## Affine Transformations

### Some notes on affine transformation conventions.

*Mar 26, 2024*

A perpetual pain point for people working on computer vision is dealing with affine transformations. Usually you have a diagram that looks something like this:

Each of the coordinate frames can be represented as a `4x4`

matrix in Numpy. For the three sensors in the image above we might have something like this:

```
T_world_to_lidar # 4x4
T_world_to_camera # 4x4
T_world_to_imu # 4x4
```

Suppose we have a point `p`

in the world frame. We can transform it to one of the other frames using matrix multiplication:

```
p = np.array([x, y, z, 1])
p_lidar = T_world_to_lidar @ p # 4x4 @ 4x1 = 4x1
```

Here’s a toy example in 2D to illustrate the concept:

```
p_a_in_world = np.array([1, 1])
p_b_in_world = np.array([2, -1])
T_world_to_lidar_a = np.array([
[1, 0, 1],
[0, 1, 2],
[0, 0, 1],
])
T_world_to_lidar_b = np.array([
[-1, 0, 2],
[0, -1, 1],
[0, 0, 1],
])
p_a_in_lidar_a = T_world_to_lidar_a @ np.append(p_a_in_world, 1) # [2, 3]
p_b_in_lidar_a = T_world_to_lidar_a @ np.append(p_b_in_world, 1) # [3, 1]
p_a_in_lidar_b = T_world_to_lidar_b @ np.append(p_a_in_world, 1) # [1, 0]
p_b_in_lidar_b = T_world_to_lidar_b @ np.append(p_b_in_world, 1) # [0, 2]
p_o_in_lidar = np.array([0, 0])
p_o_a_in_world = np.linalg.inv(T_world_to_lidar_a) @ np.append(p_o_in_lidar, 1) # [-1, -2]
p_o_b_in_world = np.linalg.inv(T_world_to_lidar_b) @ np.append(p_o_in_lidar, 1) # [2, 1]
```

In words:

`point_a`

in the world frame is`(1, 1)`

and`point_b`

in the world frame is`(2, -1)`

- After multiplying the point by the transformation matrix, we get where the points are from each LiDAR’s perspective:
`point_a`

in LiDAR A’s frame is`(2, 3)`

`point_b`

in LiDAR A’s frame is`(3, 1)`

`point_a`

in LiDAR B’s frame is`(1, 0)`

`point_b`

in LiDAR B’s frame is`(0, 2)`

- We can get each LiDAR’s position in the world frame by applying the inverse transformation to the origin:
- LiDAR A is at
`(-1, -2)`

- LiDAR B is at
`(2, 1)`

- LiDAR A is at

Here’s how this looks visually, where the axis has the origin at the world frame: