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Asked: 2026-06-15T11:13:58+00:00

I’m trying to derive the matrix of a rigid transform to map between two

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I’m trying to derive the matrix of a rigid transform to map between two coordinate spaces. I have the origin and the axis directions of the target coordinate space in terms of the known coordinate space; does anyone know how I can solve for the 4×4 rigid transformation matrix given these?

So, in other words, I have two coordinate spaces, A and B, and I know 

Point3D originOfBInA;
Vector3D xAxisOfBInA; // Unit vector
Vector3D yAxisOfBInA; // Unit vector
Vector3D yAxisOfBInA; // Unit vector

And I’m trying to find the 4×4 matrix 

Matrix4x4 AtoB;
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1 Answer

  1. Editorial Team

    First construct the 4×4 matrix for the change of basis (call it M) by using your unit vectors (Ax, Ay, Az) and the origin (T) as column vectors:

    M = 
   [Ax Ay Az T]    <-- 3x4
   [0  0  0  1]

    To transform the coordinates of a point p (specified with respect to frame A) to q (with respect to frame B), just multiply by the inverse of M:

    q = M-1p

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