What is Lambert's cosine law?

Lambert's cosine law is a fundamental principle in the field of photometryMeasurement of light in terms of its perceived brightness to the human eye.. It has significant applications in computer graphics, particularly in the area of shading and lighting models.

Definition

The law states that the apparent brightness of a lambertian surfaceA surface that appears uniformly bright from all directions of view., is directly proportional to the cosine of the angle between the surface normalA perpendicular vector to the surface at the point of interest. and the direction of the light source.

• The intensity of light on the surface decreases with the cosine of the angle between the light source and the surface normal.

• The brightness depends on the orientation of the surface.

Mathematical expression

The mathematical expression for Lambert's cosine law is as follows.

Where,

• $I$ is the intensity of the light observed.

• $I_0$ is the maximum intensity of the light.

• $\theta$ is the angle between the direction of the light and the surface normal.

Reflectance model and cosine law in 3D graphics

• Lambertian reflectance model: Lambert's cosine law is often used in the lambertian reflectance model.

  • It is a popular model for representing the diffuse reflection of light from a surface.

  • It assumes that the surface is rough on a microscopic level and then scatters incident light equally in all the directions.

  • The reflectance model is widely used because it provides a clear approximation of the way surfaces in the real-world reflect light.

• 3D graphics: In the context of 3D graphics, you can compute the cosine of the angle as given below.

  • Direction of the light hitting a point on a surface ($L$)

  • Normal to the surface at the given point ($N$)

  • Normalize both the vectors

  • Compute the cosine of the angle by taking the dot product of $L$ and $N$

Note:

• The angle gives the amount of light that is reflected off the surface at that point in the direction of the camera.

• It leads to the perceived brightness of the surface.

Implications of the Lambert's cosine law

• Lambert's cosine law and surface brightness: Lambert's cosine law states that the color intensity of brightness $c$ of a point is proportional to the cosine of the angle $\theta$ between the light source and the normal of the surface. Thus, the brightness of a point on the surface depends on its orientation relative to the light source.

• Maximum brightness at $\theta = 0$: When the light source is parallel to the surface ($\theta = 0$), the $cos(\theta) = 1$. It results in maximum brightness, as the maximum amount of light is being reflected towards the observer directly.

• Decreasing brightness with increasing $\theta$: As the angle $\theta$ between the light source and the surface normal increases from from $0^\circ$ to $90^\circ$, the value of $\cos \theta$ gradually decreases from $1$ to $0$. It leads to a decrease in the perceived brightness. The surface starts to appear darker as less light is reflected towards the observer directly.

• Zero brightness at $\theta = 90^\circ$: At an angle of $90^\circ$, the light source is tangential to the surface. Since $\cos(90^\circ) = 0$, it implies that no light is reflected towards the observer. As a result, the brightness of the surface is zero. The surface appears darkest at this angle.

• Angle beyond $90^\circ$: The light source is no longer incident on the surface if the $\theta > 90^\circ$ . Thus, it does not contribute to the observed brightness.

• Lambertian surfaces and Light scattering: Lambert's cosine law is applicable to lambertian surfaces, that scatters incident light equally in all directions. It means that the surface has the same apparent brightness regardless of the observer's angle of view.

• Real-world surfaces and lambertian assumptions: Although real-world surfaces do not adhere to the lambertian assumption due to the factors such as reflection and subsurface scattering, the lambertian reflection model is still widely used in computer graphics. It is due to the model's simplicity and ability to provide a sufficiently good approximation for a wide range of surfaces.

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