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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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    @gary hey, don't worry, I gave a look at the link that you posted and there's the answer, if you post it like an answer I'll accept it. Thanks a lot2011-07-27

3 Answers 3

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Look at the source:

en.wikipedia.org/wiki/Coordinate_system , and then let us know if this worked.

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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.

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This is a linear mapping problem. You need to decide if the mapping is an exact scaling (preserving shapes, but contracting size). In this case, you apply an identical scale-factor to all x and y coordinates. You'd choose the smaller of the two ratios I/M or J/N as the common scale-factor. This choice will prevent clipping of points under the mapping.

Or, a looser interpretation of the mapping is to warp shapes the same way the aspect ratio of the bounding coordinate box is warped. This mapping is also guaranteed to keep all points from being clipped. The x-coordinates would be scaled by I/M, and the y-coordinates get scaled by J/N.