Derive Rigid Transform Matrix from Axes and Origin 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 4x4 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 zAxisOfBInA; // Unit vector

And I'm trying to find the 4x4 matrix $\quad$
Matrix4x4 AtoB;

 A: Let $e_1, e_2, e_3$ be a basis of coordinate space $A$. Express your given vectors (xAxisOfBInA, etc..) in terms of the basis of A, lets label them as column vectors $v_1, v_2, v_3$.
Then coordinate transform matrix is $C = \pmatrix{v_1 & |& v_2 &|& v_3}$. But since you are shifting the origin, you are doing a 3D affine transform, which can be represented as a 4D linear transform.
Represent coordinates $(x,y,z) \in A$ as $\vec{x} = (x,y,z,1)$. Then the representation of $\vec{x}$ in $B$ is $D\vec{x}$, where $D = \pmatrix{C & x_0 \\ 0 & 1}$, where $x_0$ is the representation of originOfBInA in terms of the basis of $A$.
If this is too abstract, I can work out an example if you'd like.
A: I am sorry, I do not have enough reputation to add to your conversation. The correct transformation matrix in your case would indeed be:
$$
D=\begin{bmatrix}
C^{-1} & 0\\
0 & 1
\end{bmatrix}\cdot\begin{bmatrix}
1 & -x_0\\
0 & 1
\end{bmatrix}
$$
The $-x_0$ part will translate the vector so that origins of both frames coincide, and the $C^{-1}$ part will represent coordinates of a vector in the new basis. The change of basis part is explained in great detail for example here: https://ltcconline.net/greenl/courses/203/Vectors/changeOfBasis.htm
And the translation part should be self-evident.
P. S. Sorry for answering this six years later, this thread popped up in my Google search on the first page, and the answer by countunique is clearly wrong. 
