Lambert's cosine law
A law stating that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle between the direction of the incident light and the surface normal.
Lambert's cosine law: why surfaces look dimmer at an angle
Lambert's cosine law describes how the brightness of a diffusely reflecting surface changes depending on the viewing angle. When you look straight at a matte surface, it appears brightest; as you tilt your viewpoint away from perpendicular, the surface appears progressively dimmer. The intensity drops in direct proportion to the cosine of the angle between your line of sight and the surface normal, the imaginary line pointing straight out from the surface.
The law applies to ideal diffuse reflectors, surfaces that scatter light equally in all directions according to their apparent brightness. A perfectly diffuse surface follows the law exactly; real materials like paper, matte paint, and unpolished metal come close. Rough surfaces scatter light randomly and approximately follow the law. Specular surfaces like mirrors violate it entirely because they reflect light in a preferred direction.
The physical explanation lies in the projected area of the surface. When you view a surface head-on, you see its full apparent area. At 45 degrees, the projected area becomes cos(45°) or roughly 0.707 of the head-on value; at 60 degrees, it drops to 0.5. The radiating or reflecting surface itself does not change, but the effective area you receive light from does. This projection effect explains why the cosine term appears in the equation.
Applications in measurement and imaging
The law is fundamental to photometry and radiometry. When you measure the luminous intensity of a surface using a light meter, the angle matters critically. An illuminometer pointed directly at a surface gives a different reading than one held at 30 degrees, even though the surface has not changed. Calibration standards for light meters assume perpendicular measurement for this reason.
In imaging and display technology, Lambert's law helps predict how an image or scene will appear to viewers at different angles. A matte diffuser in a camera or projector approximates a Lambertian surface. Liquid crystal displays and other flat panels often depart from Lambertian behavior, which is why they look different when viewed from the side. Quality control in optical manufacturing routinely checks compliance with the law to verify surface diffuseness.