Editing Ray tracing (graphics) (section)

=== Calculate rays for rectangular viewport ===
On input we have (in calculation we use vector [[Euclidean vector#Length|normalization]] and [[cross product]]):

* <math>E \in \mathbb{R^3}</math> eye position
* <math>T \in \mathbb{R^3}</math> target position
* <math>\theta \in [0,\pi] </math> [[field of view]] - for humans, we can assume <math>\approx \pi/2 \text{ rad}= 90^\circ</math>
* <math>m,k \in \mathbb{N}</math> numbers of square pixels on viewport vertical and horizontal direction 
* <math>i,j  \in \mathbb{N}, 1\leq i\leq k \land 1\leq j\leq m </math> numbers of actual pixel 
* <math>\vec v \in \mathbb{R^3}</math> vertical vector which indicates where is up and down, usually <math>\vec v = [0,1,0]</math> - [[:simple:Pitch, yaw, and roll|roll]] component which determine viewport rotation around point C (where the axis of rotation is the ET section)

[[File:RaysViewportSchema.png|708px|Viewport schema with pixels, eye E and target T, viewport center C]]

The idea is to find the position of each viewport pixel center <math>P_{ij}</math> which allows us to find the line going from eye <math>E</math> through that pixel and finally get the ray described by point <math>E</math> and vector <math>\vec R_{ij} = P_{ij} -E </math> (or its normalisation <math>\vec r_{ij}</math>). First we need to find the coordinates of the bottom left viewport pixel <math>P_{1m}</math> and find the next pixel by making a shift along directions parallel to viewport (vectors <math>\vec b_n</math> , <math>\vec v_n</math>) multiplied by the size of the pixel. Below we introduce formulas which include distance <math>d</math> between the eye and the viewport. However, this value will be reduced during ray normalization <math>\vec r_{ij}</math> (so you might as well accept that <math>d=1</math> and remove it from calculations).

Pre-calculations: let's find and normalise vector <math>\vec t</math> and vectors <math>\vec b, \vec v</math> which are parallel to the viewport (all depicted on above picture)

:<math>
\vec t = T-E, \qquad \vec b = \vec t\times \vec v  
</math>

:<math>
\vec t_n = \frac{\vec t}{||\vec t||}, \qquad
\vec b_n = \frac{\vec b}{||\vec b||}, \qquad
\vec v_n = \vec t_n\times \vec b_n
</math>

note that viewport center <math>C=E+\vec t_nd</math>, next we calculate viewport sizes <math>h_x, h_y</math> divided by 2 including inverse [[aspect ratio]] <math>\frac{m-1}{k-1}</math>

:<math>
g_x=\frac{h_x}{2} =d \tan \frac{\theta}{2}, \qquad  g_y =\frac{h_y}{2} = g_x \frac{m-1}{k-1}
</math>

and then we calculate next-pixel shifting vectors <math>q_x, q_y</math> along directions parallel to viewport (<math>\vec b,\vec v</math>), and left bottom pixel center <math>p_{1m}</math>

:<math>
\vec q_x = \frac{2g_x}{k-1}\vec b_n, \qquad
\vec q_y = \frac{2g_y}{m-1}\vec v_n, \qquad
\vec p_{1m} = \vec t_n d - g_x\vec b_n - g_y\vec v_n
</math>

Calculations: note <math>P_{ij} = E + \vec p_{ij}</math> and ray <math>\vec R_{ij} = P_{ij} -E = \vec p_{ij}</math> so

:<math>
\vec p_{ij} = \vec p_{1m} + \vec q_x(i-1) + \vec q_y(j-1)
</math>
:<math>
\vec r_{ij} = \frac{\vec R_{ij}}{||\vec R_{ij}||} = \frac{\vec p_{ij}}{||\vec p_{ij}||}
</math>