Calculate coordinates in different reference frames

Question

I have a plane whose origin (ABC) and surface normal are defined in terms of a standard Cartesian coordinate system, XYZ. The plane is also constrained such that the line connecting the origin of the plane's coordinate system and the origin of the XYZ reference frame shall be defined as the x-axis of the plane's coordinate system.

I have the 2D coordinates of a point on that plane (a, b). How do I compute the coordinates of that point in terms of the XYZ reference frame?

Can you clarify this part? "The plane is also constrained such that the independent axis points at the origin of the XYZ reference frame." — FogleBird, Mar 02 '17 at 16:52
A line connecting the origin of the XYZ reference frame and the origin of the plane's coordinate system shall form the x-axis of the coordinate system on the plane. — Thom DeCarlo, Mar 02 '17 at 17:04
I'm voting to close this question as off-topic because it's about math. — n. m. could be an AI, Mar 02 '17 at 17:23
Ok, so where should it be asked? It's similar to other questions I found in SO with the same tags. — Thom DeCarlo, Mar 02 '17 at 17:30
So, that also means the plane always passes through the XYZ origin? — FogleBird, Mar 02 '17 at 17:48

score 1 · Accepted Answer · answered Mar 02 '17 at 17:55

1

You just need two orthogonal vectors to define your 2D space. You already have one, as you said, as the vector from the plane point, P, to the origin. To get the other, take the cross product of that one with the normal vector of the plane.

u = normalize(planePoint)
v = normalize(cross(planeNormal, u))
point = u * x + v * y

answered Mar 02 '17 at 17:55

FogleBird

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Nice! I'll try that. It's certainly a lot simpler than the rabbit hole of quaternians and Euler angles that I was stuck in. – Thom DeCarlo Mar 02 '17 at 18:03

Calculate coordinates in different reference frames

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