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NathanRidley

A radiometry question for those of you who own Real Time Rendering, 3rd Edition

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I'm a bit confused by the following diagram (7.4) on page 207 of Real Time rendering, 3rd Edition:

 

2014-10-26_1634.png

 

The text before and after I think makes a few assumptions and doesn't quite explain what I'm looking at well enough. What I understand:

  • dw is the change in steradians/solid angle w
  • dA is the area irradiated by the light ray with respect to dw
  • n is obviously the surface normal
  • The orange line is clearly a light ray

Why am I confused? The light ray is apparently hitting a surface (hence the surface normal pointing away from the surface), and the angle depicted is the angle between the ray hitting the surface, and the surface normal. But what is the circle at dw indicating? My understanding was that the solid angle is easily visualised as a cone apexed at the light source, with the circle representing a patch surface on a sphere enclosing the light source. Why then is that circle in the diagram apparently situated at the light source? It seems to me that the cone in the diagram should be facing the other way.

 

p.s. I hope the authors don't mind me displaying this diagram. I will remove it and replace it with my own hand-drawn version if requested to do so.

Edited by axefrog

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Actually looking at it a bit more, I'm wondering if the small orange circle is indicating that the light source is on the other side (i.e. not shown, and much further to the top left of the diagram) and in effect we're looking at the hypothetical patch surface hit by the light ray and then a vertical surface projected onto from there, or in other words, that diagram would be better drawn with the orange circle touching the blue oval. I could be completely wrong, but that's how I'm thinking it's supposed to be interpreted. Similarly, I'm wondering if the cone appears to be backwards because it's just trying to show the dispersion of the light ray (or w) with respect to the light ray projecting onto the blue surface?

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I don't have the book, but I think I might be able to help.

Why then is that circle in the diagram apparently situated at the light source?


You need to imagine multiple light sources, or light sources that are not points. You are interested in summing the contributions from light sources in the circle.

If that's not clear enough, ask again.

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You need to imagine multiple light sources, or light sources that are not points. You are interested in summing the contributions from light sources in the circle.

 

I'm not sure that's correct, is it...? The red circle is labelled dw (where w is omega) indicating that it represents a specific solid angle (or change therein), which is measured in steradians. So if what you said is correct, what's the significance of dw and why would the writer use a tiny circle to indicate some arbitrary group of light rays from just that circular area, of which the orange line is one?

Edited by axefrog

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What's the significance of dw? We are integrating some quantity (radiance) and the dw indicates what we are integrating over. People often think of dw as representing a tiny solid angle, which the author drew as a little circle, but this is mostly just a way of developing intuition for it.

If you are having a hard time understanding surface integrals, you are not alone. There is a subject called Differential Geometry that deals with all these things rigorously, but it's not for the faint of heart.

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Alvaro, I appreciate your answers, I just have a feeling that they're lacking the context of the preceding pages. The way book is structured, each chunk of content builds on what comes before it, starting off simply enough that the book is very approachable, which is one of the reasons it's so good. The diagrams and illustrations in the book are no different (so far) in this regard, hence why the question was kind of targeted at people who have a copy of the book. I'm honestly not disagreeing with your answer, I was just hoping for an answer that included some context from what comes before the diagram. The mathematics are not really a problem, and while my knowledge is patchy with regards to calculus, I have been switching out and learning the topics I need, as needed. If there's something new to learn here in that regard, I have no problem doing that.

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Hmm... I'm thinking reading ahead may have been a good idea. The red circle may actually represent a patch surface on the surface of the light source.

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You need to imagine multiple light sources, or light sources that are not points. You are interested in summing the contributions from light sources in the circle.

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Technically you really should be considering area lights. Point lights are an unphysical hack that don't sit well with radiometry and need special treatment. If you consider area lights then it makes perfect sense: you are integrating over the projected surface area of the light (over the solid angle subtended by that light source from your point of view) and hence calculating the total amount of radiation the light is emitting towards you. The dw is a differential solid angle and does not represent a surface area, dw is an infinitesimal area of the unit sphere (it has units steradians) and is usually defined as dw = sin(theta) dtheta dphi, via the following parameterization of the unit sphere (or hemisphere in the illustration below):

 

9gxCkK9.png

As Alvaro said, sometimes people think of it as a cone (or occasionally a direction) because it's more intuitive. If you are still confused, you don't integrate over the light's surface area directly, but over its surface area projected over your field of view (the hemisphere above) since that is the definition of radiance: the amount of light falling onto some surface of interest through some differential solid angle. And that's really all you are interested in.

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