By: nobody ( Nobody/Anonymous )
Left and Righthanded Cartesian Systems
20030309 00:02
Here:
https://www.euclideanspace.com/maths/geometry/coordinatesystems/cartesian
In defining nonconvertibility of handedness by simple rotation add "and
viceversa" i.e. a Lefthanded system cannot be rotated into a Righthanded
system either. This is obvious but certainly helps to state explicitly.
Instead of stating that there are "many Cartesian coordinate systems"
we should state that there are exactly 48  (24 righthanded, 24 lefthanded)
This can be easily demonstrated.
Consider your sketch of Lefthanded system.
As shown:
X to left; Y Up, Z toward observer.
Clearly, in the same octant we can also say:
X Up, Y toward viewer, Z left AND
X toward viewer, Y left, Z up
are lefthanded.
In space there are exactly 8 octants. Thus there will be exactly 24 lefthanded
and 24 righthanded cartesian coordinate systems  no more no less
Subhash
By: Martin Baker
RE: Left and Righthanded Cartesian Systems
20030309 17:32
Subhash,
Thanks for this, I will link to your message from the web page if that’s all right with you.
Just to clarify, I think what you are saying is that, we know the directions of 3 mutually perpendicular axes but we haven’t yet chosen which of them is the x, y or z coordinates or which direction of each is positive and which is negative, and in this case there are 48 ways of doing this. The reason I say that we know the directions is that there is an infinite number of other possible coordinate systems, say at 45 degrees, or some other angle, to the first? So how can we define a coordinate system in the first place? Of course we could define a coordinate system in terms of another coordinate system, but if we don’t already have a coordinate system then, I guess we need at least 2 reference directions to start from?
Cheers,
Martin
By: nobody ( Nobody/Anonymous )
RE: Left and Righthanded Cartesian Systems
20030310 20:41
Martin,
Thanks a lot for connecting me directly to your web page. I certainly like it.
To continue the discussion.
You are certainly right about orientation of initial X and Y at 45 degrees to
your picture. Then there are infinite, uncountable number of systems. You always
do need one starting direction, which in Geometrical terms I call the "Independent
variable axis", erect an orthogonal direction, call it the "dependent
variable axis" and then erect a perpendicular to the plane containing the
independent and dependent variable axes, at the origin. The first is universally
designated as X, the second is Y and the third is Z. To enumerate the 48 possible
systems I convert X, Y, and Z to 1, 2, and 3 as it is easier to distinguish
and hence more convenient to code.
I make one more simplification for conceptualizing  that is use the 6 cardinal
geographic directions. Any structural engineering plan is always identified
either by a true, magnetic, or an assumed (our lingo was Plant) north direction.
Therefore, for me, it is easier to conceive your lefthanded system picture
as West (X), Vertically Up (Y), South (Z). When a project was in the planning
stage, the plant location as yet unknown, we used the following system:
Top of page (T, same as North), Bottom of page (B, same as South), Right (R,
same as East), Left (L same as West), Up (U, same as vertically up), Down(D,
same as vertically down).
I take the time to define all of the above to pursue my life long agenda of
correlating geometry and mechanics for dummies (I am one). To do so, I would
like to have your permission to address an email with drawing at tachments
to martinb which I believe Carl had done.
Cheers and regards
Subhash
Yes, please feel free to send me any further information which I can include on the website.
So what you are saying is that, to define a cartesian coordinate system, we
need to:
1) define an initial axis (3 degrees of freedom)
2) define the second axis, perpendicular to first (1 degree of freedom)
3) third axis is fully defined by first two (0 degree of freedom)
4) choose one of 48 options for naming and direction of axes.
So altogether there are 4 degrees of freedom+48 options to define a cartesian coordinate system.
Martin
By: nobody ( Nobody/Anonymous )
RE: Left and Righthanded Cartesian Systems
20030314 12:20
Martin:
Thanks for acceptring my offer. You will receive some writeups by email with
sketch attachments for scrutiny.
Thank You. Regards
Subhash
COMMENTS
Item (3) of message from Martin 20030311 01:08
Slight Correction
There are TWO Possible 3rd Axis  not one. One defines the Righthanded system
the other the Lefthanded system
There are 24 possible Lefthanded and 24 possible Righthanded systems totalling
48 systems.
EXAMPLE:
Initial Axis EAST (Right)
Second Axis NORTH (Top of page)
Two possible 3rd axis
VERTICALLY UP OR VERTICALLY DOWN
Former (East  North  UP) Righthanded
Latter (East  North  DOWN) Lefthanded
Hope I have made my point.
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