# Planar Projection

Summary

## Contents

### Projection Parameters

Each of the parameters uses either the world-coordinate (WC) or viewing reference-coordinate (VRC) system.

• The View Reference Point (VRP) is the point (WC) from which the camera is viewing the 3D geometry.
• The View Plane Normal (VPN) is the normal (WC) ...
• The View Up Vector (VUP) is the vector (WC) that defines the orientation of the camera (i.e. which way is up)
• The Projection Reference Point (PRP) is the point (VRC) ...
• The Viewing Window is the rectangle (VRC) that defines the size of the 2D window upon which the 3D geometry will be projected.
• The projection type can be either parallel or perspective

## Parallel Projection

Summary

### Orthographic Projection

The first step is to translate the VRP to the origin, which can be achieved by multiplying its complement with the following matrix T:

$T = \left[ \begin{array}{cccc} 1 & 0 & 0 & -vrp_{x} \\ 0 & 1 & 0 & -vrp_{y} \\ 0 & 0 & 1 & -vrp_{z} \\ 0 & 0 & 0 & 1 \end{array} \right]$

The second step is to then rotate VPN to the z axis and VUP to the y axis. To do this we will calculate the following vectors:

• $R_{z} = \frac{VPN}{|VPN|}$
• $R_{x} = \frac{VUP \times R_{z}}{|VUP \times R_{z}|}$
• $R_{y} = R_{z} \times R_{x}$

The components of these vectors (e.g. $R_{x} = $) then form the rotation matrix R:

$R = \left[ \begin{array}{cccc} r_{x_{1}} & r_{x_{2}} & r_{x_{3}} & 0 \\ r_{y_{1}} & r_{y_{2}} & r_{y_{3}} & 0 \\ r_{z_{1}} & r_{z_{2}} & r_{z_{3}} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]$

The final transformation matrix for orthographic projection is then the result of the following multiplication:

$N_{par} = (S_{par} \cdot (T_{par} \cdot (SH_{par} \cdot (R \cdot T \cdot (-VRP)))))$

Summary