1. Mathematical Fundamentals
Computer Graphics
Mathematical Fundamentals

2. Matrix Math 2x4 Matrix dimensions are specified as: rows x cols 1 2 35 6 7 8
2x4

3. Matrix Addition 1 2 3 4 1 2 3 4 2 4 6 8 + =
2x2
2x2
2x2
The dimensions must match to add

4. Scalar Multiplication1 2 3 4 5 10 15 20 5 x =
The dimensions can be any size to perform scalar multiplication

5. Matrix Multiplication
col 2
row 1, col 2 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6
row 1 30 36 42 66 81 96 x =
2x3
3x3
2x3
1x2 + 2x5 + 3x8 = = 36
rows from the first matrix
columns from the second matrix

6. Matrix Multiplication
“inner” dimensions must match
result is “outer” dimension
Examples:
2x3 X 3x3 = 2x3
3x4 X 4x5 = 3x5
2x3 X 4x3 = cannot multiply
Question: Does AB = BA?
Hint: let A be 2x3 and B be 3x2

7. Matrices and Graphics Matrices are used to represent equations
Used as an efficient way to represent transformation equations in graphics
Multiple transformations can be composed into a single matrix
Matrix multiplications are often optimized in hardware

8. Transformations View a complex shape as a set of points
To transform the shape, one transforms each of the points that define the shape
Major types of transformations:
Translation
Rotation
Scaling

9. Translation
The equation to translate a single point (x, y) by tx in the x direction and ty in the y direction:
x’ = x + tx
y’ = y + ty
Resulting in the new point (x’, y’)

10. Translation These equation can be represented using matrices:
P = P’ = T =
Then using matrix math we simply have:
P’ = P + T x y x’ y’ tx ty

11. Rotation
The equations to rotate a point through an angle, q (ccw), around the origin:
x’ = x cos(q) – y sin(q)
y’ = x sin(q) + y cos(q)
In matrix form:
R =
Using matrix math: P’ = R P
cos(q)
-sin(q)
sin(q)

12. Scaling The equations to scale a point along the coordinate axes:
x’ = x sx
y’ = y sy
Using matrix math: P’ = S P
Where S = ?

13. Scaling Scaling is used to grow/shrink objects centered on the origin
s numbers bigger than 1 grow objects
s numbers smaller than 1 shrink objects
uniform scaling: both s numbers the same

14. Transformation Sequences
In general you need to perform a sequence of transformation:
This is a rotation followed by a translation
P’ = R P + T
Do a dimension check to make sure the math will work out:
R is 2x2 and P is 2x1, thus R P is 2x1
R P is 2x1 and T is 2x1, thus P’ is 2x1

15. Transformation Sequences
However, it is often more efficient if we can combine operations into a single matrix calculation
How can this be done?
Here are the equations:
x’ = (x cos(q) – y sin(q) ) + tx
y’ = (x sin(q) + y cos(q) ) + ty

16. Transformation Sequences
Put them into matrix form:
The only problem is that our original point, P, now need a 3x1 matrix while the resulting point, P’, needs a 2x1 matrix x’ y’
cos(q)
–sin(q) tx
sin(q) ty x y 1 =

17. Homogeneous Coordinates
We resolve this using homogeneous coordinates
Lots of theory here, but…
Basically boils down to all points being represented as a triple (x, y, 1)
Thus, we need to fix our transformation matrix so that P’ comes out as a triple

18. Homogeneous Coordinates
This 3x3 matrix is called a transformation matrix
This particular matrix rotates then translates
However, it is often easier to work with transformations matrices that handle one type of transformation at a time, then compose then to obtain our final transformation x’ y’ 1
cos(q)
-sin(q) tx
sin(q) ty 1 x y 1 =

19. 2D Transformation Matrices1 tx ty
Translation: T(x, y)
Rotation: R(q)
Scaling: S(x, y)
cos(q)
-sin(q)
sin(q) 1 sx sy 1

20. Transformation Composition
Example: How do you rotate point P about a random “pivot point”?
Note that normal rotation has the origin as a pivot
Translate P so pivot point lines up with the origin
Rotate P around origin
Translate P back so pivot point is back on original position

21. Rotation Around Pivot Point

22. Rotation Around Pivot Point
Assume pivot point is at (x, y)
Assume P is the point you wish to rotate
P’ = T(x, y) R(q) T(-x, -y) P
Recall that order make a huge difference!

23. Other Transformations1 -1 -1 1
Reflection
about x axis (left)
about y axis (right)
Shear
along x axis (left)
along y axis (right)
Identity 1
shx 1
shy 1

24. Transformation Between Coordinate SystemsP y' y x' x
Point P is represented in the original coordinate system. To transform it to the new coordinate system one needs to know how this system is oriented and positioned relative to the original coordinate system.

25. 3D Transformations I’ll be using a right-handed coordinate system
Positive z-axis is sticking out of the screen
Dashed lines will be used for lines that stick into the screen y x z

26. 3D Transformations Points now have 3 coordinates (x, y, z)
Written in homogeneous coordinates as (x, y, z, 1)
Since the points are now 4x1 matrices, the transformation matrices need to be 4x4

27. 3D Translation 1 tx ty tz
T(x, y, z):

28. 3D Rotation 3D rotation is a bit harder than in 2D
In 2D we always rotated about the z-axis
Now we have a choice of 3 axes, Rx, Ry, Rz
Rotation is still ccw about the axis
Looking from the positive half of the axis back towards the origin (same as it was in 2D)

29. Rz(q) Extending our 2D rotation equations to 3D:
x’ = x cos(q) – y sin(q)
y’ = x sin(q) + y cos(q) same as in 2D
z’ = z the height stays the same
Rz(q):
cos(q)
-sin(q)
sin(q) 1

30. Rx(q) and Ry(q) Rx(q): Ry(q): 1 cos(q) -sin(q) sin(q) cos(q) sin(q) 1
cos(q)
-sin(q)
sin(q)
cos(q)
sin(q) 1
-sin(q)

31. Rotation Around a General Axis
Translate P so that the rotation axis passes through the coordinate system origin
Rotate so that the axis of rotation coincides with one of the coordinate axes
In general, requires 2 rotations (about the other 2 axes)
Rotate around the axis
Perform the reverse of the rotations (2)
Perform the reverse of the translation (1)

32. Rotation Around a General Axis
P’ = T(-a,-b,-c) Rx(-y) Ry(-r) Rz(q) Ry(r) Rx(y) T(a, b, c) P
Requires 7 matrix multiplications
Each one requiring 64 multiplies and 48 adds
For a total of 448 multiplies and 336 adds
Per point!
Recall objects sometimes have >100k points

33. Pre-multiplication of Transformations
Recall that matrix multiplication is not commutative: AB != BA
However, it is associative: (AB)C = A(BC)
This implies that you don’t have to perform the matrix multiplies right-to-left
Thus, we can pre-multiply the 7 matrices that don’t change between points to obtain: A = T(-a,-b,-c) Rx(-y) Ry(-r) Rz(q) Ry(r) Rx(y) T(a, b, c)
Then we repeat the single multiplication P’= A P for each of the 100k points

34. Pre-multiplication of Transformations
The old method produced (for 100k points)
44.8M multiplies and 33.6M adds
The pre-multiplication method produces
approx 6.4M multiplies and 4.8M adds
Plus, not all the multiplies/adds need to be done since we are assuming in the last row
Plus, these multiplies/adds are often done on optimized graphics hardware

35. Vectors
Another important mathematical concept used in graphics is the Vector
If P1 = (x1, y1, z1) is the starting point and P2 = (x2, y2, z2) is the ending point, then the vector V = (x2 – x1, y2 – y1, z2 – z1)
This just defines length and direction, but not position P2 P1

36. Vector Projections Projection of v onto the x-axis y (5, 2) v x v'
(5, 0)
Projection of v onto the x-axis

37. Vector Projections Projection of v onto the xz plane y (5, 3, 2) v x
(5, 0, 2) z
Projection of v onto the xz plane

38. 2D Magnitude and Direction
The magnitude (length) of a vector:
|V| = sqrt( Vx2 + Vy2 )
Derived from the Pythagorean theorem
The direction of a vector:
a = tan-1 (Vy / Vx)
a.k.a. arctan or atan
atan vs. atan2
a is angular displacement from the x-axis y
|v|
(5, 2) a x

39. 3D Magnitude and Direction
3D magnitude is a simple extension of 2D
|V| = sqrt( Vx2 + Vy2 + Vz2 )
3D direction is a bit harder than in 2D
Need 2 angles to fully describe direction
Latitude/longitude is a real-world example
Direction Cosines are often used:
a, b, and g are the positive angles that the vector makes with each positive coordinate axes x, y, and z, respectively
cos a = Vx / |V|
cos b = Vy / |V| cos g = Vz / |V|

40. Vector Normalization
“Normalizing” a vector means shrinking or stretching it so its magnitude is 1
a.k.a. creating unit vectors
Note that this does not change the direction
Normalize by dividing by its magnitude
V = (1, 2, 3)
|V| = sqrt( ) = sqrt(14) = 3.74
Vnorm = V / |V| = (1, 2, 3) / 3.74 = (1 / 3.74, 2 / 3.74, 3 / 3.74) = (.27, .53, .80)
|Vnorm| = sqrt( ) = sqrt( .9 ) = .95
Note that the last calculation doesn’t come out to exactly 1. This is because of the error introduced by using only 2 decimal places in the calculations above.

41. Vector Addition Equation: Visually:
V3 = V1 + V2 = (V1x + V2x , V1y + V2y , V1z + V2z)
Visually: y y V2 V3 V2 V1 V1 x x

42. Vector Subtraction Equation: Visually:
V3 = V1 - V2 = (V1x - V2x , V1y - V2y , V1z - V2z)
Visually: y y V3 V2 V2 V1 V1 x x

43. Dot Product The dot product of 2 vectors is a scalar
V1 . V2 = (V1x V2x) + (V1y V2y ) + (V1z V2z )
Or, perhaps more importantly for graphics:
V1 . V2 = |V1| |V2| cos(q)
where q is the angle between the 2 vectors
and q is in the range 0  q   y V2 q V1 x

44. Dot Product Why is dot product important for graphics?
It is zero iff the 2 vectors are perpendicular
cos(90) = 0 y V2 V1 x

45. Dot Product
The Dot Product computation can be simplified when it is known that the vectors are unit vectors
V1 . V2 = cos(q)
because |V1| and |V2| are both 1
Saves 6 squares, 4 additions, and 2 sqrts

46. Cross Product The cross product of 2 vectors is a vector
V1 x V2 = ( V1y V2z V1z V2y , V1z V2x V1x V2z , V1x V2y V1y V2x )
Note that if you are big into linear algebra there is also a way to do the cross product calculation using matrices and determinants

47. Cross Product
Again, just as with the dot product, there is a more graphical definition:
V1 x V2 = u |V1| |V2| sin(q)
where q is the angle between the 2 vectors
and q is in the range 0  q   and u is the unit vector that is perpendicular to both vectors
Why u?
|V1| |V2| sin(q) produces a scalar and the result needs to be a vector

48. Cross Product The direction of u is determined by the right hand ruleV2 u V1
The direction of u is determined by the right hand rule
The perpendicular definition leaves an ambiguity in terms of the direction of u
Note that you can’t take the cross product of 2 vectors that are parallel to each other
sin(0) = sin(180) = 0  produces the vector (0, 0, 0)

49. Forming Coordinate Systems
Cross products are great for forming coordinate system frames (3 vectors that are perpendicular to each other) from 2 random vectors
Cross V1 and V2 to form V3
V3 is now perpendicular to both V1 and V2
Cross V2 and V3 to form V4
V4 is now perpendicular to both V2 and V3
Then V2, V4, and V3 form your new frame

50. Forming Coordinate SystemsV2 V2 V1 V3 V1 V2 x V3 z V1 V1 V4 y
V1 and V2 are in the new xy plane