Linear Algebra and Matrix Transformations, Slides of Computer Graphics

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Lecture 08:

Transform 1

Geometric Transform

• Apply transforms to a hierarchy of objects /

vertices…

• Specifically, translate (T), rotate (R), scale (S)

Vector and Matrix Notation

• Let’s say I need:

• 6 apples, 5 cans of soup, 1 box of tissues, 2 bags

of chips

• 4 Stores, A, B, C, and D (Stop and Shop,

Shaw’s, Trader Joe’s, and Whole Foods)

1 apple 1 can of soup 1 box of tissue 1 bag of chips

Stop and Shop $0.20 $0.93 $0.64 $1. Shaw’s $0.65 $0.82 $0.75 $1. Trader Joe’s $0.95 $1.10 $0.52 $3. Whole Foods $1.15 $0.20 $1.25 $2.

Shopping Example

• Which store do you go to?
  • Find the total cost from each store
  • Find the minimum of the four stores
• More formally, let 𝑞𝑖 denote the quantity of item 𝑖.
• Let 𝐴𝑖 be the unit price of item 𝑖 at store 𝐴.
• Then:
  • 𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐴 = (^4) 𝑖=1𝐴𝑖𝑞𝑖

q1 q2 q3 q A $0.20 $0.93 $0.64 $1. B $0.65 $0.82 $0.75 $1. C $0.95 $1.10 $0.52 $3. D $1.15 $0.20 $1.25 $2.

𝑞 1 = 𝟔 𝑞 2 = 𝟓 𝑞 3 = 𝟏 𝑞 4 = 𝟐

Total_A = (0.2 * 6) + (0.93 * 5) + (0.64 * 1) + (1.20 * 2) = 8.

Using the Matrix Notation

• 𝑃𝑎𝑙𝑙 =

𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐴 𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐵 𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐶 𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐷

=

0.20 0.93 (^) 0.64 1. 0.65 0.82 (^) 0.75 1. 0.95 1.10 (^) 0.52 3. 1.15 0.20 (^) 1.25 2.

6 5 1 2

• Determine totalCost vector using row-column multiplication
  • Dot product is the sum of the pairwise multiplications
  • Apply this operation to rows of prices and column of quantitities

Reminder: Matrix Multiplication

• Each entry in the resulting matrix L is the dot product of a row of M with a column of N:

𝐿 = 𝑀𝑁

𝑙𝑥𝑥 𝑙𝑥𝑦 𝑙𝑥𝑧 𝑙𝑥𝑤 𝑙𝑦𝑥 𝑙𝑦𝑦 𝑙𝑦𝑧 𝑙𝑦𝑤 𝑙𝑧𝑥 𝑙𝑧𝑦 𝑙𝑧𝑧 𝑙𝑧𝑤 𝑙𝑤𝑥 𝑙𝑤𝑦 𝑙𝑤𝑧 𝑙𝑤𝑤

=

𝑚𝑥𝑥 𝑚𝑥𝑦 𝑚𝑥𝑧 𝑚𝑥𝑤 𝑚𝑦𝑥 𝑚𝑦𝑦 𝑚𝑦𝑧^ 𝑚𝑦𝑤 𝑚𝑧𝑥 𝑚𝑧𝑦 𝑚𝑧𝑧 𝑚𝑧𝑤 𝑚𝑤𝑥 𝑚𝑤𝑦 𝑚𝑤𝑧 𝑚𝑤𝑤

𝑛𝑥𝑥 𝑛𝑥𝑦 𝑛𝑥𝑧 𝑛𝑥𝑤 𝑛𝑦𝑥 𝑛𝑦𝑦 𝑛𝑦𝑧^ 𝑛𝑦𝑤 𝑛𝑧𝑥 𝑛𝑧𝑦 𝑛𝑧𝑧 𝑛𝑧𝑤 𝑛𝑤𝑥 𝑛𝑤𝑦 𝑛𝑤𝑧 𝑛𝑤𝑤

𝑙𝑥𝑦 = 𝑚𝑥𝑥𝑛𝑥𝑦 + 𝑚𝑥𝑦𝑛𝑦𝑦 + 𝑚𝑥𝑧𝑛𝑧𝑦 + 𝑚𝑥𝑤𝑛𝑤𝑦

• Does 𝐿 = 𝑀𝑁 and 𝐿 = 𝑁𝑀 the same?

Identity Matrix

• What if our price matrix is of the following?

(^1 0 0 ) (^0 1 0 ) (^0 0 1 ) (^0 0 0 )

Identity Matrix

• What if our price matrix is of the following? (^1 0 0 ) (^0 1 0 ) (^0 0 1 ) (^0 0 0 )
• 𝑃𝑎𝑙𝑙 =

𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐴 𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐵 𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐶 𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐷

=

(^1 0 0 ) (^0 1 0 ) (^0 0 1 ) (^0 0 0 )

6 5 1 2

=

6 5 1 2

• We call this matrix the “identity matrix”

Identity Matrix

• What if our price matrix is of the following? (^2 0 0 ) (^0 2 0 ) (^0 0 2 ) (^0 0 0 )
• 𝑃𝑎𝑙𝑙 =

𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐴 𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐵 𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐶 𝑡𝑜𝑡𝑎𝑙𝐶𝑜𝑠𝑡𝐷

=

(^2 0 0 ) (^0 2 0 ) (^0 0 2 ) (^0 0 0 )

6 5 1 2

=

12 10 2 4

• So this is a “scaling matrix”

Note!

• In this example, notice the first column of the matrix is

only affecting the first value of the result

• Why is this important??

Matrix Multiplication Explained

(Visually)

• Suppose we have some matrix, like

1 1 1 2

, and we want to know what it’ll do to a 2D point.

• First, we multiply this matrix by two unit basis

vectors, the x-axis and the y-axis: 1 1 1 2

1 0

=

1 1

1 1 1 2

0 1

=

1 2

• Notice the results are the two columns of the matrix…
• So let’s visualize how the x and y axes have been

transformed using our matrix

Matrix Multiplication Explained

(Visually)

• Original Coordinate

System

• After we apply the

transform

Transformations in Computer Graphics

• Elemental Transformations:
  • Translation
  • Rotation
  • Scaling
  • Shearing
• Of the four, three of them are affine and linear:
  • Rotation, Scaling, and Shearing
• One is affine but non-linear
  • Translation

Definitions of Transformations

• Projective ⊃ Affine ⊃ Linear
  • Meaning that all linear transforms are also affine transforms, which are also projective transforms.
  • However, not all affine transforms are linear transforms, and not all projective transforms are affine.
• Definitions:
  • Linear Transform:
    • Preserves all parallel lines
    • Acts on a line to yield either a line or a point
    • The vector [0, 0] is always transformed to [0, 0]
    • Examples: scale and rotate