Transformation Matrix Project

Transformation Matrix Project

Allegra Minor and Gaby Turrinelli

Final Shape

Explanation

|0 1|*

|1 0|

(|17.22 16 5.68 13.22 12.44| - |0 0 0 0 0|)

(|10.55 19.08 22.32 18.79 14.38| |-5 -5 -5 -5 -5|)

- |0 0 0 0 0| = |15.55 24.08 27.32 23.79 19.38|

|5 5 5 5 5| |12.22 11 0.68 8.22 7.44|

This is the same as reflecting over y=x however much to y in order to move the line’s y-intercept to the origin (in this case add 5) and then reflect over y=x and subtract 5 again. This is to make the reflection easier and more efficient.

Explanation

6th Transformation: Reflect over y=x-5

|cos(75) -sin(75)| *

|sin(75) cos(75)|

(|14.2 22.12 22.58 21.12 | 16.66| - |2 2 2 2 2|)

(|-10.49 -7.1 3.7 -7.1 -4.88| |2 2 2 2 2|)

+ |2 2 2 2 2|

|sin(75) cos(75)| |2 2 2 2 2|

= |17.22 16 5.68 13.22 12.44|

|10.55 19.08 22.32 18.79 14.38|

5th Transformation: Rotation around a point that is not the origin (that point is (2,2) and the rotation angle is 75 counterclockwise)

This is the same as a normal rotation, but you “translate” the center point to the origin along with the shape and then move it back when you’re done with the rotation.

See first slide for rotation explanation

2nd Transformation:

Reflect over y=x

3rd Transformation: Dilate x by factor of 2

and y by factor of 3

Explanation

|10.2 18.12 18.58 17.12 12.66| +

|-8.49 -5.1 5.7 -2.49 -2.88|

|4 4 4 4 4|

|-2 -2 -2 -2 -2|

= |14.2 22.12 22.58 21.12 16.66|

|-10.49 -7.1 3.7 -7.1 -4.88|

|cos 60° ₋sin 60°| * | 3 7 9 7 5|

|sin 60° cos 60°| |5 6 3 5 4|

= |₋2.83 ₋1.69 1.90 ₋0.83 ₋0.96|

| 5.09 9.06 9.29 8.56 6.33 |

When translating, you want to add the desired amount (in regards to each axis) to each desired point. Therefore, you take the desired amount that you want to add to x and place it on top of the matrix so that when it is added to the xy matrix it will be added to x. The same goes for y.

In function form, the rotation of a point’s x value can be represented as x1= x cos(θ)+y sin(θ) based on a conceptual understanding of the unit circle. The rotation of a point’s y value can be represented as y1=-xsin(θ)+ycos(θ). Therefore when creating the matrix for rotating 60°, from a-d the inputs will be cos 60, -sin 60, sin 60, and cos 60. When multiplied by the matrix in which x is on top and y is on bottom, the resulting matrix will have x1= x cos(60)+y sin(60) on top (cos 60 (x) - -sin 60(y) = x cos(60)+y sin(60)) and y1=-xsin(θ)+ycos(θ)on bottom (based on the same process of dot multiplication displayed above)

Explanation

|2 0| * |5.09 9.06 9.29 8.56 6.33|

|0 3| |₋2.83 ₋1.69 1.90 ₋0.83 ₋0.96|

= |10.2 18.12 18.58 17.12 12.66|

|-8.49 -5.1 5.7 -2.49 -2.88|

4th Transformation: Translate with respect to the y-axis by 4 and the x axis by -2

Explanation

Dilating in both dimensions by different

amounts requires a and d to be different values,

b and c must be 0 because we just want to utilize

the values of a single row at a time, dilate x

independent of y and visa versa. The x coordinates

are multiplied by 2 so the shape is doubled with

respect to y (horizontally) and the y coordinates are multiplied by 3 so shape is tripled in respect to x coordinates

|0 1| *

|1 0|

|₋2.83 ₋1.69 1.90 ₋0.83 ₋0.96|

|5.09 9.06 9.29 8.56 6.33|

= |5.09 9.06 9.29 8.56 6.33|

|₋2.83 ₋1.69 1.90 ₋0.83 ₋0.96|

When reflecting over the line y=x, the x (top row) and y (bottom row) values are flipped meaning that the multiplication of matrices must produce a matrix in which the top and bottom are switched. Therefore the

“a” value of the left matrix must be filled with a 0 and the “b” value is filled in with 1 so that the first x value ends up being replaced with the first y value(0x + 1y=y ).”c” value of the left matrix is 1 and the “d” value is 0 to replace the first y value with the first x value (1x+ 0y=x).

First Transformation: Rotation 60° counterclockwise around origin