3D transformation between two polylines problem

Say I have 2 separate objects.

One is a line defined by two points, the other is a polyline defined by three points.

Line 1 consists of the set of two points:

$a=(0,0,0)$ and b=$(0,0,1)$

Line 2 consists of the set of three points, which in this example are:

$c=(4,1,1)$, $d=(3,3,2)$, $e=(4,4,3)$

So line 2 has two end points and one "inner" point. This "inner" point can also be two or more, till $n-2$ for arbitrary polylines.

The problem is a way to transform line 2 so that it fits in line 1. In other words, scale and rotate all three points of line 2 so that it aligns with line 1.

Point $c$ would map to point $a$ and point $e$ would map to point $b$. Point $d$ also has to be calculated and added to the point set of line 1. I have linked a figure to illustrate the two objects. The positive Z-axis is pointing up.

As shown in the linked figure, line 1 is just a unit vector in the Z-direction. This line can also start from another position such as $a_2= (1,1,1)$ till point $b_2=(1,1,2)$ if starting at $(0,0,0)$ is a problem and as long as it is of unit length in the Z-direction. Then, in the end it is just a matter of translating all the transformed points of line 2 with the vector $(b_2 - a_2)$ to get back to the starting position of $(0,0,0)$.

So, in this example the end the result would consist of a set of three transformed points, shown in the matrix below: $$\begin{bmatrix} 0&0&0 \\\\ ...&...&... \\\\ 0&0&1 \end{bmatrix}$$

To summarize, the problem is how to transform line 2 into line 1 and also how to compute the "inner" points.

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Check out Umeyama's method in the following presentation (page 8): http://campar.in.tum.de/twiki/pub/Chair/TeachingSs09CAMP2Lecture/06PointBasedRegAndOptimization.pdf

The problem can be solved using algorithms solving the general transformation estimation problem between 3D point sets with known correspondences (assuming a linear transformation between the points).

You would basically solve the transformation estimation from point correspondences between the endpoints of the polylines if you approach it the way you described.

Peter

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Let $T_1$ be the transformation that translates by the vector $a-c$.

Let $\theta$ be the angle between the vectors $b-a$ and $e-c$, and let $N$ be the normal of the plane containing these two vectors. Let $T_2$ be the transformation that rotates by an angle $\theta$ around $N$.

Let $s$ be the length of $e-c$ divided by the length of $b-a$. Let $T_3$ be the transformation that scales by $s$.

Then the transformation you want is $T_1*T_2*T_3$. Apply this transformation to all the points of the polyline.

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Do the points given by vectors $\left[b-a\right]^T$ and $\left[e-c\right]^T$ have to be coplanar for this to work? I am wondering now if incoplanarity would affect $N=\left[b-a\right]^T\times{\left[e-c\right]^T}$. Didn't try it out yet, just wondering... –  sPLUS Feb 18 '14 at 16:33