Maths - Cartesian coordinate systems - comments

By: nobody ( Nobody/Anonymous )
Left and Right-handed Cartesian Systems
2003-03-09 00:02
In defining non-convertibility of handedness by simple rotation add "and vice-versa" i.e. a Left-handed system cannot be rotated into a Right-handed 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 right-handed, 24 left-handed) This can be easily demonstrated.
Consider your sketch of Left-handed 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 left-handed.
In space there are exactly 8 octants. Thus there will be exactly 24 left-handed and 24 right-handed cartesian coordinate systems - no more no less


By: Martin Baker
RE: Left and Right-handed Cartesian Systems
2003-03-09 17:32

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?



By: nobody ( Nobody/Anonymous )
RE: Left and Right-handed Cartesian Systems
2003-03-10 20:41
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 left-handed 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 e-mail with drawing at tachments to martinb which I believe Carl had done.
Cheers and regards

By: Martin Baker
RE: Left and Right-handed Cartesian Systems
2003-03-11 09:08

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.


By: nobody ( Nobody/Anonymous )
RE: Left and Right-handed Cartesian Systems
2003-03-14 12:20
Thanks for acceptring my offer. You will receive some write-ups by e-mail with sketch attachments for scrutiny.
Thank You. Regards

Item (3) of message from Martin 2003-03-11 01:08
Slight Correction

There are TWO Possible 3rd Axis - not one. One defines the Right-handed system the other the Left-handed system
There are 24 possible Left-handed and 24 possible Right-handed systems totalling 48 systems.
Initial Axis EAST (Right)
Second Axis NORTH (Top of page)
Two possible 3rd axis
Former (East - North - UP) Right-handed
Latter (East - North - DOWN) Left-handed
Hope I have made my point.

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