1. CSC4820/6820 Computer Graphics Algorithms Ying Zhu Georgia State University
Transformations

2. Outline 3D transformations Concatenation of transformations
Transformations in OpenGL
Anatomy of a simple OpenGL program

3. Coordinate systems
Each point and vector is defined in reference to a coordinate system
Once we fix the origin of the coordinate system, we can represent all points unambiguously
The same point may have different 3D coordinates in different coordinate systems
Remember that part of the job of 3D graphics pipeline is to convert vertices from model space to 2D window space
Each vertex with go through several coordinate systems along the pipeline

4. Coordinate systems
A 3D coordinate system is defined by its three base vectors (x, y, and z axis)
Get familiar with OpenGL’s coordinate system

5. Geometric transformation
In 3D graphics, animation is achieved by transforming geometric objects in the virtual world
Transforming geometric objects means transforming points (vertices) and vectors
Transformation of points and vectors can be conveniently achieved by matrix multiplications
Introduced by Larry Roberts in 1966

6. Linear Algebra Review Matrix: An array of elements
Vectors: A n x 1 matrix is called a column vector
In computer graphics, vertex coordinates and vectors (directed line segments) and are represented by column vectors.

7. Linear Algebra Review
Matrix-column vector multiplication

8. Linear Algebra Review
Matrix-matrix multiplication V1 V2 V3

9. Linear Algebra Review Homogeneous representation
A vertex coordinate representation has 1 as the 4th component.
OpenGL uses homogeneous coordinates for all vertices.
(x, y, z) => (x, y, z, 1)
Why use homogeneous coordinates?
So that all transformations can be defined by matrix multiplication.

10. Affine Matrix An affine matrix is a 4x4 matrix with last low [0 0 0 1]
Why include the bottom row of (0,0,0,1)?
It’ll be useful in perspective transforms.
Affine matrix is used to define affine transformation.

11. Affine Transformation
Any transformation preserving
collinearity (i.e., all points lying on a line initially still lie on a line after transformation)
ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation).
Affine transformations preserve parallel lines.
We can transform vertices with affine transformation.
Affine transformations are what we use all the time in computer graphics.

12. Other Transformations
Isometry (preserves distance)
Similarity (preserves angles)
Collineation (lines remains lines)
Perspective projections
Non-linear (lines become curves)
Twists, bends, etc.
Can be used for deformation.

13. Elementary Affine Transformations
Four elementary affine transformations
Translation
Scaling
Rotation (about an axis)
Shear
Translation, scaling and rotation are the most commonly used transformations.

14. Translation
Translation P’ P d x y

15. Scaling
Scaling

16. Rotation Rotation about the origin
Counter-clock wise rotation by 90 degree.

17. 2D Transformation Summary
Translation
Scaling
Rotation

18. Composition of Affine Transforms
Most of the time we want to apply more than one transformation.
Example:
Translate by vector t
Rotate by r degree about the origin
This can be represented by multiple matrix multiplications:

19. Composition of Affine Transforms
Matrix multiplication is associative:
We can group the combined transformation as
The affine matrices are applied right side first.
We can also apply all the transformations to form a single matrix
This called composing or concatenation of transforms.
The result is another affine transformation.

20. Composition of Affine Transforms
It is important to always remember that matrix multiplication is not commutative
The order of transforms DOES matter
E.g. scaling then rotating is usually different than rotating then scaling
Keep this in mind when you write OpenGL program – you need to know in which order your OpenGL transform calls are executed

21. Decomposition of Affine Transforms
Any affine transform can be decomposed into the product of a rotation, a scale, and a rotation.

22. Coordinate Transformations
We have discussed how to use transformation matrices to move points around
We can also think of them as simply changing the coordinate system in which the point is presented
The origin of the coordinate system is transformed but the relative position of the point is unchanged
In different contexts, either interpretation may be more suitable.

23. Why use matrix formalisms?
Can’t I just hardcode transforms rather than use the matrix multiplication?
Yes. But it’s harder to derive, harder to debug, and not any more efficient
All current graphics APIs use this matrix formalism
Must understand this to use OpenGL or other graphics libraries

24. 3D Affine Transformations
3D affine transformations are an extension of the 2D transformations.
Use 4x4 matrices instead of 3x3 matrices in 2D transformations.
For example
3D scaling

25. 3D Translation 3D translation matrix
OpenGL function: glTranslatef(tx, ty, tz)

26. 3D Scaling
3D scaling matrix
OpenGL Function: glScalef(Sx, Sy, Sz)

27. 3D Rotation About the Origin
3D rotation is more complicated than 2D rotation.
Rotation about X axis
Rotation about Y axis
Rotation about Z axis

28. Arbitrary 3D Rotations
Normal practice is to decompose an arbitrary 3D rotation into rotations about x, y, z axes.
Create a single transformation matrix by composing.

29. Rotation in OpenGL (GLfloat angle, GLfloat x, GLfloat y, GLfloat z)
Equivalent to multiplying the following matrix

30. Rotation in OpenGL
Rotations are specified by a rotation axis and an angle
Normal practice is to decompose an arbitrary 3D rotation into rotations about x, y, z axes
Achieved by 3 glRotatef() calls

31. Rotation About a Fixed Point
The order is important:
Translate to the origin
Rotate about the origin
Translate back
Example:
To rotate by r about z around

32. Quaternions
A quaternion is a 4-tuple of real numbers [s,x,y,z] or [s,v],
s is a scalar and v is a three-dimensional vector.
Quaternions provide an alternative for describing and manipulating rotations.
Quaternions are more efficient than rotation matrices
Can be interpolated to obtain smooth sequence of rotations for animation
You can convert between a transformation matrix and a quaternion

33. Multiple 3D Transformations
By combining a scaling, rotation, and a translation matrix we get a single transformation matrix.
Notice that in the single matrix, translation is separated from scaling and rotation part.
We can think of an affine matrix as having a scaling-rotation part and a translation part.
If a transformation consists of only translations and rotations, then it’s called a rigid-body transformation.

34. Transformations in OpenGL
Internally each vertex in OpenGL is represented as homogeneous points
i.e. a column vector with 1 as the last item.
However you don’t need to specify the 1 in your program
Each vertex is transformed by the modelview matrix.
There is always a current modelview matrix that is applied to all vertices.

35. Model-view Matrix

36. Transformation in OpenGL
It is important to understand the relationship among
Current modelview matrix,
Vertices of 3D Object
So you can understand why OpenGL code has to be arranged in certain way

37. What happens inside OpenGL during transformation?
Final modelview matrix
Matrices are often created dynamically by the OpenGL program.
Matrix multiplications are performed in vertex shaders.
Vertex coordinates before transformation
Vertex coordinates after transformation

38. Summary Affine transformations are used for transforming 3D objects.
Three elementary affine transformations:
Translation
Scaling
Rotation
3D rotation is the most complicated among the three.
If we use homogeneous coordinates, then all affine transformations can be represented by affine matrix multiplication.
Notice the pattern of transformation in OpenGL programming.

39. OpenGL Tutorial Programs
Download OpenGL transformation tutorial program from