When working in OpenGL, the output of the vertex shader must be a 4 dimensional vector $(x_c,y_c,z_c,w_c)$. Mathematically this is a vector in 4D projective space $\mathbb P^4$, but in computer graphics, this is known as clip space.
OpenGL takes this output vector and automatically transforms it to 3D Euclidean coordinates the usual way
\[\begin{bmatrix} x_c\\y_c\\z_c\\w_c \end{bmatrix}\mapsto\begin{bmatrix} x_c/w_c\\y_c/w_c\\z_c/w_c \end{bmatrix}=:\begin{bmatrix} x_{ndc}\\ y_{ndc}\\ z_{ndc} \end{bmatrix}\]In the computer graphics community, this is often called perspective division since we are dividing by the $w$ coordinate. The resulting coordinate space is called normalized device coordinates.
Normalized device coordinates. Image source.
An object is rendered by OpenGL only if its normalized device coordinates lie between -1 and 1 i.e. $(x_{ndc},y_{ndc},z_{ndc})\in [-1,1]^3$. The $x_{ndc}$ and $y_{ndc}$ coordinates specify the horizontal and vertical positions as one would expect, with positive $x_{ndc}$ towards the right and positive $y_{ndc}$ towards the top.
Move the mouse over the app below to see this.
The $z_{ndc}$ coordinate is used to determine whether an object should be rendered in front of or behind another object. An object is rendered in front of another if it has a lower $z$ coordinate. You can think of it as $z=-1$ representing the front of the screen, and $z=1$ representing the back of the screen.
To see this, adjust the sliders to change the $z_{ndc}$ coordinates of the two coloured squares.
Note that NDC is a left-handed coordinate system.
Summary
Summary
The output of the vertex shader is a vector in 4D projective space $(x_c,y_c,z_c,w_c)\in\mathbb P^4$, also called clip space.
OpenGL automatically transforms this into 3D Euclidean coordinates $(x_{ndc},y_{ndc},z_{ndc})=(x_c/w_c,y_c/w_c,z_c/w_c)\in\mathbb R^3$; this process is called perspective division.
- the resulting coordinate space is called normalized device coordinates
- only objects with normalized device coordinates lying between -1 and 1 are rendered
- positive $x_{ndc}$ points to the right, positive $y_{ndc}$ points upwards, and objects with lower $z_{ndc}$ are rendered in front
- NDC is a left-handed coordinate system