<p>Why does the equation at the bottom of the following page hold?</p>

<p><a href="http://www.pbr-book.org/3ed-2018/Monte_Carlo_Integration/Transforming_between_Distributions.html" rel="nofollow noreferrer">http://www.pbr-book.org/3ed-2018/Monte_Carlo_Integration/Transforming_between_Distributions.html</a></p>

<p><span class="math-container">$p(\theta,\phi)d\theta d\phi=p(\omega)d\omega$</span></p>

<p>Is this a conclusion of probability and measure theory?</p>

CC BY-SA 4.0
2020-05-25T13:20:50.430Z
Right above it you have $d\omega = \sin\theta \,d\theta\,d\phi$. Substitute and see what you get.
lightxbulb
CC BY-SA 4.0
2020-05-25T16:33:57.230Z
I think the story should be like this: Premises: $d\omega=sin\theta\ d\theta\ d\phi$ and $p(\theta,\phi)d\theta\ d\phi=p(\omega)d\omega$; Conclusion: $p(\theta, \phi)=sin\theta\ p(\omega)$.
chaosink
CC BY-SA 4.0
2020-05-25T17:04:10.830Z
That's it really. And yes, the above is from probability and measure theory. You can define your pdf with respect to some measure and then find its expression with respect to a different measure.
lightxbulb
CC BY-SA 4.0
2020-05-25T17:15:17.360Z
But I couldn't find a formal statement in math books. Could you give a reference? Thanks!
chaosink
CC BY-SA 4.0
2020-05-25T20:19:13.520Z
You need a statement about what? That $d\omega = \sin\theta \,d\theta\,d\phi$? That's by definition of solid angle. But if you want to get where the sine comes from, you could just compute the coordinate transformation from cartesian to spherical coordinates and then set $r=1$ ($r^2\sin\theta$ pops from the Jacobian of the transformation).
lightxbulb
CC BY-SA 4.0
2020-05-26T04:38:22.263Z
This one: $p(\theta,\phi)d\theta d\phi=p(\omega)d\omega$
chaosink
CC BY-SA 4.0
2020-05-26T10:04:53.873Z
This is just writing the pdf with respect to different measures and equating it. If you've studied vector calculus it's similar to going from a surface integral to one in cartesian coordinates.
lightxbulb
CC BY-SA 4.0
2020-05-30T00:52:15.163Z
Re "I think the story should be like this:..." Yes, the story is exactly that. Those three lines appear in exactly that order in the text.
Nathan Reed
<distribution><pbrt><pdf><differential><pbrt-v3>
An equation about PDFs and differentials in the book of pbrt-v3
chaosink
# An equation about PDFs and differentials in the book of pbrt-v3

(Asked by: username_0 on 2020-05-25)

Why does the equation at the bottom of the following page hold?

<http://www.pbr-book.org/3ed-2018/Monte_Carlo_Integration/Transforming_between_Distributions.html>

$p(\theta,\phi)d\theta d\phi=p(\omega)d\omega$

Is this a conclusion of probability and measure theory?

**Tags:** distribution, pbrt, pdf, differential, pbrt-v3

> **Comments:**
> **username_1**: Right above it you have $d\omega = \sin\theta \,d\theta\,d\phi$. Substitute and see what you get.  
> **username_0**: I think the story should be like this: Premises: $d\omega=sin\theta\ d\theta\ d\phi$ and $p(\theta,\phi)d\theta\ d\phi=p(\omega)d\omega$; Conclusion: $p(\theta, \phi)=sin\theta\ p(\omega)$.  
> **username_1**: That's it really. And yes, the above is from probability and measure theory. You can define your pdf with respect to some measure and then find its expression with respect to a different measure.  
> **username_0**: But I couldn't find a formal statement in math books. Could you give a reference? Thanks!  
> **username_1**: You need a statement about what? That $d\omega = \sin\theta \,d\theta\,d\phi$? That's by definition of solid angle. But if you want to get where the sine comes from, you could just compute the coordinate transformation from cartesian to spherical coordinates and then set $r=1$ ($r^2\sin\theta$ pops from the Jacobian of the transformation).  
> **username_0**: This one: $p(\theta,\phi)d\theta d\phi=p(\omega)d\omega$  
> **username_1**: This is just writing the pdf with respect to different measures and equating it. If you've studied vector calculus it's similar to going from a surface integral to one in cartesian coordinates.  
> **username_2**: Re "I think the story should be like this:..." Yes, the story is exactly that. Those three lines appear in exactly that order in the text.  

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## 0 answers

