1. Overview of 3D Computer Graphics
Aknowledgements: to my best knowledges, these slides originally come from University of Washington,
and subsequently are adapted by colleagues.

2. Overview Modeling of 3D objects Polygonal representation (geometry)
Parametric represenation (curves and surfaces)
Rendering:
(lighting and illumination models)
on-line graphics pipeline
off-line global illumination methods: ray-tracing and radiosity

3. Graphics pipeline 2D and 3D Euclidean transformations
Hierarchical modeling
Introduction to projective geometry
(real) camera modeling from vision
(synthetic) camera for graphics: projective transformation
Hiden surface removal

4. Polygonal representation
Hierarchical data structure
An object is surfaces
A surface is polygons
A polygon is vertices
Mannually designed
Automatic from scanning of real objects
Automatically generated from other descriptions (implicit)

5. Rendering Triangles Developer specifies in OpenGL/DirectX
Rendering state (transformation matrices, texture mapping, etc…)
Triangles and vertices (with colors, etc…)
Graphics card processing
Vertex processing (model, view, projection transformation)
Triangle scan-conversion
Pixel processing (value set to Gouraud-interpolated colors)

6. Rendering Triangles Fully programmable!
Developer specifies in OpenGL/DirectX
Rendering state (transformation matrices, texture mapping, etc…)
Triangles and vertices (with colors, etc…)
Graphics card processing
Vertex processing (model, view, projection transformation)
Triangle scan-conversion
Pixel processing (value set to Gouraud-interpolated colors)
Fully
programmable!

7. Transform and Lighting Background
GPU Transform and Lighting of vertices – available on consumer graphics cards circa 1999
Also called fixed function pipeline
Previously normally done on CPU
Forced the use of Gouraud / Phong lighting
Other variations couldn’t be handled by the graphics hardware
Offloading T&L onto the GPU freed the CPU for other tasks (physics / collision detection / AI), but the drawback was that it forced the use of the basic Gouraud / Phong model for lighting – any other variations could not be handled by the graphics hardware.

8. Fixed Function Pipeline
Vertex Data
(Model space)
Fixed Function Transform and Lighting
Geometry Stage
Clipping and Viewport Mapping
Texture Stages
Rasterizer Stage
Fog, Alpha, Stencil Depth Testing

9. Open GL
OpenGL (Open Graphics Library) is a standard specification defining a cross-language cross-platform API for writing applications that produce 3D computer graphics (and 2D computer graphics as well).
The interface consists of over 250 different function calls which can be used to draw complex three-dimensional scenes from simple primitives. OpenGL was developed by Silicon Graphics Inc. (SGI) in 1992[1] and is popular in the video games industry where it competes with Direct3D on Microsoft Windows platforms (see Direct3D vs. OpenGL).
OpenGL is widely used in CAD, virtual reality, scientific visualization, information visualization, flight simulation and video game development.
OpenGL is a low-level, procedural API, requiring the programmer to dictate the exact steps required to render a scene.
the red book:

10. 2D Transformations

11. Reading Required: Hearn & Baker, 5.1-5.5 Optional:
Foley, et al.,
David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, 2nd ed., McGraw-Hill, New York, 1990, Chapter 2.
Linear algebra: Gilbert Strang

12. Intuitive introduction
Naturally everything starts from the known vector space
add two vectors
multiply any vector by any scalar
zero vector – origin
finite basis

13. Vector space to affine: isomorph, one-to-one
vector to Euclidean as an enrichment: scalar prod.
Pts, lines, parallelism
Angle, distances, circles
Affine and eucl are finite geometry in which we are handling only fnite pts

14. Points versus vectors (geometry vs. algebra)
Both can be represented as a list of coordinates
The distinction enables us to check validity of geometric operations.
Geometry:
(linear) algebra:
Point
A location in space
Its coordinate values depend on the choice of coordinate system
P+Q is not defined
P is not defined, but
is defined if
Vector
Has direction and magnitude
Free to move anywhere (free vectors)
v + w is defined
v is defined

15. An affine space is similar to a vector space, only viewed differently
An affine space is a vector space if we fix one pt as the origin.
A vector space is an affine space if we ‘forget’ the ‘zero’.
Affine geometry reduces to linear algebra

16. Geometric transformations
Geometric transformations map points in one space to points in another space: (x’, y’, z’) = f(x, y, z)
These transformations can be very simple, such as scaling each coordinate, or complex, such as non-linear twists and bends.
We’ll focus on transformations that can be represented easily with matrix operations: linear transformations.

17. Representation We can represent a point in the plane as
a column vector
a row vector

18. Representation, cont.
We can represent a 2D transformation M by a matrix
If p is a column vector, then M goes on the left:
If p is a row vector, then MT goes on the right:
We will use column vectors.

19. 2D Transformations
Here’s all you get with a 2x2 transformation matrix M:
So:

20. Identity Suppose we choose a=d=1, b=c=0: Gives the identity matrix:
Doesn’t move the points at all

21. Scaling
Suppose we set b=c=0, but let a and d take on any positive value:
Gives a scaling matrix:
Provides differential scaling in x and y:

22. Scaling

23. Suppose we keep b=c=0, but let either a or d go negative:
Examples:

24. Now let’s leave a=d=1 and experiment with b ….
The matrix
gives:

25. Effect on unit square
Let’s see how a general 2x2 transformation M affects the unit square:

26. Effect on unit square, cont.
Observe:
Origin invariant under M
M can be determined just by knowing how the corners (1,0) and (0,1) are mapped
a and d give x- and y-scaling
b and c give x- and y-shearing

27. Rotation

28. Inverse Rotation

29. Limitations of the 2x2 matrix
A 2x2 matrix allows
Scaling
Rotation
Reflection
Shearing
Q: What important operation does that leave out?

30. Homogeneous coordinates, just as a convenient tool!
Idea is to loft the problem up into 3-space, adding a third component to every point:
And then transform with a 3x3 matrix:

31. Homogeneous coordinates

32. Rotation around arbitrary point
With homogeneous coordinates, we can specify rotations about any point p with a matrix
Translate object so that pivot is at origin
Rotate object
Translate so that pivot is back to original position
Order is important!

33. Affine transformations
All of the transformations we have looked at so far are examples of affine transformations
Some properties of affine transformations:
Lines map to lines
Parallel lines remain parallel
Midpoints map to midpoints (in fact, ratios are always preserved)

34. Affine versus Euclidean Geometry
Affine geometry
Operations for vectors: v  w, v
Operations for points: P – Q, P  v
No concepts of length and angle
Eucliean geometry
Add one operation to affine geometry: inner product
Thus introduce angles and distances
Concepts defined: length of vector, normalize a vector, distance between points, angle between vectors, orthogonality, orthogonal projection

35. Angle
Angle between u and v = where are unit vectors u  v

36. Orthogonal projection
Orthogonal projection of a given vector u onto is u
unit vector

37. Summary What you should take away from this lecture:
The meaning of all the boldfaced terms
How points and transformations are represented
What all the elements of a 2x2 transformation matrix do
What homogeneous coordinates are and how they work for affine transformations
How to concatenate transformations
Properties of affine transformations

38. 3D Transformations

39. Reading
Required:
Hearn & Baker,
Optional
Hill

40. Basic 3D transformations: scaling
Some of the 3D transformations are just like 2D ones. For example, scaling
Scaling

41. Translation in 3D

42. Rotation in 3D
Rotation now has more possibilities in 3D:

43. Rotation in 3D
What about the inverses of 3D rotations?

44. Rotation around an axis by an angle …

45. Shearing in 3D

46. Compositing multiple transformations
Expressed as matrix multiplications:
p’ = M1 M2 p
means apply transformation M2 first, followed by M1
Historically, the vectors in graphics were row vectors,
but columns are more popular.

47. Compositing multiple transformations
3 steps
1) T(-2, -3)
2) R(30)
3) T(2, 3)
q’=T(2,3) R(30) T(-2,-3) q

48. Change of the coordinate frame, Change of basis
Specify: an origin, and two vectors
Let’s first look at a 2D example
p=(2, 1)T, what are the coordinates of p with respect to the new basis (coordinate system)? p

49. Linear dependence or independence
Spanning a space or a subspace, ‘generators’
Basis for a space or a subspace (a set of vectors)
Dimension of a subspace (a number)
A basis for V is a set of vectors: linearly independent (not too many) and span the space V (not too few)
Basis is not unique, coordinates are, when the basis is fixed.

50. New basis: O=(4,3)T, e0=(-1,0)T, e1=(0,-1)Tp

51. Let coordinates of p in new basis be (0, 1)T
New basis: O=(4,3)T, e0=(-1,0)T, e1=(0,-1)T
Represent as homogeneous coordinates:
O=(4,3,1)T, e0=(-1,0,0)T, e1=(0,-1,0)T
For vectors, last component of homogeneous coordinates is 0, for convenience only for the time being.

52. Basis change: e’_i = B e_i
Pt change, x’ = A x
x=\sum x_i e_i,
x=\sum x_i B^{-1} e’
= \sum B^{-1} x_i e’
x’ = B^{-1} x

53. Change of basis in 3D one origine, three vectors
3D case
Given an orthonormal basis (O, e0, e1, e2)
e0, e1, e2 are unit length vectors
e0, e1, e2 are mutually orthogonal
To change basis, need matrix M=(e0 e1 e2 O)
It is easy to find the inverse of such a matrix

54. We can write M as
Finding the inverse
Transformation between two coordinate systems involves a translation, followed by a general rotation.

55. Cross product The cross product of two vectors in 3-space is
An easy way to remember this is to write it in the form

56. Cross product: define coordinate system
Given two vectors u and v (not parallel)
Define a coordinate system such that
One of the axis is in the direction of u
Another axis is perpendicular to both u and v
Solution:
Direction of x-axis: e0 = u
Direction of z-axis: e2 = u x v
Direction of y-axis: e1 = e2 x e0 (note:  e0 x e2 )
The Gram-Shmidt process

57. Cross product: normal
Find the normal of a triangle

58. OpenGL Transformation functions

59. OpenGL OpenGL maintains a matrix stack called Modelview
Transformations are accumulated by postmultiplication with the top-of-the-stack matrix M
glLoadIdentity() M  I
glTranslatef(tx, ty, tz) M  MT
glRotatef(theta, x, y, z) M  MR
glScalef(sx, sy, sz) M  MS
Points are multiplied to the top-of-the-stack matrix during drawing
glMatrixMode( GL_MODELVIEW );
/* Subsequent matrix commands will affect the modelview matrix */
glLoadIdentity();
/* Initialise the modelview to identity */
glTranslatef( 0, 0, -3 );
/* Translate the modelview 3 units along the Z axis */

60. We issue a polygon – a green square oriented in the xy plane.
glBegin( GL_POLYGON ); /* Begin issuing a polygon */
glColor3f( 0, 1, 0 ); /* Set the current color to green */
glVertex3f( -1, -1, 0 ); /* Issue a vertex */
glVertex3f( -1, 1, 0 ); /* Issue a vertex */
glVertex3f( 1, 1, 0 ); /* Issue a vertex */
glVertex3f( 1, -1, 0 ); /* Issue a vertex */
glEnd(); /* Finish issuing the polygon */

61. Summary What you should take away from this lecture:
The meaning of all the boldfaced terms
How transformations are generalized from 2D to 3D
How to concatenate transformations
How to define a coordinate system from two given vectors
How to change basis
How to represent points and vectors in homogeneous coordinates
How OpenGL matrix stack works