Cosine Law Applet: This models the relationship between incidence angle, distribution area and relative intensity. This is an interactive applet so you can adjust the incidence angles by clicking and dragging them with the left mouse button or using the middle and right buttons to pan or rotate the view.


This applet allows you to interactively adjust the azimuth and altitude of incident energy and see the relative effect on both horizontal and vertical surfaces. To do this, use the azimuth and altitude nodes to interactively drag the direction of the energy beam. You can also adjust these values using the sliders along the left and bottom edge of the applet.

The width and height sliders along the top and right edges allow you to adjust the size and relative shape of the energy beam. However, you will notice that changing these values does not affect the relative intensity or the incidence area. This is because the Cosine Law is the same for all beam shapes and/or incidence locations on a planar surface. On such surfaces, the Lambert cosine law is solely governed by incidence angle.

Fundamental Concepts

Lambert’s cosine law (also known as the cosine emission law) states that the measure of radiant energy from a surface that exhibits Lambertian reflection is directly proportional to the cosine of the angle formed by the measurement point and the surface normal. It follows that irradiance or illuminance falling on a surface varies similarly with the cosine of the incidence angle.

The perceived measurement area orthogonal to the incident flux is significantly reduced at oblique angles, causing energy to be spread out over a wider area than it would if it was falling perpendicular to the surface. Thus, if you consider a fixed surface area, the amount of energy to which it is exposed will reduce significantly the closer the source is to grazing incidence.

Angle of Incidence

The angle of incidence is the angle between a ray that hits a surface and a line that is perpendicular in all directions to that surface at the point of incidence. This perpendicular line is usually called the surface normal.

Grazing Incidence

Grazing incidence is the term used to describe situations where the irradiance or illuminance is travelling almost parallel to the incident surface, meaning that the incidence angle is very close to 90 degrees. As the cosine of 90 degrees is zero, this means that the resulting relative intensity will be very low as the distribution area is very large.

Effects on Solar Radiation

The Lambert cosine law has a significant effect on the solar radiation received in different parts of the world and also on different surfaces of the same building. Radiant energy from the Sun strikes surfaces on the Earth at different angles of incidence. This varies both hourly, as the Sun passes through the sky, and seasonally as its daily average altitude changes - something especially obvious at higher latitudes away from the equator.

At the equator where the Sun passes almost directly overhead, the intensity of light and heat received on horizontal surfaces is usually much greater than on vertical surfaces. This is especially true for those facing directly North and South as the Sun travels from East to West and passes close to the zenith point. At higher latitudes, this begins to change as the Sun is usually much lower in the sky such that vertical equator-facing surfaces receive much more radiation than horizontal surfaces.

Click here to comment on this page.