= Light transport =

== Radiometry recap ==

What is radiant flux?
* total amount of energy passing through a surface (measured per second)
* Radiant flux in Watts or Joules/second

=== Why do we not use radiant flux? ===

* When we emasure a high flux val, we dont know if lots of energy through a
  small surface, or a little energy through a huge surface
* TLDR its to ambiguous

New unit irradiance
* Flux per unit area
* IE Watt/m^2

=== Why do we not use irradiance? ===

We know the surface, but we still need an angle
* Could be lost of energy in a huge angle
* little energy in a small angle

New unit Radiance
* Flux per unit area angle
* W / (m^2 * radians)

== Basic question ==

How do we calculate how much light exits a surface point in a given direction

[[maxwell_equations|Maxwell equations]]!

In practive, we dont do that. We try to think of light as a ray unless we have
to

Instead we use the rendering equation!

Terms used in diagrams

* V - direction towards the viewer
* N - surface normal
* L - vector pointing towards the light source
* R - reflected ray direction
* Theta,,i,, and Theta,,r,, - incident and reflected angles

To calcuate R `R = L - 2N(L * N)`

== Light attentuation ==

* The amount of intensity light looses as it travels farther away. Light looses
  energy as well as it reflects off of surfaces
* How can we calculate attentuation?

`L * N`

Assuming that L and N are normalized, the attentuation will be equal to
`cos(theta)` where theta is the angle between L and N. This is true because of
the dot product.

== Materials ==

How can we simluate the look of different materials?

Based on how they relfect light.

* Specular surface reflects exactly one ray
  * One incoming direction
  * One outgoing direction
* Diffuse spreads ray into smaller rays in all directions
  * On incoming direction
  * many outgoing direction
  * many outgoing intesntiy
* Spread breaks out the ray into a few smaller rays in a single direction

To simulate a surface, we can use a probability density function that takes

* incoming light direction
* point on surface

This method will output the _probability of a given outward direction_

Something like `probability of happening = f(incoming direction, point, outgoing direction)`.

This is known as the _Bidirectional reflectance distribution function (BRDF)_.


What about materials that reflect some light allow some right through
(transmitted).

Properties of a transparent material

* Helmholtz-reciprocity
  * direction of the ray of light can be reveresd
  * This means the odds of the light bouncing from A to B are the same as the
    odds of boucing B to A
* Positivity
  * It is impossible for an exit direction to have a negative probability
* Energy conservation
  * An object may reflect OR absorb incoming light, but no more light can come
    out than the incoming amount

== Rendering equation ==

Sum of all light reflections and absorbtions at a point

Note that an object can emit light itself. It also receives light from
different directions, which it will either reflect or absorb. Therefore

`Light exiting = Emitted light + reflected incoming light`

AKA

{{{
Light output(x, vector w) = Light emitted(x, vector w)
  + Integral of (Light incoming to x from vector w * BRFD(vector w, x vector w
  prime) cos theta dw
  }}}