<p>It’s kind of inaccurately stated—by comparing the SDF with a linearly increasing radius, the shape that’s being traced out is a cone with a hemisphere on the end, something like this:</p>

<p><a href="https://i.sstatic.net/zTeTH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/zTeTH.png" alt="diagram"></a></p>

<p>At any point along the horizontal center line, if the distance to the nearest surface is less than the current test radius, the cone-plus-hemisphere has intersected that surface.</p>

CC BY-SA 4.0
2020-01-24T04:01:39.467Z
Although I agree that that shape does indeed match the description, under which criteria are you saying that's what the document is talking about? I didn't see the mention of the spherical component in the thesis
Makogan
CC BY-SA 4.0
2020-01-24T19:51:45.990Z
What the document’s referring to is an approximation to the actual shape—there isn’t a good word for “the shape formed by sweeping a sphere of linearly increasing radius along a line”, but that shape *looks* basically like a cone and has similar characteristics, so “cone” is close enough.
Noah Witherspoon
CC BY-SA 4.0
2020-01-25T00:43:56.760Z
Granted—two outer lines should be tangent to the circle rather than attached to its diameter. Harder to draw that, though. :)
Noah Witherspoon
CC BY-SA 4.0
2020-01-24T00:37:10.107Z
2020-01-24T00:37:10.107Z
Noah Witherspoon
<p>I am reading <a href="https://static1.squarespace.com/static/559921a3e4b02c1d7480f8f4/t/596773e9f7e0ab3d29bb102f/1499952110561/Mroz+Michael_746.pdf" rel="nofollow noreferrer">this document</a> that talks about signed distance field rendering.</p>

<p>In that document section 6.3, on the topic of cone tracing, it says:</p>

<blockquote>
  <p>Cone tracing is an extension of the sphere tracing algorithm. In
  addition to the ray, a cone is created and expanded in the marching
  direction. At each sampling step, the current cone radius is compared
  with the minimum distance from the distance field. If the SDF value is
  smaller than the current radius, the cone has intersected an object
  [14]. This incurs very little additional cost to the normal sphere
  trace, while providing useful intersection information.</p>
</blockquote>

<p>In particular: <code>If the SDF value is smaller than the current radius, the cone has intersected an object</code>.</p>

<p>Am I reading this wrong? there's no way that's true. Consider the following diagram:</p>

<p><a href="https://i.sstatic.net/wGL7T.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/wGL7T.png" alt="enter image description here"></a></p>

<p>From that diagram, it seems obvious that for any cone radius <span class="math-container">$R$</span> there exists an SDF value <span class="math-container">$\epsilon$</span> such that <span class="math-container">$R &gt; \epsilon$</span> and the cone doesn't intersect the sphere.</p>

<p>Am I interpreting the document wrong?</p>

<raytracing><mathematics><geometry><raymarching><ray>
Reading thesis on cone tracing, did the author make a mistake?
Makogan
# Reading thesis on cone tracing, did the author make a mistake?

(Asked by: username_0 on 2020-01-23)

I am reading [this document](https://static1.squarespace.com/static/559921a3e4b02c1d7480f8f4/t/596773e9f7e0ab3d29bb102f/1499952110561/Mroz+Michael_746.pdf) that talks about signed distance field rendering.

In that document section 6.3, on the topic of cone tracing, it says:

> Cone tracing is an extension of the sphere tracing algorithm. In
> addition to the ray, a cone is created and expanded in the marching
> direction. At each sampling step, the current cone radius is compared
> with the minimum distance from the distance field. If the SDF value is
> smaller than the current radius, the cone has intersected an object
> [14]. This incurs very little additional cost to the normal sphere
> trace, while providing useful intersection information.

In particular: `If the SDF value is smaller than the current radius, the cone has intersected an object`.

Am I reading this wrong? there's no way that's true. Consider the following diagram:

[![enter image description here](https://i.sstatic.net/wGL7T.png)](https://i.sstatic.net/wGL7T.png)

From that diagram, it seems obvious that for any cone radius $R$ there exists an SDF value $\epsilon$ such that $R > \epsilon$ and the cone doesn't intersect the sphere.

Am I interpreting the document wrong?

**Tags:** raytracing, mathematics, geometry, raymarching, ray


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## 1 answers

### Answer by username_1 on 2020-01-24. Score: 2 | Accepted answer

It’s kind of inaccurately stated—by comparing the SDF with a linearly increasing radius, the shape that’s being traced out is a cone with a hemisphere on the end, something like this:

[![diagram](https://i.sstatic.net/zTeTH.png)](https://i.sstatic.net/zTeTH.png)

At any point along the horizontal center line, if the distance to the nearest surface is less than the current test radius, the cone-plus-hemisphere has intersected that surface.

> **Comments:**
> **username_0**: Although I agree that that shape does indeed match the description, under which criteria are you saying that's what the document is talking about? I didn't see the mention of the spherical component in the thesis  
> **username_1**: What the document’s referring to is an approximation to the actual shape—there isn’t a good word for “the shape formed by sweeping a sphere of linearly increasing radius along a line”, but that shape *looks* basically like a cone and has similar characteristics, so “cone” is close enough.  
> **username_1**: Granted—two outer lines should be tangent to the circle rather than attached to its diameter. Harder to draw that, though. :)  

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