# Homogeneous Linear Transformation

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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Instead of four steps, it seems like it should be three. Move to origin, rotate to its final orientation, then translate to second position. But I guess if your scheme requires it, you could rotate to a "standard" orientation, and then rotate it to the final orientation from standard. Still, those rotations should probably take place next to each other, not alternating with the translations. – rschwieb Sep 27 '12 at 18:23
That is why the final equation I stated only has three terms: translation from 1 to origin, rotation from 1 to 2 and translation from origin to 2. However, if you could provide an example of how to do this it would be great. – user Sep 27 '12 at 18:26
Oh ok, great :) Sorry, sometimes without TeX things like that pass me by. Can you produce a rotation to align two oriented spheres centered at the origin? That's the hardest part, and I think you understand the translation, right? – rschwieb Sep 27 '12 at 18:40
BTW translation is not a linear transformation. – rschwieb Sep 27 '12 at 18:42
I understand the translation conceptually. The translation matrix is T = [1 0 0 d(x)| 0 1 0 d(y) | 0 0 1 d(z) | 0 0 0 1]. What I don't understand is if you had a triangle for example, do you have to compute the distances for each point relative to the origin or how does that work? Let's say I had a triangle with points (1,5),(2,8),(1,11), how would that work with the translation matrix? – user Sep 27 '12 at 18:45

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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I'm doing this all in my head, so apologies for any typos that might have slipped in. – rschwieb Sep 27 '12 at 19:20
Hey, this helps but can you show the actual math for the first translation? As I mentioned in one of my comments, I know what the 4x4 translation matrix is. I'm just not sure how to utilize it in problems. – user Sep 27 '12 at 19:25
The "translation matrix" is $1\times 2$. The math is: $[0,1]-[0,1]=[0,0]$,$[0,2]-[0,1]=[0,1]$,$[1,2]-[0,1]=[0,1]$, $[1,1]-[0,1]=[1,0]$. Those are the new vertices of the translated square. – rschwieb Sep 27 '12 at 19:27
Oh right, for your example it is not 4x4 but is it possible for you to show a translation of an rigid object in 3-space? – user Sep 27 '12 at 19:28
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 $[a,b,c]$. – rschwieb Sep 27 '12 at 19:30

Here is a valuable reference of geometric transformations:

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

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