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31. Mathematics of Lighting and Shading part 5

Dated: 02-07-2025

Physically Based Illumination

Reflection

Reflection of a light wave is the change in direction of light ray when it bounces off the boundary between two media.

\[\theta_\text{incident} = \theta_\text{reflected}\]

Refraction

Refraction happens when a light wave goes from one medium into another. Because of the difference in the speed of light of the media, light bends when it crosses the boundary. Snell's law gives the change in angles.

\[n_a \sin(\theta_a) = n_b \sin(\theta_b)\]

Here \(n_a\) and \(n_b\) are the refractive indices of the materials.

The refractive index of vacuum is \(1\).

The critical angle an an angle at which we observe the phenomenon of total internal reflection. At any angle greater than this, light will not pass though a boundary but will be reflected internally.

cs602_i_31_1.png

Refractive indices for sapphire

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Refractive indices for quartz

\[\text{red} = 780 - 622 \ nm\]
\[\text{green} = 577 - 492 \ nm\]
\[\text{blue} = 492 - 455 \ nm\]

Temperature Correction for Refractive Index

\[\eta^t = \eta^x + (x - t)(0.00045)\]

Where \(x\) is the given temperature and \(t\) is the new temperature for which we are finding the new refractive index.