Mistakes I perceived:
the emitter is always at normal incidence to the detector's view angle
no it isn't
radiance measured at the emitter is the same as radiance measured at the detector
this doesn't mean the radiance is the same when measured from a DIFFERENT ANGLE
it means the radiance doesn't change over DISTANCE
also it means radiance is the same for both combinations of terms:
- solid angle of emitter and projected area of detector
- solid angle of detector and projected area of emitter
The key point is that the cosine term in the definition of radiance does not use the detector's surface normal but the emitter's surface normal
it can use either, see my point above
apparent size of the source is quite clearly constant due to the nature of your setup
no it isn't, the apparent size of the source is SMALLER for the angled detector
you calculated radiance using the following terms:
- power is constant, since the emitter is obviously still emitting the same amount of light in both cases
- area is constant, as the emitter's surface area hasn't changed
- solid angle is constant, because the emitter is held facing the detector at normal incidence in both cases
but you missed PROJECTED AREA (or the cosine term)
Therefore, measured radiance is the same in both cases
not once you factor in the projected area term you missed
but for all practical purposes ™ it’s a point and a direction
no it isn't
you can't just ignore the differential projected area and solid angles ... otherwise what's the point in defining radiance at all?
you CAN simplify like this inside a video game using the simplified Lambertian dot product lighting we use ... but not in Radiometry and definitely not in the example I gave!
that is actually the WHOLE POINT of my post, to clarify why this is the case!
Assuming the receiver is left and the emitter right in your images
Irradiance at every point of the receiver is smaller for the angled configuration
agreed
but the emitter is on the left
so irradiance at the detector is the same
as the flux is the same and the detector area is the same
Think about what happens when you aim the emitter perpendicular to the receiver, the projected area goes to zero and no light moves towards the receiver so radiance is zero, as expected
your logic is flawed
when the emitter is perpendicular there is nothing to discuss, nothing to measure ...
it is the same as if I removed the emitter completely!
you are making the assumption that because the radiance is zero when perpendicular
that it must change from 1 to 0 as the viewing angle increases
this makes no sense
if instead you considered what happened at a viewing angle of 89.99999999 degrees - then you would have seen my point!
The orientation of the emitter doesn't matter since it's defined to emit equally in all directions
it doesn't matter when considering intensity or power ... it DOES matter when measuring radiance
as the differential projected area is LESS
Radiance (radiometry) clarification
If you carefully read these 2 articles
http://en.wikipedia.org/wiki/Lambert%27s_cosine_law
See the section "Details of equal brightness effect"
http://www.oceanopticsbook.info/view/radiative_transfer_theory/level_2/the_lambertian_brdf
You will see that Lambertian cosine * Radiance's 1/cosine _combine_ to give constant radiance
Obviously a 1/cosine term tends to INFINITY when the angle tends to 90°
Also the wikipedia Talk comments are agreeing with ME
you are misinterpreting them
http://en.wikipedia.org/wiki/Talk:Radiance#cosine_term
read them again carefully!
And another explanation:
In the figure on the right, if the source is Lambertian and equally bright in all directions, then the amount of energy that falls each the detectors is not the same for each of the detectors, but depends on the angle ? as cos?.
[Note: A source that is equally bright in all direction does not emit the same amount of energy in all directions. When determining the brightness of a small source we measure the energy that falls on a detector subtending a given solid angle at the source and then divide by the apparent size (area) of the source. A small source emitting the same amount of energy into the given solid angle as a larger source is brighter than the larger source.]
http://electron6.phys.utk.edu/optics421/modules/m4/radiometry.htm
Note:
A small source emitting the same amount of energy into the given solid angle as a larger source is brighter than the larger source
Maybe if you stopped attacking everyone and turning this into a "I'm right and you're wrong" argument, people would actually be able to discuss this to benefit everyone's understanding. And it is entirely possible to misinterpret the definitions in such a way that everything is still consistent, but incorrect with respect to the accepted definitions, so the distinction between "right" and "wrong" is meaningless in itself - the real question should be correct with respect to what? Radiometry is a very confusing field where everyone seems to use different definitions (I think this thread is a living example).
But as far as I am concerned, any physical quantity which goes to infinity like this is implausible and absolutely useless as it cannot be effectively measured. Regarding the Wikipedia comments - a physical source is defined as a source with an area (i.e. not a point light source). Now Srleffler says:
"The radiant flux for physical sources falls off at least as fast with angle as cos(?)."
And the radiant flux term in the radiance definition is at the top of the fraction, effectively cancelling the cosine term in the denominator. Radiance does not approach infinity as the angle approaches 90 degrees. The 1 / cosine term does, obviously, but the radiant flux term approaches zero at a rate at least inversely proportional to that. Radiant flux is the amount of radiant power crossing some imaginary cross-section (so, energy per unit of time). And guess what - the radiant flux considered here must pass through the projected area of the light source. And guess what again - this projected area is proportional to cos(theta). So radiant flux is proportional to cos(theta). Lambert's Law! The number of photons/sec (power) directed into any wedge is proportional to the area of the wedge!
And for the record, I see your point about intensity per unit solid angle. But your interpretation that radiance tends to infinity seems inconsistent with other equations, such as, say, the rendering equation. With your current interpretation of "infinite radiance", perceived brightness would increase without bounds when looking at anything that emits or reflect light in your direction (not just a light source) at close to 90 degree angles. That is clearly not the case.
apparent size of the source is quite clearly constant due to the nature of your setup
no it isn't, the apparent size of the source is SMALLER for the angled detector
Nice strawman. But that was my first post where I assumed emitter was on the right and receiver was on the left. Since you said later on that you meant something else, this is obviously not relevant anymore, and I acknowledged it. Good job quoting my post out of context, it is really helping this thread move forward.
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Either way, your tone and attitude are not making me want to reply to you, and I believe I speak for the other people in the thread as well. You created this thread asking a question. Now it seems you know all the answers, and are literally challenging anyone to prove you wrong. Now it is possible I am wrong. It is possible you are wrong. It is possible everyone in this thread is wrong, or everyone is right (and just talking past one another). It could just be because of some conceptual difference between the definition of radiance as used in computer graphics (where "radiance" might in fact mean "radiant exitance", for instance, which would not surprise me) and in physics (which you are using). You can probably come up with a set of definitions which makes any statement correct. That doesn't mean they will be of any practical use for whatever you are trying to do. Different applications sometimes call for different definitions (which should probably have different names, to avoid confusion, but sometimes that doesn't happen)
If radiance, which is a measure of energy passing through some part of space, approaches infinity as the viewing angle approaches 90º, then we could build infinite energy machines by placing an LED panel and a photovoltaic panel at 89.999 degress relative to each other. 
A small source emitting the same amount of energy into the given solid angle as a larger source is brighter than the larger source
That should just be common sense following from geometry though.
e.g. a big and a small emitter on the same plane, viewed by two detectors on a different angled plane. For the left detector in the image to read the same amount of energy as the right detector, then the small/left source will have to be outputting much more energy per m2 than the big/right source, because more area of the right source is visible to the detector.
If radiance is watts per steridian per area, and area increases with angle, then received watts should decrease as angle increases, because you divide by area. So, we should think that these energy measurements approach zero as the angle increases, because the area approaches infinity...
However, it's important to note that as the detector angle increases and the area visible to the detector increases, the solid angle occupied by the emitter also decreases, which balances things out somewhat.
e.g. above, the top detector, which is at a more shallow angle can see a larger area, but emitter is a smaller fraction of it's visible area. If we draw the wedge taken up by the emitter, it shrinks as the visible area grows.
You will see that Lambertian cosine * Radiance's 1/cosine _combine_ to give constant radiance
Obviously a 1/cosine term tends to INFINITY when the angle tends to 90°
If there's a term that approaches infinity, but the final result remains constant... then there must be another term that approaches zero to balance it out? And then it goes without saying that if radiance remains constant, then radiance can't approach infinity? So which is it, constant or increasing with angle?
I just want everybody to find the right answer
I don't care if I'm right or not
But right now I strongly believe my view is the correct one
and that radiance measured from the angled detector will be 1.414 times greater
All the points that have been made giving a radiance of 1.0 or less than 1.0 seem flawed to me
I don't mean to attack anybody, but let me point out where I think you are going wrong:
And the radiant flux term in the radiance definition is at the top of the fraction
yes it is, but the cosine term at the top comes from the Lambertian reflectance NOT from the radiance equation
If you consider the radiance equation alone (which I am doing by using a non-Lambertian surface)
you can see that radiance increases as the viewing angle increases
The "confusing" part of the radiance equation is that brightness increases as the projected area of the emitter decreases
this is because the power density increases (same number of photons/sec in a smaller area = brighter)
Only when you combine this with a Lambertian surface, where the radiant intensity decreases as viewing angle increases
do you get constant radiance when measured from different viewing angles
I hope I managed to make my point clearly this time
If radiance, which is a measure of energy passing through some part of space
you are going wrong right there
radiance is not a measure of energy passing through some part of space - that is flux
radiance is an abstract concept that ALLOWS you to measure energy passing through some part of space if you combine radiance with the viewing geometry
radiance = flux / solid_angle / projected_area
so
flux = radiance * solid_angle * projected_area
because you divide by area
the mistake you are making here is that of "dividing by area" means it will be smaller
but the cosine term is between 0 and 1
dividing something by 0.5 results in an increase, not a decrease
So which is it, constant or increasing with angle
in my diagram the radiance increases as viewing angle increases
IF the surface in my diagram was Lambertian the radiance would be constant
Hodgman: your top diagram is missing the point
you are showing a smaller emitter area
I am talking about a smaller emitter PROJECTED area
This doesn't make sense intuitively because Radiance is not a physical quantity and in real life surfaces are close to Lambertian
Hodgman
very interesting point about the area of the emitter that is visible being greater
however it just confirms the radiance of 1.414
here I show the radiance calculation from both points of view, and the result is 1.414 for both:
Your idea that the solid angle changes is wrong ...
