Re: translation and rotation in Euclidean space


"John Polasek" <jpolasek@xxxxxxxxxx> wrote in message
news:iu15g4l5mani1g36mmnrav75eha5avemfv@xxxxxxxxxx
On Mon, 20 Oct 2008 18:08:40 -0700 (PDT), athan <metanosis@xxxxxxxxx>
wrote:

Clearly taking a vector and rotating it, then translating it, does
nothing to the distance. So, under translation and rotations, the
Euclidean standard metric is invariant. Am seeking to show this in the
plane. clearly,
(x,y,1)dot(x,y,1)=x^2+y^2+1 Now I wish to translate and rotate a
vector, and check that the distance is the same.

(a -b 0) (1 0 u)
(b a 0) (0 1 v)
(0 0 1) (0 0 1) are the rotation and translation
matrices, the result of multiplying them is

(a -b au-bv )
(b a bu+av)
(0 0 1 ) now to apply them to a vector (x,y,1) we
get


(ax-by+au-bv )
(bx+ay+bu+av )
( 1 ) now we dot this vector with itself and
do some simplification to the resulting scalar...



(a^2+b^2)[ (x+u)^2 + (y+v)^2)] +1 now (a^2+b^2)=1 so we have (x
+u)^2 +(y+v)^2 + 1

the question, am supposed to get a scalar equal to x^2 + y^2 +1.
Clearly we have only translated
the vector, but I must be getting a definition wrong, somewhere mixing
scalars and vectors in a wrong way.

Is this correct, to translate a vector,


(1 0 u) (x) (x+u)
(0 1 v) x (y) = (y+v)
(0 0 1) (1) ( 1 ) but to me, this vector isn't the vector
(x,y,1) translated, it's the addition
of two vectors, and will have a different, longer distance. Thanks in
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

In the second place a single rotation only requires a 2x2 matrix.
[cos -sin]
[ sin cos]

In the third place a 3x3 matrix permits rotation in pitch, roll and yaw,
in the fourth place a 4x4 matrix can be used to include translation.

In the fifth place your first place is wrong.






.

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