# 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 $(-1, -1, -1)$ to $(1, 1, 1)$). 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 $\mathbf{z}$ 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 $\mathbf{x}$-coordinate of the cuboid leftmost face parallel to the $\mathbf{yz}$-plane.
right The $\mathbf{x}$-coordinate of the cuboid rightmost face parallel to the $\mathbf{yz}$-plane.
bottom The $\mathbf{y}$-coordinate of the cuboid leftmost face parallel to the $\mathbf{xz}$-plane.
top The $\mathbf{y}$-coordinate of the cuboid leftmost face parallel to the $\mathbf{xz}$-plane.
znear The distance between the viewer (the origin) and the closest face of the cuboid parallel to the $\mathbf{xy}$-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 $\mathbf{xy}$-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 $l$, $r$, $b$, $t$, $zn$, and $zf$:

// 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 frustum 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 $-z$ 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 frustum faces on the $\mathbf{xy}$-plane. This is division of the width by the height of any section (parallel to the $\mathbf{xy}$-plane) of the frustum .
fovy The field of view along the $\mathbf{y}$ axis. This is the angle between uppermost and lowermost faces of the frustum.
znear The distance between the viewer (the origin) and the closest face of the frustum parallel to the $\mathbf{xy}$-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 frustum parallel to the $\mathbf{xy}$-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.