3D Transformations

3D Transformations

3D Rotations

• Rotation matrices
• Euler angles
• Three rotations about x-, y- and z-axis
• 3D homogeneous and affine transformations
• Euler angle problems – “gimbal lock”

Recap: 2D Rotation

• Homogenous transformation:
  $\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} cos \theta & -sin \theta \ sin \theta & cos \theta \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$

  $\begin{bmatrix} x' \ y' \ 1 \end{bmatrix} = \begin{bmatrix} cos \theta & -sin \theta & 0 \ sin \theta & cos \theta & 0 \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ 1 \end{bmatrix}$

• (x', y') > (x, y)

• $\theta$ is a positive angle along the counter-clockwise direction

3D Coordinate System Conventions

• Positive Z direction for left-handed coordinates.
• Positive Z direction for right-handed coordinates.
• We use right-handed coordinates unless otherwise noted.

3D Rotation Conventions

• Right-hand screw rule:
  • Thumb aligns with axis rotated about.
  • Other fingers indicate positive rotation.
• For right-handed coordinates.
• 2D Rotation about origin.
• 3D Rotation about an axis or arbitrary line.
• Positive angle along the counter-clockwise direction.

3D Rotations about Z-axis

• Find matrix for a positive rotation of $\theta$ about the z-axis.

  $\begin{bmatrix} x' \ y' \ z' \end{bmatrix} = \begin{bmatrix} cos \theta & -sin \theta & 0 \ sin \theta & cos \theta & 0 \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix}$

• (x', y') > (x, y)

• (x', y', z') > (x, y, z)

• The point only moves in the X-Y plane that is orthogonal to Z axis, z value remains the same.

• Not homogeneous coordinates (z value is not affected).

3D Rotations – by Swapping Axes

• Rotation about Z-axis:

  $\begin{bmatrix} z' \ x' \ y' \end{bmatrix} = \begin{bmatrix} cos \theta & -sin \theta & 0 \ sin \theta & cos \theta & 0 \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} z \ x \ y \end{bmatrix}$

• $x \rightarrow z, y \rightarrow x, z \rightarrow y$

• Final look of the rotation matrix about Y axis:

  $\begin{bmatrix} x' \ y' \ z' \end{bmatrix} = \begin{bmatrix} cos \theta & 0 & sin \theta \ 0 & 1 & 0 \ -sin \theta & 0 & cos \theta \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix}$

Reordering the Matrix

Step 1: Rearrange the Input Columns

• The input vector order changes from:

  $\begin{bmatrix} z \ x \ y \end{bmatrix}$

• To achieve this:

  • The 1st column of the matrix (corresponding to z) becomes the 3rd column.
  • The 2nd column of the matrix (corresponding to x) becomes the 1st column.
  • The 3rd column of the matrix (corresponding to y) becomes the 2nd column.

• This yields the Intermediate Matrix:

  $Intermediate Matrix = \begin{bmatrix} cos \theta & 0 & -sin \theta \ 0 & 1 & 0 \ sin \theta & 0 & cos \theta \end{bmatrix}$

Step 2: Rearrange the Output Rows

• The output vector order changes from:

  $\begin{bmatrix} z' \ x' \ y' \end{bmatrix} to \begin{bmatrix} x' \ y' \ z' \end{bmatrix}$

• To achieve this:

  • The 1st row of the matrix (corresponding to z') becomes the 3rd row.
  • The 2nd row of the matrix (corresponding to x') becomes the 1st row.
  • The 3rd row of the matrix (corresponding to y') becomes the 2nd row.

• This yields the Final Matrix:

  $Final Matrix = \begin{bmatrix} cos \theta & 0 & sin \theta \ 0 & 1 & 0 \ -sin \theta & 0 & cos \theta \end{bmatrix}$

• Rotation about X-axis:

  $R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \ 0 & cos \theta & -sin \theta \ 0 & sin \theta & cos \theta \end{bmatrix}$

• Hints: $y \rightarrow z, z \rightarrow x, x \rightarrow y$

3D Rotation Matrices

• Rotation about Y-axis:

  $R_y(\theta) = \begin{bmatrix} cos \theta & 0 & sin \theta \ 0 & 1 & 0 \ -sin \theta & 0 & cos \theta \end{bmatrix}$

• Rotation about X-axis:

  $R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \ 0 & cos \theta & -sin \theta \ 0 & sin \theta & cos \theta \end{bmatrix}$

• Rotation about Z-axis:

  $R_z(\theta) = \begin{bmatrix} cos \theta & -sin \theta & 0 \ sin \theta & cos \theta & 0 \ 0 & 0 & 1 \end{bmatrix}$

• The rows with “1” preserve the values of related components.

  • e.g., z component of a point does not change after the point rotated about z axis

Euler Angles

• We need 3 angles to describe the orientation of a rigid body.

• Number of possible rotations:

  • XXX, XXY, XXZ, XYX, XYY, XYZ, XZX, XZY, XZZ,
  • YXX, YXY, YXZ, YYX, YYY, YYZ, YZX, YZY, YZZ,
  • ZXX, ZXY, ZXZ, ZYX, ZYY, ZYZ, ZZX, ZZY, ZZZ

• Generates an arbitrary rotation matrix (rotation about an arbitrary axis passing through the origin).

• Simple idea: use an ordered combination of rotations about the x-, y- and z-axes, e.g. first X, then Y then Z (all variations used).

• NB: order matters!

  $R = R<em>z(\gamma)R</em>y(\beta)R_x(\alpha)$

3D Rotation Strangeness

• Suitable (ordered) combination of rotations about any two axes can generate any 3D rotation.

  $R<em>z(90^\degree) = R</em>x(90^\degree)R<em>y(90^\degree)R</em>x(-90^\degree)$

  $R<em>z(\theta) = R</em>x(90^\degree)R<em>y(\theta)R</em>x(-90^\degree)$

Steps to combine the rotations

1. Represent the rotation as matrices

   • Rotation Matrix around X-axis ($R_x(\theta)$) :

     $R_x(\theta) = \begin{bmatrix} 1 &amp; 0 &amp; 0 \ 0 &amp; cos \theta &amp; -sin \theta \ 0 &amp; sin \theta &amp; cos \theta \end{bmatrix}$

   • Rotation Matrix around Y-axis ($R_y(\theta)$) :

     $R_y(\theta) = \begin{bmatrix} cos \theta &amp; 0 &amp; sin \theta \ 0 &amp; 1 &amp; 0 \ -sin \theta &amp; 0 &amp; cos \theta \end{bmatrix}$

2. Define the Rotations

   • First rotation: $R_x(-90^\degree)$

     $R_x(-90^\degree) = \begin{bmatrix} 1 &amp; 0 &amp; 0 \ 0 &amp; 0 &amp; 1 \ 0 &amp; -1 &amp; 0 \end{bmatrix}$

   • Second rotation: $R_y(90^\degree)$

     $R_y(90^\degree) = \begin{bmatrix} 0 &amp; 0 &amp; 1 \ 0 &amp; 1 &amp; 0 \ -1 &amp; 0 &amp; 0 \end{bmatrix}$

3. Combine the Rotations

   • The combined rotation matrix $R<em>{combined}$ is: $R</em>{combined} = R<em>x(90^\degree) \cdot R</em>y(90^\degree) \cdot R_x(-90^\degree)$

     • Compute $R<em>y(90^\degree) \cdot R</em>x(-90^\degree)$

       $\begin{bmatrix} 0 &amp; 0 &amp; 1 \ 0 &amp; 1 &amp; 0 \ -1 &amp; 0 &amp; 0 \end{bmatrix} \cdot \begin{bmatrix} 1 &amp; 0 &amp; 0 \ 0 &amp; 0 &amp; 1 \ 0 &amp; -1 &amp; 0 \end{bmatrix} = \begin{bmatrix} 0 &amp; -1 &amp; 0 \ 0 &amp; 0 &amp; 1 \ -1 &amp; 0 &amp; 0 \end{bmatrix}$

     • Compute $R<em>x(90^\degree) \cdot (R</em>y(90^\degree) \cdot R_x(-90^\degree))$
       $\begin{bmatrix} 1 &amp; 0 &amp; 0 \ 0 &amp; 0 &amp; -1 \ 0 &amp; 1 &amp; 0 \end{bmatrix} \cdot \begin{bmatrix} 0 &amp; -1 &amp; 0 \ 0 &amp; 0 &amp; 1 \ -1 &amp; 0 &amp; 0 \end{bmatrix} = \begin{bmatrix} 0 &amp; -1 &amp; 0 \ 1 &amp; 0 &amp; 0 \ 0 &amp; 0 &amp; 1 \end{bmatrix}$

4. Apply the Combined Rotation

   • Let's assume the initial point is $P = [x, y, z]^T$. After applying $R<em>{combined}$, the final point becomes: $P' = R</em>{combined} \cdot P$

   • For an example point $[1, 0, 0]$:

     $P' = \begin{bmatrix} 0 &amp; -1 &amp; 0 \ 1 &amp; 0 &amp; 0 \ 0 &amp; 0 &amp; 1 \end{bmatrix} \cdot \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix} = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}$

Euler Angle Troubles

• Gimbal lock occurs when one axis of rotation lines up with another, losing a degree of freedom.

Gimbal Lock

• $p' = R<em>z(\gamma)R</em>y(\beta)R_x(\alpha)p$
• $\beta = 90^\degree$
• The rotation axis of $R<em>x$ is made to line up with the rotation axis of $R</em>z$. We are no longer able to move in the direction that $R_x$ previously enabled us to.

Homogeneous Coordinates in 3D

• Add a “1” to the last coordinate to make translation a linear operation (i.e. a matrix multiplication).

  $\begin{bmatrix} p'<em>1 \ p'</em>2 \ p'<em>3 \end{bmatrix} = \begin{bmatrix} p</em>1 \ p<em>2 \ p</em>3 \end{bmatrix} + \begin{bmatrix} t<em>1 \ t</em>2 \ t<em>3 \end{bmatrix}$$\begin{bmatrix} p'</em>1 \ p'<em>2 \ p'</em>3 \ 1 \end{bmatrix} = \begin{bmatrix} 1 &amp; 0 &amp; 0 &amp; t<em>1 \ 0 & 1 & 0 & t</em>2 \ 0 &amp; 0 &amp; 1 &amp; t<em>3 \ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} p</em>1 \ p<em>2 \ p</em>3 \ 1 \end{bmatrix} = \begin{bmatrix} p<em>1 + t</em>1 \ p<em>2 + t</em>2 \ p<em>3 + t</em>3 \ 1 \end{bmatrix}$

• Represent T as a matrix.

3D Affine Transformations

• Recall 2D:

  $\begin{bmatrix} p'<em>1 \ p'</em>2 \ 1 \end{bmatrix} = \begin{bmatrix} m<em>{11} & m</em>{12} &amp; m<em>{13} \ m</em>{21} &amp; m<em>{22} & m</em>{23} \ 0 &amp; 0 &amp; 1 \end{bmatrix} \begin{bmatrix} p<em>1 \ p</em>2 \ 1 \end{bmatrix}$

• So in 3D:

  $\begin{bmatrix} p'<em>1 \ p'</em>2 \ p'<em>3 \ 1 \end{bmatrix} = \begin{bmatrix} m</em>{11} &amp; m<em>{12} & m</em>{13} &amp; m<em>{14} \ m</em>{21} &amp; m<em>{22} & m</em>{23} &amp; m<em>{24} \ m</em>{31} &amp; m<em>{32} & m</em>{33} &amp; m<em>{34} \ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} p</em>1 \ p<em>2 \ p</em>3 \ 1 \end{bmatrix}$

• 12 degrees of freedom

• 6 degrees of freedom

Affine Transformation in Homogeneous Coordinates

• Example: Transformation Decomposition

  Given Transformation Matrix

  $T = \begin{bmatrix} 2 &amp; 0 &amp; 0 &amp; 5 \ 0 &amp; 3 &amp; 0 &amp; 2 \ 0 &amp; 0 &amp; 4 &amp; -3 \ 0 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix}$

  Components:

  1. Scaling:
     • The diagonal elements [2, 3, 4] indicate scaling along the x, y, and z axes, respectively.
  2. Translation:
     • The last column [5, 2, -3] represents a translation by (tx = 5, ty = 2, tz = −3).
  3. No Rotation or Shearing:
     • The off-diagonal elements are all 0, indicating no rotation or shearing.

Applying to a Point

• Given a point $[1, 1, 1, 1]^T$:

  $T \cdot \begin{bmatrix} 1 \ 1 \ 1 \ 1 \end{bmatrix} = \begin{bmatrix} 2 \cdot 1 + 0 \cdot 1 + 0 \cdot 1 + 5 \cdot 1 \ 0 \cdot 1 + 3 \cdot 1 + 0 \cdot 1 + 2 \cdot 1 \ 0 \cdot 1 + 0 \cdot 1 + 4 \cdot 1 + (-3) \cdot 1 \ 1 \end{bmatrix} = \begin{bmatrix} 7 \ 5 \ 1 \ 1 \end{bmatrix}$

Compound Transformations

• Homogeneous transformations in 3D are just as easily concatenated as in 2D:

  $p' = T<em>3 T</em>2 T_1 p$

• Remember that the order of operations matters in general:

  $T<em>1 T</em>2 \neq T<em>2 T</em>1$

Order of Transformations

2D Local/Global Coordinates

• Transformation matrix that combines rotation and translation.

3D Local/Global Coordinates

• $p<em>{world} = T</em>1 p_{wall_e}$
• $\tilde{p}<em>{world} = R</em>1 \tilde{p}<em>{wall_e} + t</em>1$

Coordinate Transformations

• Hand coordinates to arm coordinates: $p<em>{arm} = T</em>3 p_{hand}$
• Arm coordinates to Wall-e coordinates: $p<em>{wall_e} = T</em>2 p_{arm}$
• Wall-e coordinates to world coordinates: $p<em>{world} = T</em>1 p_{wall_e}$
• Hand coordinates to world coordinates: $p<em>{world} = T</em>1 T<em>2 T</em>3 p_{hand}$
• Order of Transformations is important. These transformations are in a hierarchy.
• For example, if the hand moves, only the $T_3$ matrix would change because the hand is relative to the arm. The other transformations will remain the same.