# Is a Lambertian reflector illuminated by a smaller fraction of the incident radiation when it's tilted?

In reading about Lambertian reflectance on Wikipedia I found the following phrase (in bold) which doesn't sound right to me:

In computer graphics, Lambertian reflection is often used as a model for diffuse reflection. This technique causes all closed polygons (such as a triangle within a 3D mesh) to reflect light equally in all directions when rendered. In effect, a point rotated around its normal vector will not change the way it reflects light. However, the point will change the way it reflects light if it is tilted away from its initial normal vector since the area is illuminated by a smaller fraction of the incident radiation.

The way I picture the situation described in the paragraph, only tilting away from the light source would cause less light to be incident in a given area. In general, tilting away from the initial normal vector could lead to either an increase or a decrease in incident light per area, as this says nothing about the location of the light source.

Have I misunderstood the context, or is this something that should be rewritten on Wikipedia?

• The cross sectional area visible dimishes as your incidence angle grows. Aug 27, 2015 at 13:01
• @joojaa I follow that bit, but the bit in bold seems to be talking about tilting the surface away from its initial normal vector, which would only make sense for the specific case that the incident light is perpendicular to the surface, or I'm missing something. Aug 27, 2015 at 13:10
• Yes, the wording is quite strange (what is a rotating point for instance ? :-) ). It's not a mistake, it's poor wording. I'm afraid that once a guy quicky made the whole "basic computer graphics" content of wikipedia for some reason, letting a lot of polishing (or more) to be done. Seems like hot topics are well edited and completed (by academics and master/PhD students ?), but not basic topics (I did, for a very few). Oct 9, 2015 at 7:44

I see some problems in the quote you posted.

In effect, a point rotated around its normal vector will not change the way it reflects light.

This is true, because a Lambertian reflector will never change the way it reflects light. The underlying principle stays the same. Also, Lambertian surfaces are isotropic, so the amount of reflected light won't change either (which is probably what this sentence is aiming for).

However, the point will change the way it reflects light if it is tilted away from its initial normal vector since the area is illuminated by a smaller fraction of the incident radiation.

Again not true, because the principle doesn't change. The amount may change, except for the special case that the cosine is <= 0 before and after the tilting. The amount does not necessarily grow, except if we define that the cosine equals 1 before, i.e. that the normal points directly towards the light source.

This whole paragraph should probably be rewritten to be less ambiguous. Including isotropy could make it more complete.

You are correct, it's badly worded. Illumination falls off with the cosine of the angle between the surface normal and the inverse light direction, so the wording implies the light is shining down the original surface normal, and so any tilting away would be tilting away from the lighting direction.

• Illumination does not actually fall of the surface area towards the light is just smaller Sep 4, 2015 at 17:11
• You're right I think, "fall-off" to me is anything that makes the surface area smaller from the lights perspective, so distance from and rotation away from the light have the same effect to me, but my definition of "fall-off" is probably not mathematically correct :P Sep 5, 2015 at 15:27
• well yes but that would be hard for a layman to understand. lots of things can fall off. Sep 5, 2015 at 17:38

What one has to do in this matter is first define the quantities that are physically at play here, so that everybody speaks about the same thing.

There is:

Flux emitted by a surface per solid angle per projected area.

Unit is W·sr−1·m−2
Origin of radiance, you take the surface area out of the unit.
Unit is W·sr−1
• intensity (wikipedia)
A perception-based unit of power per solid angle.
Unit is candela
• luminance (wikipedia)
The luminance is normally obtained by dividing the luminous intensity by the light source area (source)
therefore this is also perception based.
Unit is cd·m−2
• luminuous flux (wikipedia)
Same thing but not related to solid angle.
quote:
Luminous flux is a measure of the total amount of light a lamp puts out. The luminous intensity (in candelas) is a measure of how bright the beam in a particular direction is
Unit is lumen