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
 A: 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.
A: Here is a valuable reference of geometric transformations:
http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html
