Converting between two frames of reference given two points

Question

I have two points ($P_1$ & $P_2$) with their coordinates given in two different frames of reference ($A$ & $B$). Given these, what I'd like to do is derive the transformation to be able to transform any point $P$ ssfrom one to the other.

There is no third point, but there is an extra constraint, which is that the y axis of Frame $B$ is parallel to the $X$-$Y$ plane of Frame $A$ (see sketch below). I believe that is enough information to be able to do the transformation.

Also:

• The points are the same distance apart in both frames (no scaling).
• The points don't coincide.
• The origins don't necessarily coincide.

As you may have gathered, I'm not a mathematician (ultimately this will end up as code), so please be gentle...

_{I've seen this question (Finding a Rotation Transformation from two Coordinate Frames in 3-Space), but it's not quite the same as my problem, and unfortunately I'm not good enough at math to extrapolate from that case to mine.}

EDIT I've updated the diagram, which makes it a bit cluttered, but (I hope) shows all the 'bits': $P3_B$ is what I'm trying to calculate...

Answer 1

The problem is to find a rotation matrix $R$ and a translation vector $\vec t$ such that

$$R\vec p_{1B}+\vec t=\vec p_{1A}\;,\tag1$$ $$R\vec p_{2B}+\vec t=\vec p_{2A}\;.\tag2$$

Subtracting these yields

$$R\left(\vec p_{1B}-\vec p_{2B}\right)=\vec p_{1A}-\vec p_{2A}\;.\tag3$$

Since the $y$ axis of system $B$ is parallel to the $x$-$y$ plane of system $A$, we can obtain the rotation by first rotating around the $y$ axis and then rotating around the $z$ axis, so $R$ must have the form

$$ \begin{pmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\\&&1\end{pmatrix} \begin{pmatrix}\cos\beta&&\sin\beta\\&1&\\-\sin\beta&&\cos\beta\end{pmatrix} = \begin{pmatrix} \cos\alpha\cos\beta&\sin\alpha&\cos\alpha\sin\beta\\ -\sin\alpha\cos\beta&\cos\alpha&-\sin\alpha\sin\beta\\ -\sin\beta&&\cos\beta \end{pmatrix} \;. $$

So the third row depends only on $\beta$, and writing out the corresponding row of $(3)$ yields a trigonometric equation for $\beta$:

$$-\sin\beta(p_{1Bx}-p_{2Bx})+\cos\beta(p_{1Bz}-p_{2Bz})=p_{1Az}-p_{2Az}\;.$$

Since we'll get another equation of the form

$$a\sin\phi+b\cos\phi=c\tag4$$

shortly, I'll solve it in that general form, and you can substitute

$$ \begin{eqnarray} a&=&p_{2Bx}-p_{1Bx}\;,\\ b&=&p_{1Bz}-p_{2Bz}\;,\\ c&=&p_{1Az}-p_{2Az} \end{eqnarray} $$

to solve this one. Writing $a$ and $b$ in polar form, $a=r\cos\xi$, $b=r\sin\xi$, leads to

$$r\cos\xi\sin\phi+r\sin\xi\cos\phi=c\;,$$

$$\sin(\xi+\phi)=\frac cr\;.$$

You can get one solution from

$$\phi_1=\arcsin\frac cr-\xi\;,$$

but note that in general there's a second one, $\phi_2=\pi-\phi_1$, which makes sense, since there are in general two different angles through which you can turn a vector around the $y$ axis to give it a certain $z$ component.

You can determine $r$ and $\xi$ using $r=\sqrt{a^2+b^2}$ and $\xi=\text{atan}(b,a)$, where $\text{atan}$ is the two-argument arctangent function found in many programming environments, which takes into account the signs of both arguments to disambiguate the arctangent on the full range of angles.

Now you have two values of $\beta$, and you can substitute them into the rotation matrix. For instance, substituting into the first row of $(3)$ yields

$$ \cos\alpha\cos\beta(p_{1Bx}-p_{2Bx})+\sin\alpha(p_{1By}-p_{2By})+\cos\alpha\sin\beta(p_{1Bz}-p_{2Bz})=p_{1Ax}-p_{2Ax}\;, $$

which is again a trigonometric equation for $\alpha$ of the form $(4)$, with

$$ \begin{eqnarray} a &=& p_{1By}-p_{2By}\;, \\ b &=& \cos\beta(p_{1Bx}-p_{2Bx})+\sin\beta(p_{1Bz}-p_{2Bz}) \;, \\ c &=& p_{1Ax}-p_{2Ax} \;. \end{eqnarray} $$

You can solve it just like the other one, to obtain two values of $\alpha$ for each of the two values of $\beta$. Again, this makes sense, since we've only used the first row of $(3)$ so far and there are two different angles through which you can turn a vector around the $z$ axis to give it a certain $x$ component. However, only one of each of these two pairs of values of $\alpha$ will also solve the second row of $(3)$ (that is, the other one would produce a wrong $y$ component), so in the end you obtain two sets of solutions for $\alpha$ and $\beta$. You can substitute each of these into either $(1)$ or $(2)$ to get the corresponding value of $\vec t$.

So in general you get two different affine transformations for valid point coordinates, though sometimes, e. g. if $a=b=c=0$ in either trigonometric equation, the solution will be a lot more underdetermined than that, and if you substitute invalid point coordinates (e. g. for points at different distances from each other, or such that $|c|>r$ in either trigonometric equation), there will be no solutions at all.

I wrote the affine transform for transforming from $B$ to $A$ to simplify the calculations, but you can easily obtain the transform in the other direction as

$$\vec p_B=R^T(\vec p_A-\vec t)\;.$$

Answer 2

So given two points, an initial frame of reference, and the fact that the 'new y axis', which I will designate $y'$, is in the $xy$ plane, we do not have enough information to determine a unique new frame of reference around the new point. Even if $y'$ were parallel to $y$, we still don't quite have enough information. We need one more constraint, somewhere.

But let's say for a moment that you know both the new and the old frame of reference. By this, I mean that you know the vectors of the three mutually perpendicular basis elements (the vectors along the axes). Then, you are in business. As long as there is no scaling (vectors in the first frame don't get shorter or longer in the second), this is a simple process. I don't know what math you do or don't know, so I'll say it in some sort of general way and if it doesn't make sense to you, I can edit in more.

I assume throughout that the first frame of reference is at the origin (this is not unreasonable, and if we can't keep it then that's okay too). Assume for a moment that the two frames of reference are actually at the origin (i.e. that the second frame is at the origin too, but only for a moment). Then your task is to find the unique matrix transformation that rotates the original basis to the new basis. There are a few ways to do this, but the big idea is to remember that the columns of a matrix take one basis element to another. Good. Call this matrix $T$, so I can refer to it later.

Now find the difference between the two points. If $x_1$ is in the old frame, and $x_2$ is in the new frame, then find the vector $d = x_2 - x_1$. Then any element, given by the vector $v$ in the old frame will be given by the new vector $v' = [T][v] - d$, where I used brackets to emphasize the fact that there is a matrix multiplication.

Does that make any sense?

Answer 3

I think that the two points should be enough information even without the additional information about the $y$-axis of the second frame. EDIT: joriki pointed out that this isn't true (see comments). The additional piece of information about the $y$-axis is needed, but then the problem should be solvable.

The transformation you want is affine because it sends lines to lines and its linear part is orthogonal because it preserves distances. This means it will have the form

$$\begin{align} x' &= a_{11}x+a_{12}y+a_{13}z +b_1\\ y' &= a_{21}x+a_{22}y+a_{23}z +b_2\\ z' &= a_{31}x+a_{32}y+a_{33}z +b_3\\ \end{align}$$

This form makes it affine. To make sure it is orthogonal the $a$'s are subject to the constraints

$$\begin{align} &a_{11}^2+a_{21}^2+a_{31}^2=\\ &a_{12}^2+a_{22}^2+a_{32}^2=\\ &a_{13}^2+a_{23}^2+a_{33}^2=1 \end{align}$$

and

$$\begin{align} &a_{11}a_{12}+a_{21}a_{22}+a_{31}a_{32}=\\ &a_{11}a_{13}+a_{21}a_{23}+a_{31}a_{33}=\\ &a_{12}a_{13}+a_{22}a_{23}+a_{32}a_{33}=0 \end{align}$$

EDIT: The fact that the new $y$-axis is parallel to the old $xy$-plane means that $a_{23}=0$.

Right now there are 11 unknown quantities (the $a$'s and $b$'s).

If you plug in one of your points (with both old and new coordinates) to the transformation at the top, you will get 3 equations. If you plug in the other point (both old and new coordinates) you will get 3 more. The constraints on the $a$'s give 6 more equations; there are 12 in all. EDIT: Due to joriki's point, one of these equations will be redundant, but we should still have enough information to solve them. (There may be a small finite number of solutions because the equations in the $a$'s are quadratic.)

Of course, solving this system of equations is its own challenge. A general method is elimination using Grobner bases, which is too elaborate for me to explain quickly. But the problem has many special features that make it seem like a shortcut should be available. I'll look for a shortcut and add it to this answer if I find one.

(P.S. I am taking your question as primarily practical, so not justifying my claims above about the form of the transformation. If you would like some theoretical background let me know and I will either explain briefly if I can or point you to good sources.)

Answer 4

OK, I've been thinking about this (like an engineer, not a mathematician), and here's my (half-baked) take:

I take Frame A, and translate it (TA) such that it's origin is at P1, then rotate it (RA) around Z and Y such that P2 is on the X axis: this gives me a new Frame A'.

I do the same thing with Frame B, translate (TB), and rotate (RB), which gives me Frame B'.

At this point, Frame A' = Frame B', and I have a 'route' from A to B:

$$TA \rightarrow RA \rightarrow -RB \rightarrow -TA$$

It's not the answer, but it's a start. Please tell me if I'm completely up the creek.