beojan wrote:The labels x, y, and z aren't special. They do not inherently mean horizontal, vertical, and into plane. We simply usually define them that way. When we have more than one coordinate system, we use different labels for the second coordinate system, sometimes x', y', and z', sometimes u, v, w, sometimes others.steve waterman wrote:Indeed. S'(x-d,0,0) = S(x,0,0), That math would be fine.
Apparently we all agree. This is what we were all trying to show is true (for translations).
It's simply that, for the sake of ease of communication, we use x' to refer to the first argument of S', y' for the second, and z' for the third. Hence, if S'(x-d,0,0) = S(x,0,0), x' = the first argument of S' = x - d.
NOT TRUE! if S'(x-d,0,0) = S(x,0,0) does not mathematically extrapolate to x' = x-d.
This is apparently what others may believe as well. It would seem obvious that it should, but it does not.
x is not manifold material, only S(x,0,0) as point P is, and the equation x' = x-d has no mention nor connections to a point.
Someone needs to make this distinction between x the coordinate and (x,0,0) the point. Thanks for dropping the vt requirement. Now a further request, please...to drop anything/logic re manifold, mapping, point or in the point format (a,b,c). The discussion/equation/thread evolution is now about x the coordinate. P had left the building.
So. No point P, an ignored manifold and given two coincident Cartesian systems S(x,y,z) and S'(x',y',z') with x = x', y = y', z= z'. Then, we reposition so that the two origins are separated by d along the common x/x' axis.
x = x'.