• Rendering: mapping 3D data on to a 2D canvas
  • Picking: mapping a 2D point to a 3D world
  • Inverse: need to run pipeline backwards
  • Normalized Device Coordinates
    • standardese for Canonical View Volume
    • Visible volume that gets rasterized finally
    • NDC $(x, y, z) \in [-1, 1]$

\begin{gather*} View \xrightarrow{projection} Clip \xrightarrow[Z]{divide} NDC \xrightarrow{viewport} Screen \end{gather*}

Pick Ray

Credit: James M. Van Verth, Lars M. Bishop
Essential Mathematics for Games and Interactive Applications

Spaces without Z

Viewing plane, NDC and screen
Credit: Ganovelli, Corsini, Pattanaik, Benedetto
Introduction to Computer Graphics ~ A Practical Learning Approach

Screen → View

  • Pick ray constructed in view space but got $x_{scr}$ and $y_{scr}$
  • Map Screen → NDC → Clip → View
  • Clip space (4D) can be ignored
    • Space to set up $z$ in $w$ for perspective divide
    • Used only for clipping otherwise
  • Essentially find Screen → NDC → View (with $z$ dropped)
  • Rectangle mapping: $[w, h] \mapsto [2, 2] \mapsto [2a, 2]$
  • Should require only scaling and translation

Screen → NDC

  • Rendering: NDC cube $\mapsto$ screen space cuboid such that
    • $x_{scr} \in [0, w]$
    • $y_{scr} \in [0, h]$
  • Rectangle dimensions do not vary with depth for both volumes
  • Straight forward inversion; one rectangle to another
  • \begin{align} \require{enclose} \enclose{box}{x_{ndc} = \frac{2 x_{scr}}{w} - 1} && \enclose{box}{y_{ndc} = -\frac{2 y_{scr}}{h} + 1} \qquad (1) \end{align}

NDC → View

  • Rendering: view frustum $\mapsto$ NDC cube
    • Rectangle dimension varies based on depth in frustum
    • Depth already chosen: view plane's z = focal length
  • View window dimensions, where $a$ is aspect ratio \begin{align}x_v \in [-a, a] && y_v \in [-1, 1] \end{align}
  • Map one rectangle to another like before \begin{align} x_v = a x_{ndc} && y_v = y_{ndc} \qquad(2) \end{align}
  • Final Screen $\mapsto$ View by substituting $(1)$ in $(2)$

\begin{align} \enclose{box}{x_v = \frac{2ax_{scr}}{w} - a} && \enclose{box}{y_v = -\frac{2y_{scr}}{h} + 1} \end{align}