Find coordonates of a point relative to a second plane in 3D

dot_binary

In a 3D plane, there is a point with it's X,Y and Z coordonates known and a second 3D plane within the first plane with it's origins X,Y and Z coordonates relative to the parent plane origin.

How do I find the X,Y and Z coordonates of that point relative to the origin of the second 3D plane taken in consideration that the second one may rotate on all axis and change position.

HallsofIvy

"Change position", I take it, means to translate the origin. If the new coordinate sytem has center at [itex](x_0, y_0, z_0)[/itex] in the old coordinates then a point that has coordinates (x, y, z) in the old coordinate system will have coordinates [itex](x- x_0, y- y_0, z-z_0)[/itex] in the new coordinate system.

"Rotate on all axis" is harder. A rotation about any axis can be reduced to a series of rotations about the coordinate axes and each of those can be written as a matrix product.

Rotation about the z-axis through angle [itex]\theta[/itex] is given by
[tex]\begin{bmatrix}cos(\theta) & -sin(\theta) & 0 \\ sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1\end{bmatrix}[/tex]

Rotation about the yaxis through angle [itex]\theta[/itex] is given by
[tex]\begin{bmatrix}cos(\theta) & 0 &-sin(\theta) \\ 0 & 1 & 0 \\ sin(\theta) & 0 & cos(\theta) \end{bmatrix}[/tex]

Rotation about the x-axis through angle [itex]\theta[/itex] is given by
[tex]\begin{bmatrix}1 & 0 & 0 \\ 0 & cos(\theta) & -sin(\theta)\\ 0 & sin(\theta) & cos(\theta)\end{bmatrix}[/tex]