Cranium Cruncher
By Douglas Yazell, Councilor
June 2004 puzzle:
Given an aircraft's velocity vector's coordinates in two coordinate
systems with the same origin, can we find the orientation between the
two coordinate systems? Is it unique? If not, describe all solutions.
June 2004 answer:
There are an infinite number of solutions which can be pictured easily
(though it certainly took me and others a lot of work). Imagine one
coordinate system and a sketch of the given vector projected into the XY
plane and from there to the X and Y axes, to emphasize the given
coordinates. Then imagine the other coordinate system with a different
orientation. Project the same vector to this XY plane and from there to
these X and Y axes to picture those given coordinates. Now allow this
coordinate system to rotate about the given vector. That describes all
possible orientations between the two coordinate systems. In this
moving coordinate system, the vector coordinates do not change during
that rotation.
Calculations were not needed to obtain full credit for this puzzle, but
here is one way to find any of those direction cosine matrices
describing the orientation between the two coordinate systems. Given
the vector coordinates in one coordinate system, we rotate another
coordinate system from that orientation by a Z-axis rotation, then a
Y-axis rotation so that its X-axis lies along the given vector. Then
rotate it about that X-axis unit basis vector by any value (call that
rotation angle delta), then use the other given coordinates to rotate
this coordinate system through a Y-axis rotation, then a Z-axis
rotation. The product of those 5 rotation matrices is the direction
cosine matrix. It is a simple function of delta.
Here are the names of those who sent correct answers to me: Judah
Richardson of Rice University. In the fall, he will be a senior in
mechanical engineering.
This month's puzzle:
Given an aircraft velocity vector's coordinates before and after a
rotation of that vector, find the rotation axis. Is it unique? If not,
describe all possible solutions.
Hints:
- There are more than the two obvious rotation axes.
- Full credit can be obtained without equations, using words like
lines, axes, vectors, rotations, circles, etc.
- We are not finding matrix solutions here (rotation matrices),
but the answer could be expressed in that format, and they are the same
answers presented above for last month's puzzle (direction cosine matrices).
Send solutions to me at douglas.yazellAIAA@honeywell.com (remove 'AIAA'
before sending), or call me at 281-244-3925 (to deliver solutions, ask
for more hints, talk it over, etc.).
My thanks go the same people as noted in
last month's article. I submit
this month's puzzle as a member of two technical committees in our
section. The first is our International Space Activities Committee
(ISAC). ISAC is chaired this year by Padraig Moloney (NASA), and the
other members are Dr. Zafar Taqvi (Dynacs) and Elizabeth Blome (NASA).
The second is our Astrodynamics technical committee, chaired by Dr.
Albert Jackson (Lockheed Martin), an engineer at NASA and a visiting
scientist at the Lunar Planetary Institute. Like all of our committees
in the technical branch of our section, we are always looking for new
members: students, young professionals, experts, etc. We don't impose
much on our members' time, but there are many potential benefits.
