# Projections§

Projections in **nalgebra** are projections as commonly defined by the computer
graphics community. In particular, they are **not idempotent** as some
may be used to. Instead they are bijective mappings that transform a given
6-faced convex shape to the double unit cube centered at the origin (i.e. the
axis-aligned cube composed of points with coordinates ranging from to ). The resulting coordinates are usually called *Normalized
Device Coordinates* (corresponding to the *clip-space*) by the computer
graphics community:

The actual shape to be transformed depends on the projection itself. Note that
projections implemented on **nalgebra** also flip the axis. This
is a common convention in computer graphics applications for rendering with,
e.g., OpenGL, because the coordinate system of the screen is left-handed.

Currently, **nalgebra** defines only the 3D orthographic
projection and the 3D perspective
projection, aka., `Orthographic3`

and `Perspective3`

.
They both store a 4x4 homogeneous transformation matrix internally which can be
retrieved by-value using the `.unwrap()`

or `.to_homogeneous()`

methods. A
reference can be obtained with `.as_matrix()`

. The projection matrix inverse
can be computed with the projection `.inverse()`

method. Note that this will be
much more efficient than calling the inverse method on the raw homogeneous
`Matrix4`

.

Projection types can transform points and vectors using the
`.project_point(...)`

and `.project_vector(...)`

methods. The latter ignores
the translational part of the projection because the input is a vector
(remember the semantic difference
between points and vectors). Because projections following our convention are
invertible, it is possible to apply the inverse projection to points using
`.unproject_point(...)`

. This is typically used for screen-space coordinates to
view-space coordinates conversion.

## Orthographic projection§

An orthographic projection `Orthographic3`

maps a rectangular axis-aligned
cuboid to the double unit cube centered at the origin. This is basically a
translation followed by a non-uniform scaling. An orthographic projection is
characterized by:

Property | Meaning |
---|---|

`left` |
The -coordinate of the cuboid leftmost face parallel to the -plane. |

`right` |
The -coordinate of the cuboid rightmost face parallel to the -plane. |

`bottom` |
The -coordinate of the cuboid leftmost face parallel to the -plane. |

`top` |
The -coordinate of the cuboid leftmost face parallel to the -plane. |

`znear` |
The distance between the viewer (the origin) and the closest face of the cuboid parallel to the -plane. If used for a 3D rendering application, this is the closest clipping plane. |

`zfar` |
The distance between the viewer (the origin) and the furthest face of the cuboid parallel to the -plane. If used for a 3D rendering application, this is the furthest clipping plane. |

The following example, shows the effect of an orthographic projections with its
`left`

, `right`

, `bottom`

, `top`

, `znear`

, and `zfar`

properties noted
respectively as , , , , , and :

```
// Arguments order: left, right, bottom, top, znear, zfar.
let proj = Orthographic3::new(1.0, 2.0, -3.0, -2.5, 10.0, 900.0);
let pt = Point3::new(1.0, -3.0, 10.0);
let vec = Vector3::new(21.0, 0.0, 0.0);
assert_eq!(orth.project_point(&pt), Point3::new(-1.0, -1.0, -1.0));
assert_eq!(orth.project_vector(&vec), Vector3::new(42.0, 0.0, 0.0));
```

All properties can be read and modified. In-place modification is done with
methods starting with the `set_`

name prefix, e.g., `.set_right(...)`

. Instead
of recomputing the whole projection matrix, this will modify only the relevant
entries. Some setters combine two modifications at once for better efficiency:

Setter | Meaning |
---|---|

`.set_left_and_right(...)` |
Sets both left and right cuboid face coordinates simultaneously. |

`.set_bottom_and_top(...)` |
Sets both bottom and top cuboid face coordinates simultaneously. |

`.set_znear_and_zfar(...)` |
Sets both clipping planes simultaneously. |

## Perspective projection§

A perspective projection `Perspective3`

maps a frustrum to the double unit cube
centered at the origin. It is a non-linear transformation that uses homogeneous
coordinates to apply to each point a scale factor that depends on its distance
to the viewer.

The viewer of the perspective projection is always assumed to be located at the origin and to look toward the axis. Changing the viewer position and orientation requires an additional isometry (in a separate data structure) to form a view-projection transformation. A perspective projection is characterized by:

Property | Meaning |
---|---|

`aspect` |
The aspect ratio of the frustrum faces on the -plane. This is division of the width by the height of any section (parallel to the -plane) of the frustrum . |

`fovy` |
The field of view along the axis. This is the angle between uppermost and lowermost faces of the frustrum. |

`znear` |
The distance between the viewer (the origin) and the closest face of the frustrum parallel to the -plane. If used for a 3D rendering application, this is the closest clipping plane. |

`zfar` |
The distance between the viewer (the origin) and the furthest face of the frustrum parallel to the -plane. If used for a 3D rendering application, this is the furthest clipping plane. |

```
// Arguments order: aspect, fovy, znear, zfar.
let proj = Perspective3::new(16.0 / 9.0, 3.14 / 4.0, 1.0, 10000.0);
```

All properties can be read and modified. In-place modification is done with
methods starting with the `set_`

name prefix, e.g., `.set_fovy(...)`

. Instead
of recomputing the whole projection matrix, this will modify only the relevant
entries. The setter `.set_znear_and_zfar(...)`

modify both clipping planes
simultaneously. This is more efficient than calling both `.set_znear(...)`

and
`.set_zfar(...)`

separately.

Points and transformations Computer graphics recipes