Content
- World Coordinate System
- Combined Transformations
- Articulated Structures
Combined Transformations
Graphics 1 CMP-5010B
Dr. David Greenwood
February, 2022
Combining 2D Transformations
Order matters!
Recap
So far we have looked at individual 2D transformations applied to the vertices of a 2D polygon.
Recap
Having put in place a uniform method, applicable to all transformations, we can now look at combining transformations.
Recap
We will show now that the order of the applied transformations is absolutely crucial to obtain the desired results!
World Coordinate System
Where in the world do we start?
World Coordinate System
Most graphics systems adopt a World Coordinate System (WCS), with a camera in a particular position and orientation.
World Coordinate System
For example in OpenGL, the camera is at the origin of the WCS pointing in the negative z-direction with its “up” vector pointing in the y-direction.
World Coordinate System
So far we have been working with 2D transformations.
Given the description of the OpenGL coordinate system, what has been the significance of the z-axis?
- Answer: It is the axis of rotation.
World Coordinate System
The WCS is represented by a right-handed coordinate system, with the z-axis popping out of the screen.
- For 2D, we draw in the x-y plane, and rotate about the z-axis.
Combined Transformations
Order matters…
Recap: Matrix Multiplication
Matrix multiplication is associative:
\[ABC=A(BC)=(AB)C\]
Matrix multiplication is not commutative:
\[AB \neq BA\]
Combined Transformations
Start with two common concatenated transformations:
- Rotate the model, then translate it.
- Translate the model, then rotate it.
Combined Transformations
Start with two common concatenated transformations:
\[ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & ~0& ~t_x \\ 0 & ~1& ~t_y \\ 0 & ~0& ~1 \end{bmatrix} \begin{bmatrix} \cos \alpha& -\sin \alpha& ~0 \\ \sin \alpha&~ ~\cos \alpha& ~0 \\ 0& 0& ~1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]
\[ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} \cos \alpha& -\sin \alpha& ~0 \\ \sin \alpha&~ ~\cos \alpha& ~0 \\ 0& 0& ~1 \end{bmatrix} \begin{bmatrix} 1 & ~0& ~t_x \\ 0 & ~1& ~t_y \\ 0 & ~0& ~1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]
Multiplying out the first example from right to left:
- \[ \begin{aligned} \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} &= % \begin{bmatrix} 1 & ~0& ~t_x \\ 0 & ~1& ~t_y \\ 0 & ~0& ~1 \end{bmatrix} % \begin{bmatrix} x \cos \alpha - y \sin \alpha \\ x \sin \alpha + y \cos \alpha \\ 1 \end{bmatrix} \\[10pt] % \Rightarrow % \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} &= \begin{bmatrix} (x \cos \alpha - y \sin \alpha) + t_x \\ (x \sin \alpha + y \cos \alpha) + t_y \\ 1 \end{bmatrix} \end{aligned} \]
Multiplying out the second example from right to left:
- \[ \begin{aligned} \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} &= % \begin{bmatrix} \cos \alpha& -\sin \alpha& ~0 \\ \sin \alpha&~ ~\cos \alpha& ~0 \\ 0& 0& ~1 \end{bmatrix} % \begin{bmatrix} x + t_x \\ y + t_y \\ 1 \end{bmatrix} \\[10pt] % \Rightarrow % \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} &= \begin{bmatrix} (x + t_x) \cos \alpha - (y + t_y) \sin \alpha \\ (x + t_x) \sin \alpha + (y + t_y) \cos \alpha \\ 1 \end{bmatrix} \end{aligned} \]
Combined Transformations
\[ RTv \neq TRv \]
The order of the transformations is important.
Articulated Structures
Hierarchical transformations…
Make the articulated structure from only the square polygon.
Assumptions
- The objects are in order of their position in the hierarchy, i.e. we start with the first object in the hierarchy, then the next and so on.
- The articulations should be operative rather than static, i.e for different value angles, the structure will still be connected.
- We only use the metric \(s\) in our transformation matrices.
- The solution may not be unique.
The base
The base
Order of transformations:
- scale in y, \(~s_y = 2\)
- translate in y, \(~t_y = 1s\)
The base
\[ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1s \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]
Link 1
Link 1
Order of transformations:
- scale in x, \(~s_x = 3\)
- translate in x, \(~t_x = 1.5s\)
- rotate, \(~\alpha = 30^{\circ}\)
- translate in y, \(~t_y = 2\)
Link 1
\[ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 2s \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 1.5s \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]
Link 2
Link 2
Order of transformations:
- scale in x, \(~s_x = 2\)
- translate in x, \(~t_x = s\)
- rotate, \(~\beta = 45^{\circ}\)
- translate in x, \(~t_x = 3s\)
- rotate, \(~\alpha = 30^{\circ}\)
- translate in y, \(~t_y = 2s\)
Link 2
\[ \begin{aligned} \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} =~ &\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 2s \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 3s \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \\[10pt] &\begin{bmatrix} \cos \beta & - \sin \beta & 0 \\ \sin \beta & \cos \beta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & s \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \end{aligned} \]
NB: equation runs over two lines.
Articulated Arm Template
There is a template solution for each of the three parts.
Articulated Arm Template
Base:
\[ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & ? \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & ? & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]
Articulated Arm Template
Link 1:
\[ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & ? \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & ?\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} ? & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \]
\[ \alpha = ?^{\circ} \]
Articulated Arm Template
Link 2
\[ \begin{aligned} \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} =~ &\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & ? \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & ? \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \\[10pt] &\begin{bmatrix} \cos \beta & - \sin \beta & 0 \\ \sin \beta & \cos \beta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & ? \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} ? & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \end{aligned} \]
\[\alpha = ?^{\circ} ~, ~ \beta = ?^{\circ}\]
Summary
- World Coordinate System
- Combined Transformations
- Articulated Structures
Reading:
- Hearn, D. et al. (2004). Computer Graphics with OpenGL.
- Strang, Gilbert, et al. (1993) Introduction to linear algebra.