I have two coordinate systems, like so
How can I transform a point on the one of the coordinate system to other ?
Pxyz = M . Px'y'z' what is M ?
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\$\begingroup\$ @kshahar hi, do you know the answer ? \$\endgroup\$– gveafCommented Nov 13, 2012 at 10:53
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\$\begingroup\$ M is the transformation matrix.. kwon3d.com/theory/transform/transform.html \$\endgroup\$– teodronCommented Nov 13, 2012 at 11:04
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\$\begingroup\$ @teodron can you write the answer \$\endgroup\$– gveafCommented Nov 13, 2012 at 11:07
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1 Answer
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From the image it looks like both your coordinate systems are cartesian coordinates, where the only difference between the two is that one has a different origin from the other.
If this is the case then to translate from xyz coordinates to x'y'z' coordinates all you need is a translation, i.e.
x' = x + dx
y' = y + dy
z' = z + dz
Where [dx, dy, dz] is the difference between the origins of the two coordinate systems.
Matrix multiplications always have the origin as a fixed point and so you can't perform translations using a 3x3 matrix, however as a sort of workaround you can express your 3-dimensional coordinates as the 4-dimensional vector [x, y, z, 1] and use a 4x4 transformation matrix that includes a translation, e.g.
|1 0 0 dx | |x| |x + dx|
|0 1 0 dy | |y| |y + dy|
|0 0 1 dz | |z| = |z + dz|
|0 0 0 1 | |1| | 1 |
You can then ignore the 4th component to get the translated 3D coordinates. See more on Wikipedia here