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This is a general question but can someone provide a worked example of a 3d transformation? Or a link that has a worked example of one? I've looked on the internet for a long time and couldn't find anything except explanations on how to do a transformation but no mathematical examples.

I am trying to transform an object from one location to another where initial (location 1) and final coordinates (location 2) are given. I know that you have to translate the object to the origin (which I'll call O for origin), rotate is in alignment with the origin, translate it to the new location and then rotate it to the final coordinates. I understand conceptually what's going on but not clear on the math.

So basically: t(O-->2).R(O-->2).R(1-->O).t(1-->O) = F (frame transformation)

I also know that the two rotation matrices can be calculated simply as R(1-2) so the equation becomes: t(O-->2).R(1-->2).t(1-->O)=F

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    Hope the addition to my answer helps.2012-09-27

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Suppose you have an oriented sphere $O_1$ initialy, which moves to become the oriented sphere $O_2$ finally.

Suppose $O_1-w$ and $O_2-v$ are the translations of those spheres to the origin. The spheres can be rotated with a rotation $R$ so that they become the same, that is:

$R(O_1-w)=O_2-v$

Then $R(O_1-w)+v=O_2$.

This is where I'm getting the "shift rotatate shift" scheme from.


Here is an example in the plane. Suppose you want to move the oriented square with vertices $A,B,C,D$ at $(0,1),(0,2),(1,2),(1,1)$ to $(-1,-1),(-1,-2),(-2,-2),(-2,-1)$.

First, translate $A$ to the origin by transforming the plane with $T_1(x)=x-[0,1]$.

Then rotate the square around $A$ so that its orientation matches the second square: $R(x)=x\begin{bmatrix}0&-1\\-1&0\end{bmatrix}=M$ where $M$ is the rotation matrix rotating the plane 180 degrees around the origin.

Then, translate the result to the final position with the translation $T_2(x)=x+[-1,-1]$.

So the final transformation is $M(x-[0,1])+[-1,-1]$, which takes in any coordinate pair $[x_1,x_2]$ and outputs its new location.

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    Yes: every translation on 3-space is affected by adding whatever fixed $[a,b,c]$ you want to every point in the space. This translation moves $[0,0,0]$ to $[a,b,c]$.2012-09-27
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Here is a valuable reference of geometric transformations:

http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html