Question

find the 3x3 matrix that produce the copostie 2d transofrmation using homogenous coordiates. translate by (3,1) then rotate 45 about the origin

Answer 1

Explanation:

Answer 2

To find the 3x3 matrix that produces the composite 2D transformation of translating by (3,1) and then rotating 45 degrees about the origin using homogeneous coordinates, we can use the following steps:

1. Write the translation and rotation matrices in homogeneous coordinates:

Translation matrix:
```
[1 0 3]
[0 1 1]
[0 0 1]
```

Rotation matrix:
```
[cos(45) -sin(45) 0]
[sin(45) cos(45) 0]
[0 0 1]
```

2. Multiply the translation matrix by the rotation matrix in that order:

```
[cos(45) -sin(45) 0] [1 0 3] [cos(45) -sin(45) 3cos(45)-sin(45)]
[sin(45) cos(45) 0] * [0 1 1] = [sin(45) cos(45) 1+3sin(45)+cos(45)]
[0 0 1] [0 0 1] [0 0 1 ]
```

3. The resulting matrix is the 3x3 matrix that produces the composite transformation:

```
[cos(45) -sin(45) 3cos(45)-sin(45)]
[sin(45)cos(45) 1+3sin(45)+cos(45)]
[0 0 1 ]
```

Therefore, the 3x3 matrix that produces the composite 2D transformation of translating by (3,1) and then rotating 45 degrees about the origin using homogeneous coordinates is:

```
[cos(45) -sin(45) 3cos(45)-sin(45)]
[sin(45) cos(45) 1+3sin(45)+cos(45)]
[0 0 1 ]
```