Intro to 3D graphics week 2 Matrices

Matrices are a useful way of presenting data in a neat tabular form. A matrix is a rectangular array of numbers with each element being a number of the array. n rows and m columns

A = 1,2,3
4,5,6
7,8,9

Adding two matrices together

3,-2,5         7,8,4    =    3+7,-2+8,5+4    =     10,6,9
1,3,4     +   3,2,5    =    1+3,3+2,4+5      =      4,5,9
5,-4,6         3,6,4    =    5+3,-4+6,6+4    =    8,2,10

Taking away

3,-2,5          7,8,4     =     3-7,-2-8,5-4      =       -4,-10,1
1,3,4     –    3,2,5     =     1-3,3-2,4-5         =      -2,1,-1
5,-4,6         3,6,4     =    5-3,-4-6,6-4       =       2,-10,2

Multiplying by a scalar

k = 3

3,-2,5      = (3)3,(3)-2,(3)5             9, -6,15
1,3,4        = (3)1,(3)3,(3)4       =     3, 9, 12
5,-4,6      = (3)5,(3)-4,(3)6             15,-12,18

premultiply of A by B means forming BA and postmultiplication of A by B means forming AB

Transpose of a matrix

first row becomes first column and second row becomes the second column

A,D,G               A,B,C
B,E,H      =       D,E,F
C,F,I                G,H,I

Matrix Identity

Square matrix with 1s on the main diagonal and zeros everywhere else

100
010
001

Multiply Matrices

Row by column

3,-2,5           7,8,4
1,3,4     x      3,2,5
5,-4,6          3,6,4

=    (3×7 + -2×3 + 5×3),(3×8 + -2×2 + 5×6),(3×4 + -2×5 + 5×4) = 10,50,2
=    (1×7 + 3×3 + 4×3),(1×8 + 3×2 + 4×6),(1×4 + 3×5 + 4×4) = 28,38,35
=    (5×7 + -4×3 + 6×3),(5×8 + -4×2 + 6×6),(5×4+-4×5+6×4) = 41,68,24

Obtaining the inverse of a 3×3 matrix

determinant A must be non-zero

2, 4, 6
2, 7, 8      =    2   |7, 8| -4|2, 8| + 6|2, 7| = (-2)-(40)+(54) = 12 Det
1, 8, 9                 | 8, 9|     |1, 9|      |1, 8|

|7, 8| (7×9-8×8 )   |2, 8|               |2, 7|
|8, 9|  = -1                |1, 9|  = 10    |1, 8| = 9

|4, 6|                          |2, 6|               |2, 4|
|8, 9|  = -12             |1, 9|  = 12    |1, 8| = 12

|4, 6|                          |2, 6|               |2, 4|
|7, 8|  = -10             |2, 8|  = 4    |2, 7| = 6

-1, 10, 9                  + – +                -1, -10, 9                                -1, 12, -10
-12, 12, 12      x       – + -        =      12, 12, -12         Transpose > -10, 12, -4
-10, 4, 6                  + – +               -10, -4, 6                                   9, -12, 6

Inverse =
-1/12, 12/12, -10/12                     -0.833, 1, -0.833
-10/12, 12/12, -4/12             =     -0.833, 1, -0.333
9/12, -12/12, 6/12                          0.750, -1, 0.500

Dot Product(Scalar Product)

The dot product is used to calculate the angles between two vectors used mainly in lighting calculations. It is the sum of the products of the corresponding components.

[4, 6] . [-3, 7] = (4)(-3) + (6)(7) = 30

Cross Product

Creates a vector which is perpendicular to the plane containing the two input vectors.

a = 8i – 3j + 6k

b = -2i – 4j – 8k

I    J    K

8   -3   6

-2  -4  -8

I = (-3 x-8 ) – (-4×6) = 24 – -24 = 48

J=(8x-8)-(6x-2)= -64 – -12 = -52 > 52   REVERSE J SIGN

K=(8x-4)-(-2x-3)=-32-6= -38