challenge to currently applied mathematical procedure for Cartesian coordinate transformations
wherein
"any origin repositioning from initial coincindence,
alters coordinations wrt their own repositioned system."

Whereas,
it is proposed that coordinates and selected points stay fixed wrt to their own system,
during, or after any origin relocation; indeed, forever fixed in relationship to their own system.

GIVEN
SCENARIO / CONDITION
MATH LOGIC
coincident coordinate systems
A with (x,y,z) and B with (x',y',z')
1
A (x,y,z) = B (x',y',z')
A was relocated left 3
or
B was relocated right 3

along x/x' direction
2
It is impossible to know
which has occurred
from the depiction.
A (x,y,z) = B (x',y',z')
origin relocation

random

3
A (x,y,z) = B (x',y',z')
select

point P in A = (2,0,0)

4
.
select

point P in A = (2,0,0)
point P' in B = (2,0,0)

5
This is not
a coordinate transformation
between systems.
method 1 origin relocation
take diagram 5 and relocate
B right 3
along the x/x' direction

6
point P' in B is mysteriously missing?
method 2 origin relocation
take diagram 5 and relocate
B right 3 or A left 3
along the x/x' direction
7
This is not
a coordinate transformation yet...
not until an actual transfer
between systems occurs.
coordinate transformation
related Physics terms
8
These are not
pertinent to a
mathematical analysis
given that
Velocity x Time equals
a purely mathematical relocation Distance.


The transformation of selected points between systems.

Given


relocate A left 3 compared to relocate B right 3



transformation method 1
[ ignores axiom 2 ]
transformation method 2
[ applies axiom 2 ]
variant transformations
dependending upon which
system is chosen as stationary

B relocated right 3...A stationary

commencing with depiction 6
the transformation between systems is...
invariant transformation
independent upon which system chosen as stationary
B relocated right 3 or A left 3

commencing with depiction 7
the transformation between systems is...
A relocated left 3...B stationary
this time...
it commences with the relocated singular point in the opposite system.




conclusion

method 1 variant transformation results

The value of P in A is dependent upon which system was relocated
IF A left 3, THEN P in A = 5      IF B right 3, THEN = P in A = 2

The value of P' in B is dependent upon which system was relocated
IF A left 3 = THEN P' in B = 2      IF B right 3 = THEN P' in B = -1

method 2 invariant transformation results

coordinates and selected points stayed fixed wrt their origin, P in A = 2 P' in B = 2,
regardless of origin relocation.
As well,
P in A TRANSFOMS to P in B - ( relocated distance )
P' in B TRANFORMS to P' in A + ( relocated distance )



Conclusion Galilean A left 3 is variant because that result is different than B right 3 results.

additional original poems inspired by 3 xkcd comics....related directly to this particular math challenge


all rights reserved steve waterman February 13, 2012.

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