Re: translation and rotation in Euclidean space

On Sat, 25 Oct 2008 18:06:06 +0100, "Androcles"
<Headmaster@xxxxxxxxxxxxxxxx> wrote:


"John Polasek" <jpolasek@xxxxxxxxxx> wrote in message
news:b4g6g4llu7j8ssojcpiupsdrkuq89a742t@xxxxxxxxxx
On Sat, 25 Oct 2008 04:54:31 +0100, "Androcles"
<Headmaster@xxxxxxxxxxxxxxxx> wrote:
snip
advance for any help.
Best...Frank
Another flagrant case of the vanishing OP. He posts a poorly framed
problem, confusing distance with length, introducing a, b, u, v
without defining them, and now we have a half dozen kindly experts
poring and proposing amongst themselves, with nary any evidence that I
have seen, that the OP is still there or even interested.

In the first place the rotation can only be done by multiplying the
vector by a 3x3 matrix
which is a 2nd rank tensor, isomorphic to one axis of a gimbal set.

In the second place a single rotation only requires a 2x2 matrix.
[cos -sin]
[ sin cos]
This is mathematics, not physics. In physics we need to represent the
axis of rotation which is blatantly missing. To make an honest job out
of it we would take the outer product or projection operator of the
rotation axis (0 0 1):
(0) (0 0 1) to generate the 3x3 matrix into which you put your 2x2.
(0)
(1)
In the third place a 3x3 matrix permits rotation in pitch, roll and yaw,

A rotation in any of pitch. roll or yaw can be done with a 2x2 if you
want to play that way.

Of course, but when you have a full database such as many computer
games like "Flight Simulator" or even Google Earth or Google Sketchup
do and wish to rotate and translate a scene it is helpful if you process
all
relevant points (vectors) with a single 4x4 matrix. That way you don't
need to recompute.



in the fourth place a 4x4 matrix can be used to include translation.

In the fifth place your first place is wrong.

I didnt notice he was stuck in 2 dimensions. But a rotation matrix is
a 2d rank tensor, say Lij, a vector is a 1st rank tensor say Xi and
they cannot be combined in some kind of 4x4 and still be a tensor.
This 4x4 tensor cannot be rotated, just as the metric tensor cannot be
rotated. It's a donkey matrix.

Ok.


Be a good guy and show Frank how to write your 4x4 matrix.

That's all over the web:
http://tinyurl.com/5vsnqs


This is obviously a translation ; (-) is a dummy variable:

[ 0 0 0 0 ] [ a ] [ d ]
[ 0 0 0 0 ] * [ b ] = [ e ]
[ 0 0 0 0 ] [ c ] [ f ]
[d/a, e/b, f/c, 0 ] [ (-) ] [ 0 ]

Likewise a null rotation is obvious:

[ 1 0 0 0 ] [ a ] [ a ]
[ 0 1 0 0 ] * [ b ] = [ b ]
[ 0 0 1 0 ] [ c ] [ c ]
[ 0, 0, 0, 0 ] [ (-) ] [ 0 ]

Now the question becomes :
Do we want to translate and then rotate or
rotate first and then translate?
Bakers URL is a great little site to explain this and other math
tools. His treatment with abc1 in column 4 is just like this 1927
article I mentioned before, where the 3x3 told the direction cosines
of the output ray in optics, and column 4 just added the object
distance.
It's really just a question of how the scene changes as
you move or an object in the scene moves.

If you are watching a car move along a highway then
that's different to being in a car watching the world
passing by you. And you might be doing both.

http://www.androcles01.pwp.blueyonder.co.uk/Vector/RP.gif


(By the way, where is Frank/Athan anyway?)
No idea. Probably cheesed off by all the cranks mumbling crap
they couldn't program a computer to do. sci.physics.relativity
is a kook NG. <shrug>
I only post here to take the piss out of them, which is why nobody
likes me.
http://www.androcles01.pwp.blueyonder.co.uk/QUESTION.htm

Frank's is a chronic case; even taunting won't cause him to blow his
cover.
John Polasek
.

Relevant Pages

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  • Re: translation and rotation in Euclidean space
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