter, equatorial radius, and angular velocity. Experience with
source code from numerous high, medium, and low fidelity astrodynamics
related modeling tools clearly demonstrates this quandary. Confusion is
understandable given past approaches assumed only a single value for
each constant. In addition, references that precisely define these
values are often difficult to interpret. While the details covered here
may be pedantic in nature, the devil is quite often in the details when
it comes to modeling astrodynamics related problems.
Quaternion to DCM and Back Again
Given the definition of a quaternion reference frame transformation, the
conversion from a unit quaternion to a direction cosine matrix and back is
derived. The reader is expected to be familiar with basic
quaternion properties and algebra. A method of selecting the initial
quaternion element to solve for utilizing the quaternion norm condition
to guarantee numerical stability is presented and compared to Shepperd's
method of inspecting components of the transformation matrix.