The previous thread identified a lot of things that could go
wrong ... but now let's focus on how to do things right.
Suppose you are building an autopilot system or a flight simulator.
(Those two things have a lot in common.)
The internal representation of the /attitude/ of the airplane
is in terms of bivectors. This concept is based on Clifford
algebra. These are the same bivectors that make cross products
obsolete in three dimensions. Special cases include complex
numbers (in two dimensions) and quaternions (in three dimensions).
They even work for representing rotations (including boosts) in
four dimensional spacetime. http://www.av8n.com/physics/clifford-intro.htm http://www.av8n.com/physics/complex-clifford.htm http://www.av8n.com/physics/rotations.htm
The attitude bivector represents the orientation of the craft
relative to some "standard" attitude that is fixed in space.
"Help stomp out cross products."
Switching now from attitude to location, one good representation
for the location of the craft uses Cartesian rectangular coordinates.
Conventionally, the X axis passes through the (latitude, longitude)
point (0n, 0e) which is in the Gulf of Guinea, south of Ghana and
west of Gabon. The Y axis passes through the (latitude, longitude)
point (0n, 90e) which is in the Indian Ocean, southeast of Sri Lanka
and west of Indonesia. The Z axis passes through the north pole. http://www.av8n.com/physics/coords.htm
Let's be clear: The surface of the earth is topologically two
dimensional, but this representation uses three variables: X, Y,
and Z. Being "on the surface" is a constraint within this system.
Vectors such as displacements, velocities, and accelerations can
be represented using the vector basis induced by that system: dX,
dY, and dZ. This has the disadvantage of being a non-inertial
system, but the centrifugal and Coriolis forces can be handled
using well-known techniques.
If we treat that representation as primary, we can *sometimes*
project things onto this-or-that secondary coordinate system.
One such secondary system is the local Cartesian North-East-Down
system. Closely related to that is the local cylindrical system,
r-θ-z, where z is altitude, θ is azimuth, and r is distance in the
local tangent plane. Both of these secondary systems are singular
at the poles and impractical near the poles ... which is why you
don't want them to be primary.
It must be emphasized that the intrinsic physics is *not* singular
near the poles. It is not even badly behaved. If you and I are
standing somewhere near the pole, possibly on opposite sides of
the pole, we can play catch, and the singularity in the coordinate
system doesn't affect us at all. We can even toss a gyroscope back
and forth, and the singularity does not affect the orientation of
the gyro axis or anything else. We can draw a picture of what we
are doing, and the picture looks perfectly ordinary. Physical
concepts such as "throw it to my left" or "throw it to my right"
continue to work just fine. It is difficult or impossible to
express things in terms of northward, southward, eastward, westward,
or other heading-related notions ... but we don't need to do that.
There are other (better) representations we can use.