## Spaces

- **Object Space**  
  Local object coordinates.
- **World Space `(world_matrix * object_space)`**  
  The position relatively to the origin of the world (point `(0,0)` below).  
  <img width="400" src="https://user-images.githubusercontent.com/1623053/85816645-37e00280-b76c-11ea-9831-e6ae7378830e.png"/>
- **Eye Space**  
  The position relatively to the placement of the camera.
  ```rust
  let view_matrix = camera_matrix.inverse();
  let eye_space   = view_matrix * world_space;
  ```
  <img width="400" src="https://user-images.githubusercontent.com/1623053/85816908-d40a0980-b76c-11ea-8be6-6c982b1d8ce5.png"/>
- **Clip Space**  
  The position inside of the Normalized Device Coordinates (NDC) cube. In
  perspective projection, a 3D point in a truncated pyramid frustum (eye
  coordinates) is mapped to the NDC cube. The range of x-coordinate from `[l,r]`
  to `[-1,1]`, the y-coordinate from `[b,t]` to `[-1,1]` and the z-coordinate
  from `[-n,-f]` to `[-1,1]`. Note that the eye coordinates are defined in the
  right-handed coordinate system, but NDC uses the left-handed coordinate
  system. That is, the camera at the origin is looking along -Z axis in eye
  space, but it is looking along +Z axis in NDC.

  ```rust
  let clip_space = projection_matrix * eye_space;
  ```

  <img width="600" src="https://user-images.githubusercontent.com/1623053/85817711-0caae280-b76f-11ea-9111-1357195cf580.png"/>
  <img width="600" src="https://user-images.githubusercontent.com/1623053/85817751-22b8a300-b76f-11ea-8f18-f3e78f3139c1.png"/>
  <img width="600" src="https://user-images.githubusercontent.com/1623053/85817783-3e23ae00-b76f-11ea-8972-c90f1eb6ba1e.png"/>

## Examples

```rust
use nalgebra as n;

let object_space      = n::Vector4::new(3.0,5.0,7.0,1.0);
let object_matrix     = n::Matrix4::identity().append_translation(&n::Vector3::new(10.0,0.0,0.0));
let inv_object_matrix = object_matrix.try_inverse().unwrap();

let camera_matrix   = n::Matrix4::identity().append_translation(&n::Vector3::new(0.0,0.0,100.0));
let inv_view_matrix = camera_matrix;
let view_matrix     = camera_matrix.try_inverse().unwrap();

let aspect = 1.0;
let fov    = 90.0f32.to_radians();
let near   = 0.1;
let far    = 100.0;
let perspective           = n::Perspective3::new(aspect,fov,near,far);
let projection_matrix     = *perspective.as_matrix();
let inv_projection_matrix = perspective.inverse();

// let left   = -100.0;
// let right  = 100.0;
// let bottom = -100.0;
// let top    = 100.0;
// let near   = 0.0;
// let far    = 100.0;
// let orthographic          = n::Orthographic3::new(left,right,bottom,top,near,far);
// let projection_matrix     = *orthographic.as_matrix();
// let inv_projection_matrix = orthographic.inverse();

let world_space   = object_matrix * object_space;
let eye_space     = view_matrix * world_space;
let clip_space    = projection_matrix * eye_space;

let eye_space2    = inv_projection_matrix * clip_space;
let world_space2  = inv_view_matrix * eye_space2;
let object_space2 = inv_object_matrix * world_space2;

println!("---------------------------------");
println!("object_space: {:?}", object_space);
println!("world_space: {:?}", world_space);
println!("eye_space: {:?}", eye_space);
println!("clip_space: {:?}", clip_space);
println!("eye_space2: {:?}", eye_space2);
println!("world_space2: {:?}", world_space2);
println!("object_space2: {:?}", object_space2);
```

## Sources

Images and fragments used here are parts of the following articles:

- https://webglfundamentals.org/webgl/lessons/webgl-3d-camera.html
- http://www.songho.ca/opengl/gl_transform.html
- http://www.songho.ca/opengl/gl_projectionmatrix.html
- http://www.songho.ca/math/homogeneous/homogeneous.html