# Calculate coordinates in smaller cartesian system

sorry for my english, I need to know, that, given a set of points x, y in a M x N cartesian system, how can I calculate the same set of point in another system I x J, being M > I and N > J? Is this possible? If it is, which is the name of the method?

Example:

Being a plane with 100 units of width and 150 units of height, and a set of point A= {(30, 20), (78, 56), (56, 18)}, is there a method that given a plane of 20 x 60, get a set of points equivalent with A in this second plane?

Thanks and sorry about my english.

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Scale the x-coordinates by I/M and the y-coordinates by J/N... – J. M. Jul 27 '11 at 16:08
I don't know if this is overkill, but the general method would be one of switcjing between charts on a manifold ;the manifold here being $\mathbb R^2$ itself – gary Jul 27 '11 at 16:13
@gary How come? Sorry, I'm not a mathematician, could you please, be more especific? – Cheluis Jul 27 '11 at 16:23
Look at parts 7,9 here: en.wikipedia.org/wiki/Coordinate_systemand I'll get back to you with an answer. Basically, a coordinate is a way of expressing points in a space called a manifold in relation to $\mathbb R^n$. For example, points in the sphere can be given coordinates, in a way that makes points in the sphere have , locally, properties similar to those of the plane $\mathbb R^2$ – gary Jul 27 '11 at 16:29
Cheluis: I'm sorry, the answer is kind of involved, and I'm kind of rushed at the moment. Hopefully, I will have more time later for a better answer. – gary Jul 27 '11 at 18:27

I don't know what you mean by an $M\times N$ coordinate system, but I suggest the following: Draw in black a horizontal and a vertical axis, make a few ticks and label these with the values they should denote in your "$M\times N$-system". Then draw in red another pair of axes offset $1$mm with respect to the black axes, make a few ticks and label these with the values they should denote in the "$I\times J$-system". This means that now any point $x$ on a virtual horizontal axis has an $M$-value $x_M$ and an $I$-value $x_I$ assigned, and there is a certain relationship between the numbers $x_M$ and $x_I$ which is independent of the chosen point $x$. In the same way any point $y$ on a virtual vertical axis has an $N$-value $y_N$ and a $J$-value $y_J$ assigned, and there is a certain relationship between the numbers $y_N$ and $y_J$ which is independent of the chosen point $y$. Finally, if you have an arbitrary point $P$ in the plane then it has a vertical projection to the horizontal axis whereby two numbers $x_M$ and $x_I$ are generated, and a horizontal projection onto the vertical axis whereby two numbers $y_N$ and $y_J$ are generated. It now should be easy to write down the formulas by which the pairs $(x_M,y_N)$ and $(x_I,y_J)$ are related.