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Book 3 figure captions
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hollasch committed Nov 2, 2019
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problem being a classic case study. We’ll do a variation inspired by that. Suppose you have a circle
inscribed inside a square:

![Figure 2-1](../images/fig-3-02-1.jpg)
![Figure [circ-sq]: Estimating π with a circle inside a square](../images/fig.circ-sq.jpg)

</div>

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We can mitigate this diminishing return by *stratifying* the samples (often called *jittering*),
where instead of taking random samples, we take a grid and take one sample within each:
![Figure 2-2](../images/fig-3-02-2.jpg)
![Figure [jitter]: Sampling areas with jittered points](../images/fig.jitter.jpg)
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First, what is a _density function_? It’s just a continuous form of a histogram. Here’s an example
from the histogram Wikipedia page:

![Figure 3-1](../images/fig-3-03-1.jpg)
![Figure [histogram]: Histogram example](../images/fig.histogram.jpg)

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between 0 and 2 whose probability is proportional to itself: $r$. We would expect the pdf $p(r)$
to look something like the figure below. But how high should it be?

![Figure 3-2](../images/fig-3-03-2.jpg)
![Figure [linear-pdf]: A linear PDF](../images/fig.linear-pdf.jpg)

<div class='together'>
The height is just $p(2)$. What should that be? We could reasonably make it anything by
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goes through. The solid angle $\omega$ and the projected area $A$ on the unit sphere are the same
thing.

![Figure 4-1](../images/fig-3-04-1.jpg)
![Figure [solid-angle]: Solid angle / projected area of a sphere](../images/fig.solid-angle.jpg)

Now let’s go on to the light transport equation we are solving.

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<div class='together'>
And plot them for free on plot.ly (a great site with 3D scatterplot support):

![Image 7-1](../images/fig-3-07-1.jpg)
![Figure [pt-uni-sphere]: Random points on the unit sphere](../images/fig.pt-uni-sphere.jpg)

</div>

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$1/A$. But what is it on the area of the unit sphere that defines directions? Fortunately, there is
a simple correspondence, as outlined in the diagram:

![Figure 9-1](../images/fig-3-09-1.jpg)
![Figure [light-pdf]: Projection of light shape onto PDF](../images/fig.light-pdf.jpg)

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<div class='together'>
Now what is $\theta_{max}$?

![Figure 12-1](../images/fig-3-12-1.jpg)
![Figure [sphere-cone]: A sphere enclosing cone](../images/fig.sphere-cone.jpg)

</div>

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