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On 6/10/19 1:03 PM, Francois Primeau wrote:
Do you plan on posting notes regarding the use of bivectors for dealingJust now I rewrote that passage, adding considerably more detail:
with coordinate singularities for motion confined to the sphere as
mentioned in your tangential remark?
Tangential remarks: If you switch to a better representation, you can get good results near the poles (and everywhere else). The physics doesn???t care what coordinate system ??? if any ??? you choose, and the physics is not singular. You can play catch near the north pole, and the ball will nicely obey all of Newton???s laws, just like anywhere else. Examples of better representations include:
*) An airplane can maneuver in three dimensions. So we do not have a 2D spherical world embedded in an imaginary 3D space; we have a genuine 3D world. So it makes sense to represent the aircraft???s position as a 3D vector, namely (X, Y, Z), for all computational purposes. The velocity is also a 3D Cartesian vector. If desired, the (X, Y, Z) representation can be displayed as (latitude, longitude, altitude), provided the latitude is not too high. For polar travel, the internal (X, Y, Z) computations remain the same, but other display schemes are used; for example, there is a well-defined notion of ???West Antarctica???.
*) The attitude (i.e. orientation) of the aircraft is sometimes described in terms of three Euler angles ??? yaw, pitch, and roll ??? but those have terrible singularities at pitch=??90??. The smart move is to switch to the bivector representation (aka quaternions) and then use Clifford algebra to compute changes in attitude. For the next level of detail on how this works, see reference 6 and reference 7. Every autopilot and every nontrivial flight simulator in the world does it this way.
There is no Cartesian representation of rotations in 3D or higher, because rotations around different axes do not commute. So you have to use bivectors (or matrices, or some other higher-dimensional representation).
*) To calculate the great-circle route between point A and point B, it is super-easy to construct two 3D vectors (from the center of the earth to A and to B), then convert that into a bivector, and then use Clifford algebra to rotate things in the plane defined by that bivector. Again, see reference 6 and reference 7. To compute straight lines (i.e. geodesics) on the real earth, which is significantly ellipsoidal, takes a bit more work, but there are well-known formulas for that.
*) In a sailboat or other conveyance restricted to the surface of a sphere, you can represent position using bivectors (aka quaternions). This works better than the (latitude, longitude) representation, especially at high latitudes. In particular, it accounts for the fact that ???lines??? of constant latitude are not straight lines. Another option is to embed the sphere in 3D space and use the (X, Y, Z) representation.
===================
The references are:
6. ???Introduction to Clifford Algebra???
www.av8n.com/physics/clifford-intro.htm
7. "Multi-Dimensional Rotations, Including Boosts"
www.av8n.com/physics/rotations.htm
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