"Relative" transformations - from a coordinate system to another

Question

So, I'm having a memory lapse. I'm looking for the concept of transforming a vector space to another vector space, relative to itself. (not relative to identity) In other words, the moves I make from one 3D coordinate system, to another, relative to the 1st system.

I have no problem finding the linear transformation. It's the "relative" transformation that eludes me. I'm just trying to find the steps I have to take to get from one system to the next, without going all the way back to identity.

Thanks in advance.

Answer 1

I think I figured this one out for myself. If I know the inverse of the first vector space, back to identity, then applying the inverse to the second vector space, preserves the relationship, but returns me a relative transformation - since anything times the identity results in... the anything.

Please comment if this isn't correct, but it sure seems to be.

Answer 2

If you want to go from any vector space to a given vector space, you need the following information. Let your original space be $V_1$, with a known basis $B_1$, and your target space be $V_2$ with a basis $B_2$.

Now, to get a transformation that takes you from $V_1$ to $V_2$, you need to express the elements of $T(B_1)$ using the elements of $B_2$ and construct the matrix approriately (column-wise). This does not require knowing the standard identity basis, just your target space and original space

EDIT

Another approach is to use a $4 \times 4$ transformation matrix to describe how to get $V_2$ from $V_1$ purely geometrically. A transformation matrix is of the form

$$[T]_{V_1V_2} = \begin{bmatrix}R & p \\ 0 & 1\end{bmatrix}$$

Here, $R$ is the rigid rotation matrix (3 by 3) required to align the axes of $V_2$ to $V_1$, $p$ is the linear displacement vector of the origin of the frame of $V_2$ with respect to the origin of $V_1$ (3 by 1)

If you have any point expressed in $V_1$ coordinates, you can express it in $V_2$ coordinates as follows

$$q_2 = [T]q_1$$

Expressed in homogenous coordinates (just append th 3D vector with a fourth entry equal to 1 for both points and disregard the final entry after the multiplacation)