Rotation and Translation
Practical Exercises (Part II)

Finish Part I

If you haven't already, finish part I of this assignment. This assignment contains a solution to part I, so you should be sure to finish that assignment before starting this one!

Download Project

First, download Translate2.zip, and unzip it. Open up the solution Translate, compile, and run. You should see 3 cubes orbiting each other, and a red crosshair, as follows:

You can move and the crosshair around using the arrow keys (left/right and up/down) and the keys a and s (forward and backward) and the keys q, w, and e (yaw / pitch / roll)

. If you press the 1, 2, or 3 keys, then the world position and orientation of the crosshair will be sent to the corresponding object, calling the methods "setPositionFromWorldPostion" and "setOrientationFromWorldOrentation". Right now, the 1, 2, and 3 keys do nothing, since those methods have yet to be written!

Local space to parent space, Parent space to local space

Let's assume we have two objects, Object 1 and Object 2. We have the position and orientation of Object 1 in world space (as a rotational matrix M₁ and vector p₁) and the position and orientation of Object 2 in Object 1's space (as a rotational matrix matrix M₂ and vector p₂). That is, M₂ and p₂ are relative to the position and orientation of Object 1.

When it is time to render Object 2 to the screen, we will need its position and orientation in world space. How can we get the position and orientation of Object 2 in world space? By using the position and orientation of Object 1 as follows (assuming column major matrices and column vectors):

Position of Object 2 in World space p_{2_world} = M₁p₂ + p₁
Orientation of Object 2 in World space: M₁M₂

what if we have the position and rotation of Object 2 in world space, and we want the position and rotation of Object 2 in local space (that is, relative to Object 1)? Well, from above we have we have p_{2_world} = M₁p₂ + p₁, so with a little algebra we get:
p_{2_world} = M₁p₂ + p₁
p_{2_world} - p₁= M₁ p₂
M₁^-1(p_{2_world} - p₁) = M₁^-1 M₁ p₂
M₁^-1(p_{2_world} - p₁) = p₂

where M^-1 is the inverse of the matrix M. As it turns out, finding the inverse of a rotational matrix is very easy: the inverse of a rotational matrix is just its transpose. That is, for a rotational matrix M, M^-1 = M^T.

We can get the transpose of an Ogre::Matrix3 using the transpose method:

Ogre::Matrix3 rotationalMatrix;
// Code to create rotational matrix ...
Ogre::Matrix3 inverse = rotationalMatrix.Transpose()

Object parented to objects parented to objects ...

We can repeat this process as many times as we like. For instance, if we have the position and orientation of Object 1 world space, and the position and orientation of Object 2 relative to Object 1, and the position and orientation of Object 3 relative to object 2, then:

Position of Object 3 in world space: M₁(M₂p₃ + p₂) + p₁
Orientation of Object 3 in world space: M₁ M₂M₃

Of course, we can use similar logic to go from the world space back to local space as well ...

What to do

Ready to get to work? Here we go!

After you download Translate2.zip, you need to implement two methods:
- ```
void setPositionFromWorldPosition(Ogre::Vector3 worldPosition)
```
  Modifies the local position of the object so that it has the correct world position
- ```
 void setOrentationFromWorldOrientation(Ogre::Matrix3 worldOrientation)
```
  Modifies the local orientation of the object so that it has the correct world orientation
Recursion is your friend. I recommend writing some recursive helper methods that will make writing the above two methods much easier. While Part I of this assignment was relatively easy to do iteratively, this will be much easier to do recursively
As with Part I, you may wish to look at: Changing Coordinate Systems in 2D and Changing Coordinate Systems in 3D

Supporting files

Translate2.zip Everything you need
MovingObject.h Moving object header file
MovingObject.cpp Moving object source file. Note that this file contains the solution to Part I.
World.cpp World source file, so you can see how these moving objects are used