Ray Tracing

Key formulas

• $$\vec{P}(t) = \vec{O} + t\,\hat{D}$$

  Parametric ray

  $\vec{P}(t)$

  Point on the ray

  $\vec{O}$

  Ray origin (the camera)

  $\hat{D}$

  Direction (unit vector)

  $t$

  Distance along the raym

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• $$\Delta = b^2 - 4ac,\quad b = 2\,\hat{D}\cdot(\vec{O}-\vec{C}),\quad c = |\vec{O}-\vec{C}|^2 - r^2$$

  Ray–sphere intersection

  $\Delta$

  Discriminant

  $\vec{C}$

  Sphere centre

  $r$

  Sphere radiusm

  $a$

  \hat{D}\cdot\hat{D}, equal to 1 for a unit direction

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• $$E = \dfrac{I_0}{d^2}$$

  Inverse-square law of light

  $E$

  Irradiance on the surfaceW/m²

  $I_0$

  Source intensityW/sr

  $d$

  Distance from the sourcem

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• $$I_d = k_d\,E\,\max(0,\ \hat{N}\cdot\hat{L})$$

  Lambert's cosine law

  $I_d$

  Diffuse brightnessW/m²

  $k_d$

  Diffuse reflectance (albedo)

  $E$

  Incoming irradianceW/m²

  $\theta$

  Angle between normal \hat{N} and light \hat{L}°

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• $$\hat{D}_r = \hat{D} - 2(\hat{D}\cdot\hat{N})\,\hat{N}$$

  Law of reflection

  $\hat{D}$

  Incoming direction (unit vector)

  $\hat{N}$

  Surface normal (unit vector)

  $\hat{D}_r$

  Reflected direction

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