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

In summary, to find the X,Y and Z coordinates of a point relative to the origin of a second 3D plane that may rotate on all axis and change position, you can use a series of rotations about the coordinate axes and a matrix product to calculate the new coordinates. If the new coordinate system has a center at (x_0, y_0, z_0) in the old coordinates, then the coordinates of the point in the new system will be (x-x_0, y-y_0, z-z_0).
  • #1
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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.
 
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  • #2
"Change position", I take it, means to translate the origin. If the new coordinate system 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]
 

1. How do you find the coordinates of a point relative to a second plane in 3D?

To find the coordinates of a point relative to a second plane in 3D, you will need to use the concept of vector projection. This involves finding the perpendicular distance from the point to the second plane, and then projecting that distance onto the axes of the plane to determine the coordinates.

2. What is the equation for finding the coordinates of a point relative to a second plane in 3D?

The equation for finding the coordinates of a point relative to a second plane in 3D is:
x = (P · u)u + (P · v)v + (P · w)w
where P is the vector representing the point, u, v, and w are vectors representing the axes of the second plane, and · represents the dot product operation.

3. Can I use this method to find the coordinates of a point on any plane in 3D?

Yes, this method can be used to find the coordinates of a point on any plane in 3D. However, it is important to note that the vectors representing the axes of the plane must be orthogonal (perpendicular) to each other.

4. What are some real-world applications of finding coordinates of a point relative to a second plane in 3D?

This method is commonly used in fields such as engineering, architecture, and computer graphics. It can be used to determine the position of objects in 3D space, such as the location of a building on a city map or the orientation of a 3D model in a computer program.

5. Are there any limitations or assumptions when using this method?

One limitation of this method is that it assumes the second plane is a flat, infinite surface. This may not always be the case in real-world scenarios. Additionally, the accuracy of the coordinates will depend on the accuracy of the vectors representing the axes of the plane.

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