|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
- Real things exist within a 3 dimensional space continuum, but it can be argued that translation is only a 1 dimension phenomenon. Since translation is defined as a displacement in a single direction. The problem gets complicated when we refuse to arrange our space dimension coordinate system so that the directional displacement coincides in direction with one of our space direction ordinates. So we correct that by determining the combined result of the displacement in all three directions, (using the square root of the sum of squares formula) to get the amount of the single direct displacement. Now when we start talking about translation related to spacetime considerations, we complicate the issue considerably by throwing in all those 3 dimensional resultant calculations into the middle of the spacetime interdependence calculations. And the question is basically about what is the interdependence of the single dimensional space and time measurement values under translation conditions.
- Now it is to be noted that when we consolidated the 3 space dimension ordinates into a single displacement value we added the squares of the values because the directions were at right angles (orthogonal) to each other. Now we're trying to combine a distance value with a time value and the question is how we do that. And the answer is that instead of the time and space vectors being added at right angles to each other like in ths 3 dimension consolidation process, The vectors are in this instance considered as representing the hypotenuse and one side of a right triangle, with the combined value being the length vector of the other side of the right triangle. So we have the time vector shortening the space vector and vise versa, and that's hard to explain conceptually.
- Now if we reexamine our time and space (direction) values, we note that the space direction value is that that we move in the considered time period. And thus we're comparing 2 space values namely, (1) How far we have moved, and (2) how far light would move in the same time period. And the light velocity moving distance is the amount that we need to use to correct our measured move distance to some combined value. And the vector difference between these two translation values is evidently the (hypothesized) true dimension quantity value of our move through the combined spacetime continuum. And, of course, if these two values are the same, the difference becomes zero. And since we're also considering a single translation direction, we accomplish this by rotating the direction of the time vector to be opposite in direction to that of the translation move vector. But in doing so we have completely fouled up the orthogonality of the time vector to the space (dimension) vector.WFPM (talk) 03:49, 21 August 2009 (UTC)
Copied text from translation (physics) to translation (geometry)
I have moved the text from translation (physics) to translation (geometry). It seems to fit in well. The article translation (physics) can be set to redirect to translation (geometry). Prof McCarthy (talk) 18:25, 29 December 2011 (UTC)
I removed the merge template because this article now contains everything in translation (physics), however translation (physics) has not yet been set to redirect here. Prof McCarthy (talk) 18:38, 29 December 2011 (UTC)
not only Euclidean
Translation also exists in hyperbolic space; one way to define it is as the composition of two reflections in ultraparallel (=neither intersecting nor asymptotic) planes. Or is it called something else there? —Tamfang (talk) 05:14, 28 May 2012 (UTC)
translation is not a linear transformation
Translation is not a linear transformation even in homogeneous coordinates, since it does not transform origin, for example (0,0,1), to origin. From another perspective, the fact that a transformation is linear or not does not depend on the choice of basis or coordinates, so translation in homogeneous coordinates is not linear since it's not linear in Cartesian coordinates. Even in homogeneous coordinates translation can be represented by matrix multiplication, but it doesn't mean it's linear, because matrix multiplications can also represent non-linear transformations, like affine or perspective transfromations. So I think the last parts of the first sentence in the "Matrix representation" section, i.e. quote "and thus to make it linear", should be removed.
Let me know if I'm getting anything wrong here, but I think
is not equal to
We are comparing a 4d vector to a 3d one which is not technically possible.
I think this should be a better answer:
So, now, people will know that: