Intersection of a ray and a plane

I previously showed the derivation of how to determine the intersection of a plane and a cone. At the time I had to solve that equation, so after doing so I decided to publish it for anyone to use. Given the positive feedback, it seems this was useful, so I might as well continue with a few more.

Here is probably the most basic intersection: a ray and a plane. Solving it is straightforward, which I hope can be seen below. Like last time, I am using vector notation.

  1. We define a ray with its origin $O$ and its direction as a unit vector $\hat{D}$.
    Any point $X$ on the ray at a signed distance $t$ from the origin of the ray verifies: $\vec{X} = \vec{O} + t\vec{D}$.
    When $t$ is positive $X$ is in the direction of the ray, and when $t$ is negative $X$ is in the opposite direction.
  2. We define a plane with a point $S$ on that plane and the normal unit vector $\hat{N}$, perpendicular to the plane.
    The distance between any point $X$ and the plane is $d = \lvert (\vec{X} – \vec{S}) \cdot \vec{N} \rvert$. If this equality is not obvious for you, you can think of it as the distance between $X$ and $S$ along the $\vec{N}$ direction. When $d=0$, it means $X$ is on the plane itself.
  3. We define $P$ the intersection or the ray and the plane, and which we are interested in finding.

Since $P$ is both on the ray and on the plane, we can write: $$ \left\{ \begin{array}{l} \vec{P}=\vec{O} + t\vec{D} \\ \lvert (\vec{P} – \vec{S}) \cdot \vec{N} \rvert = 0 \end{array} \right. $$ Because the distance is $0$, the absolute value is irrelevant here. We can just write: $$ \left\{ \begin{array}{l} \vec{P}=\vec{O} + t\vec{D} \\
(\vec{P} – \vec{S}) \cdot \vec{N} = 0 \end{array} \right. $$ All we have to do is replace $P$ with $\vec{O} + t\vec{D}$ in the second equation, and reorder the terms to get $t$ on one side.
$$ (\vec{O} + t\vec{D} – \vec{S}) \cdot \vec{N} = 0 $$ $$ \vec{O} \cdot \vec{N} + t\vec{D} \cdot \vec{N} – \vec{S} \cdot \vec{N} = 0 $$ $$ t\vec{D} \cdot \vec{N} = \vec{S} \cdot \vec{N} – \vec{O} \cdot \vec{N} $$ $$ t = \frac{(\vec{S} – \vec{O}) \cdot \vec{N}}{ \vec{D} \cdot \vec{N} } $$

A question to ask ourselves is: what about the division by $0$? Looking at the diagram, we can see that $\vec{D} \cdot \vec{N} = 0$ means the ray is parallel to the plane, and there is no solution. Otherwise, the ray intersects the plane for the value of $t$ written above. That’s it, we’re done.

Note: There are several, equivalent, ways of representing a plane. If your plane is not defined by a point $S$ and a normal vector $\hat{N}$, but rather with a distance to the origin $s$ and a normal vector $\hat{N}$, you can notice that $s = \vec{S} \cdot \vec{N}$ and simplify the result above, which becomes: $$ t = \frac{s – \vec{O} \cdot \vec{N}}{ \vec{D} \cdot \vec{N} } $$


Signed distance to a plane

For the sake of simplicity, in the above we defined the distance to the plane as an absolute value. It is possible however to define it as a signed value: $d = (\vec{X} – \vec{S}) \cdot \vec{N}$. In this case $d>0$ means $X$ is somewhere on the side of the plane pointed by $\vec{N}$, while $d<0$ means $X$ is on the opposite side of the plane.

Distances that can be negative are called signed distances, and they are a foundation of Signed Distance Fields (SDF).

Interview with competitive live coders

Last week my friend LLB and I wrote an article about live coding. Or rather, given what the task consisted in, it would be more accurate to say that we copy-pasted the answers of six interviewees until the order felt right.

In their comments, they remember how and when they’ve discovered live coding and got involved, explain how they prepare for a competition, talk about their state of mind during a match, share their esteem for fellow live coders, and reflect on this new kind of e-sport.

You can read them here: A new e-sport: live coding competitions.

Implementing a Physically Based Shading without locking yourself in

Over the last few months I have been trying to push my understanding of Physically Based Shading, by actively exploring every corner and turning over every stone, to uncover any area where I lack knowledge. Although this is still an ongoing process and I still have a lot to do, I thought I could already share some of what I have learned in the process.

Last weekend the Easter demoparty event Revision took place, as an online version due to the current pandemic situation. There, I presented a talk on Physically Based Shading, in which I went into electromagnetism, existing models, and an brief overview of a prototype I am working on.

The presentation goes into a lot of detail about interaction of light with matter from a physics point of view, then builds its way up to the Cook-Torrance specular BRDF model. The diffuse BRDF and the Image Based Lighting were skipped due to time constraints. I am considering doing a Part 2 to address those topics, but I haven’t decided anything yet.

In the mean time, please leave a comment or contact me if you notice any mistake or inaccuracy.

Abstract

How do you implement a Physically Based Shading for your demos yet keep the possibility to try something completely different without having to rewrite everything?
In this talk we will first get an intuitive understanding of what makes matter look the way it looks, with as much detail as we can given the time we have. We will then see how this is modeled by a BRDF (Bidirectional Reflectance Distribution Function) and review some of the available models.
We will also see what makes it challenging for design and for real-time implementation. Finally we will discuss a possible implementation that allows to experiment with different models, can work in a variety of cases, and remains compatible with size coding constraints.

Slides

Here are the slides, together with the text of the talk and the link to the references:
Implementing a Physically Based Shading model without locking yourself in.

Video

And finally here is the recording of the talk, including a quick demonstration of the prototype:

Interference shader

Here is the shader used during the presentation to illustrate light interaction at the interface between to media:

Acknowledgements

Thanks again to Alan Wolfe for reviewing the text, Alkama for the motivation and questions upfront and help in the video department, Scoup and the Revision crew for organizing the seminars, Ronny and Siana for the help in the sound department, and everyone who provided feedback on my previous article on Physically Based Shading.

Addendum

Following the publication of this article, Nathan Reed gave several comments on Twitter:

FWIW – I think the model of refraction by the electromagnetic field causing electrons to oscillate is the better one. This explains not only refraction but reflection as well, and even total internal reflection. Feynman does out the wave calculations: https://feynmanlectures.caltech.edu/II_33.html

It also explains better IMO why a light wave keeps its direction in a material. If an atom absorbs and re-emits the photon there is no reason why it should be going in the same direction as before (conservation of momentum is maintained if the atom recoils). Besides which, the lifetime of an excited atomic state is many orders of magnitude longer than the time needed for a light wave to propagate across the diameter of the atom (even at an IOR-reduced speed).

Moreover, in the comments of the shader above, CG researcher Fabrice Neyret mentioned a presentation of his from 2019, which lists interactions of light with matter: Colors of the universe.
Quoting his summarized comment:

In short: the notion of photons (and their speed) in matter is a macroscopic deceiving representation, since it’s about interference between incident and reactive fields (reemitted by the dipoles, at least for dielectrics).


From Maxwell’s equations to Fresnel’s equations

This series of short videos shows how to derive Maxwell’s equations all the way to Fresnel’s equations. Each one is about 10 to 15mn long.

The first four videos show how to use boundary conditions to deduce the relationship between the electromagnetic field on both sides of a surface (or interface between two different media).

The next four videos use the previous results to obtain the Fresnel equations, for S-polarized and P-polarized cases.

The rest of the series then dives into other topics like thin film interference.

The series assumes the viewer to be already familiar with the Maxwell equations, so it can be helpful to first see the explanation by Grant Sanderson of 3Blue1Brown on Maxwell’s equations.

Physically based shading references, at the end of 2019

A lot has happened in the graphics community in the last ten years, especially when it comes to physically based rendering (PBR). It started to become popular around 2009, as hardware made more powerful models affordable, and really took over between 2010 and 2014. Real-time engines started replacing Phong and Blinn-Phong with a normalized Blinn-Phong, until pretty much everyone switched to GGX and its long trailing reflections. Researchers explored how to make it work with image based lighting (IBL), area lights, and it seems that nowadays everyone is looking at the secondary bounce problem.

I am not sure why adoption also happened at the same in the film industry (instead of much earlier), despite having different constraints than real-time. Films made before 2010 were mostly ad hoc, until a wave converted nearly the entire industry to unbiased path tracing.

I gathered a first PBR reading list back in 2011, but since then, the community has collectively made strides of progress. I also have a better understanding of the topic myself. So I think it is time to revisit it with a new, updated (and unfortunately, longer) reading list.

However, covering the entire PBR pipeline would be way too vast, so I am going to focus on physically based shading instead, and ignore topics like physical lighting units, physically based camera or photogrammetry, even though some of the links cover those topics.

Note: If you see mistakes, inaccuracies or missing important pieces, please let me know. I expect to update this article accordingly during the next few weeks.

Courses and tutorials

  • Physically Based Shading in Theory and Practice (formerly “Practical Physically Based Shading in Film and Game Production”)
    2010, (no 2011?), 2012, 2013, 2014, 2015, 2016, 2017.
    This recurring SIGGRAPH course by the leading actors of the field is a fantastic resource and a must see for anyone interested in the topic. Naty Hoffman then Stephen Hill have been hosting on their websites the course material for several years. Some of the presentations are also available on Youtube.
  • Physically Based Rendering: From Theory To Implementation, Third edition, 2016, Matt Pharr, Wenzel Jakob, and Greg Humphreys
    As of 2018, the content of this reference book is entirely available online.
  • Implementation Notes: Runtime Environment Map Filtering for Image Based Lighting, 2015, Padraic Hennessy.
    Details how to implement the environment map filtering described in Karis and Lagarde publications (see below), then how to optimize it by reducing the number of samples thanks to importance sampling and rejecting samples that don’t contribute.
  • Image Based Lighting, 2015, Chetan Jaggi.
    Focused on specular reflections, the article presents the implementation of image based lighting (IBL) using the split sum approximation from Unreal Engine 4 (described below), and how to improve quality for several cases.
  • Physically Based Rendering Algorithms: A Comprehensive Study In Unity3D, 2017?, Jordan Stevens.
    This tutorial explains what the different parts of the Bidirectional Reflection Distribution Function (BRDF) mean, lists many available bricks, and shows them in isolation. It is directed at Unity, but translates easily to other environments.
  • LearnOpenGL’s PBR series (theory, lighting, diffuse irradiance, specular IBL), 2017, Joey de Vries.
    An excellent introduction that explains the basics and walks the reader through the implementation of a shader based on the same model as Unreal Engine 4 (detailed below). There seems to be a confusion between albedo and base colour, but it’s otherwise clear and well structured.

Real world references

In depth overviews

The following publications all describe the work done by teams who had to do an inventory of the existing options, and choose a model for their particular needs.

  • Physically-Based Shading at Disney (slides), 2012, Brent Burley et al.
    What came to be known as the “Disney BRDF” was a milestone in PBR literature, and a reference many other works are built upon. It compares different existing models to the MERL database (see previous section), notes their strengths and weaknesses, discusses in length the observed behaviour, especially the diffuse response at grazing angles, and proceeds to define their own empirical shading model to mimic that behaviour. The Disney BRDF is designed to be robust and expressive but also simple and intuitive for artists.
    In the annex, a brief overview of the history of BRDF is given.
    The publication proposes a tool, the BRDF Explorer, to visualize and compare analytic BRDF models or measured ones.
    (In 2015, a follow up publication extended their model to a full BSDF in order to support refraction and scattering, but this falls out of the scope of this already long list.)
  • Real Shading in Unreal Engine 4 (slides), 2013, Brian Karis.
    Strongly inspired by the Disney BRDF, it presents a similar shading model. It prefers a simple Lambert diffuse BRDF due to both the cost and the integration with spherical harmonics, and uses other approximations for realtime. The course notes mention that a lot of work was done to compare the various available bricks, but doesn’t list them. Karis lists them in a separate publication, listed next.
    For image based lighting, the famous “split sum” approximation of the integral is introduced, allowing to convolve a part of the integral, and precompute the rest in a 2D look-up table (LUT).
    When explaining how the workflow adapted to this new model, the course notes stress the importance of having linear parameters for material interpolation.
    Warning: I was told a year ago by Yusuke Tokuyoshi that there was an error in a derivation, but my understanding of it is not sufficient to spot it. Apparently the error is only in the publication, and was fixed in the actual code though.
  • Specular BRDF Reference, 2013, Brian Karis.
    Lists various available bricks for the Cook-Torrance BRDF, using the same naming convention.
  • Moving Frostbite to Physically Based Rendering 3.0, 2014, Sébastien Lagarde.
    The biggest publication in this list, with over 120 pages of course notes. I haven’t finished reading it yet, but this is an outstanding piece of work, that goes deep into details for many of the aspects involved.
  • Physically Based Rendering in Filament, 2018, Romain Guy et al.
    This documentation presents the shading model used in Filament, the choices that were made, which are similar to Frostbite in many ways, and the alternatives that were available. The quality of this document is outstanding, and it seems it is becoming a reference for PBR implementations.
  • MaterialX Physically-Based Shading Nodes, 2019, Niklas Harrysson, Doug Smythe and Jonathan Stone.
    This specification is meant as a transfer format in the VFX industry. It describes a wide range of materials, not limited to BRDF, but also including emissive and volumetric materials, and allows to choose between a variety of such functions.
    Reading this document can help solidify or confirm the understanding of how all these different functions contribute to the rendering ecosystem. However I would only recommend it to readers who already have a fairly good understanding of the PBR models.

Disney BRDF and BRDF Explorer implementations

The Disney BRDF is so popular that many implementations can be found in the wild. Here are a few of them.

Diffuse

It seems that in litterature, diffuse BRDF are a lot less covered than specular ones. I suppose this is because it is harder to solve, while the low frequency nature of the diffuse component makes its quality less noticeable. Therefore, many realtime implementations consider the Lambert model sufficient. However, the following publications explore the topic.

  • Physically-Based Shading at Disney (slides), 2012, Brent Burley et al.
    One of the contribution of the “Disney BRDF” is its diffuse model. It compares several existing diffuse models with the measured data of the MERL database (see earlier section) but, unsatisfied with their response, it proposes its own, empirical one. One of the features of that model is the retroreflection at grazing angles.
    I have read multiple times that this model is not energy conserving. Yet Disney uses it for offline rendering, which I assume is path tracing (?), so I am not sure of what is the impact of that decision.
  • Moving Frostbite to Physically Based Rendering 3.0, 2014, Sébastien Lagarde.
    The diffuse BRDF described is a normalized version of the Disney BRDF to make it energy conserving.
  • Designing Reflectance Models for New Consoles (slides), 2014, Yoshiharu Gotanda.
    Gotanda explains here several weaknesses of the Oren-Nayar model for PBR (its geometry term is different than the one used for the specular term, and it’s not energy conserving), and proceeds to propose a modified version. Since there is no analytic solution, he suggests a fitted approximation.
    He also reminds his own improvement over Schlick’s Fresnel approximation, but concludes that both models fail for complex indices of refraction.
  • PBR Diffuse Lighting for GGX+Smith Microsurfaces, 2017, Earl Hammon, Jr.
    This presentation tries to combine the Oren-Nayar diffuse model (originally a Gaussian normal distribution) with the GGX normal distribution. It studies the Smith geometry fonction (G), proposes a BRDF to use for testing with path tracing, and concludes with an approximation for a diffuse GGX.
    On a side note, a few final slides give some identities that are useful for shader optimization.

Energy conservation

One of the pillars of PBR is to make sure to respect the energy conservation law. When designing a BRDF, there shouldn’t be more energy out than comes in. This is especially important for path tracing to converge. The following links explain how to take that constraint into account.

  • Energy Conservation In Games, 2009, Rory Driscoll.
    Explains briefly the problem of energy conservation, and details how to obtain the normalization factor for diffuse Lambert. It’s a good example of how to get started.
    The comments discuss the case of the Phong and Blinn-Phong specular lobe.
  • Phong Normalization Factor derivation, 2009, Fabian Giesen.
    Demonstrates the derivation to obtain the normalization factor for the Phong and Blinn-Phong specular lobe.
  • The Blinn-Phong Normalization Zoo, 2011, Christian Schüler.
    Lists various normalization factors that exist for variants of Phong and Blinn-Phong.
    Also proposes a crude approximation for Cook-Torrance.
  • How Is The NDF Really Defined?, 2013, Nathan Reed.
    Explains conceptually what the Normal Distribution Function (NDF) is, and how this affects the region to integrate over for normalization.
  • Adopting a physically based shading model, 2011, Sébastien Lagarde et al.
    Starts by reminding a few normalization factors (Lambert, Phong and Blinn-Phong). Includes a quick paragraph on the factor to use to combine diffuse with specular.
  • How to properly combine the diffuse and specular terms?, 2016, CG Stack Exchange.
    A question I candidly asked on how to combine diffuse and specular so the energy lost to specular is taken into account in the diffuse term.
  • Designing Reflectance Models for New Consoles (slides), 2014, Yoshiharu Gotanda.
    The third section explains that a Fresnel term should be taken into account for the diffuse part, but also why this is problematic. This Fresnel term should take into account all microfacets, not only the perfect reflection ones that contribute to the specular component.
    This is an answer to my Stack Exchange question above.
  • PBR Diffuse Lighting for GGX+Smith Microsurfaces, 2017, Earl Hammon, Jr.
    Among the other topics it covers, this presentation shows the derivation to normalize a BRDF.
  • Physically Based Shading at DreamWorks Animation, 2017, Feng Xie and Jon Lanz.
    In the appendix of these course notes, the derivation to normalize their fabric BRDF is shown.

Wrapped diffuse

It is a common trick in video games to represent certain diffuse materials that have a lot of scattering with a custom diffuse that “wraps” around and brings light in the shadowed part. When PBR became popular, several people looked into how to make their wrapped diffuse PBR compliant.

Multiple scattering

A recently tackled problem is the energy loss due to ignoring multiple scattering. In many BRDF models, rays occluded by geometry are simply discarded. This tends to cause a noticeable darkening as roughness increases, visible in many of the charts showing material appearance for various roughnesses. However the trend is changing and this is why we see more and more references to the “furnace test”, which is a way to highlight energy loss.

  • Multiple-Scattering Microfacet BSDFs with the Smith Model, 2016, Eric Heitz et al.
    I haven’t read that paper except for the abstract, but the reception it received indicates that it’s an important publication. Recently, Morgan McGuire even said about it:

    “It is such a beautifully complete piece of work, a short, careful, and clear book on microfacets of the form that typically only arrives out of a complete Ph.D. thesis.”.

    If I understand correctly, they extended the Smith model to take multiple scattering into account, and compared their results with a simulation, by raytracing a surface at the micro-facet level.
    Károly Zsolnai-Fehér of Two Minute Papers did a video abstract of their paper.

  • A Multi-Faceted Exploration, part 1, part 2, part 3, part 4  2018-2019, Stephen Hill.
    This series of articles explores the feasibility of using in real-time rendering a model used by Sony Pictures Imageworks for offline rendering. The first part explains and illustrates what the problem is. The second part presents the solution from Heitz, and uses it as a ground truth reference, before presenting the Sony Pictures Imageworks solution and comparing the two. It then proposes an improvement of the latter. The third part gives a brief and clear reminder of the idea behind the split integral technique from UE4 and others, and uses it to propose a further improvement by precomputing a 2D LUT (instead of a 3D one). The fourth part details the precomputation step and shows the results in a WebGL demo.
    The series is not concluded yet, so I imagine one or more articles are coming.
  • Advances in Rendering, Graphics Research and Video Game Production (PDF version, video), 2019, Steve McAuley.
    This presentation shows the steps that were involved in implementing multiscattering BRDF and area lights for diffuse and specular, in FarCry, which uses a complex rendering engine that has to support a variety of combination of cases. It’s a reminder that such task can become more involved that expected. It’s also a case for academic papers that highlight their main insight, and have code available.

Area lights

I haven’t explored this topic yet, but I bookmarked some publications that spent time on the topic.

  • Lighting of KillZone Shadowfall, 2013, Michal Drobot.
    A part of the presentation is dedicated to area lights. It observes that point lights are inadequate for artists and that they tend to tweak roughness to compensate. It then briefly explains the technique, which consist in analytically integrating over the area light. Unfortunately the full derivation is not shown.
  • Real Shading in Unreal Engine 4 (slides), 2013, Brian Karis.
    A part covers the area lights. Like Drobot, Karis observes a tendency of artists to use roughness to compensate the small reflection highlight of point lights. The course notes list their requirements, some solutions that were considered (including Drobot’s) and why they were rejected. They then present a method based on a “representative point”, and how it applies to spherical lights and tube lights.
  • Real-Time Polygonal-Light Shading with Linearly Transformed Cosines, 2016, Eric Heitz et al.
    The current state of the art. This technique approximates physically based lighting from polygonal lights by transforming a cosine distribution (which is simpler to integrate) so it matches the BRDF properties. A demo with the code as well as a WebGL demo showing the result are provided.

Acknowledgements

Many thanks to Calvin Simpson, Dimitri Diakopoulos, Jeremy Cowles, Jonathan Stone, Julian Fong, Sébastien Lagarde, Stefan Werner, Yining Karl Li and the computer graphics community at large for your contributions and suggestions of material to read.

Long hiatus

Last week I was lucky enough to attend SIGGRAPH 2018, in Vancouver. My colleagues and I were presenting on a booth the work we had done, a VR story with a distinctive comic book look. I was also invited to participate to a panel session on demoscene, where I shared some lessons learned while making the 64k intro H – Immersion. The event brought a certain sense of conclusion to this work, aside from filling me with inspiration and motivation to try new things.

It has been a long time since I last posted anything here. For the last two years the majority of my spare time went into making that 64k intro. In fact the last post, “Intersection of a ray and a cone”, was related to it. I was implementing volumetric lighting for the underwater scenes, and wanted to resolve cones of light with ray tracing, before marching inside those cones. LLB and I have talked about the creation process in two making-of articles: “A dive into the making of Immersion”, and “Texturing in a 64kB intro”.

During that time, a lot of new things have happened in the computer graphics community. It has been difficult to keep track of everything. The last topic I started experimenting with is point cloud and mesh capture from photos; I might expend on it here in the future. I also want to experiment with DIY motion capture. Anyway, it’s time to resume posting here.

Intersection of a ray and a cone

Some time ago I needed to solve analytically the intersection of a ray and a cone. I was surprised to see that there are not that many resources available; there are some, but not nearly as many as on the intersection of a ray and a sphere for example. Add to it that they all use their own notation and that I lack math exercise, after a bit of browsing I decided I needed to write a proof by myself to get a good grasp of the result.

So here goes, the solution to the intersection of a ray and a cone, in vector notation.

  1. We define a ray with its origin $O$ and its direction as a unit vector $\hat{D}$.
    Any point $X$ on the ray at a signed distance $t$ from the origin of the ray verifies: $\vec{X} = \vec{O} + t\vec{D}$.
    When $t$ is positive $X$ is in the direction of the ray, and when $t$ is negative $X$ is in the opposite direction.
  2. We define a cone with its tip $C$, its axis as a unit vector $\hat{V}$ in the direction of increasing radius, and $\theta$ the half angle between the axis and the surface.
    Any point $X$ on the cone verifies: $(\vec{X} – \vec{C}) \cdot \vec{V} = \lVert \vec{X} – \vec{C} \rVert \cos\theta$
  3. Finally we define $P$ the intersection or the ray and the cone, and which we are interested in finding.

$P$ verifies both equations, so we can write:

$$
\left\{
\begin{array}{l}
\vec{P}=\vec{O} + t\vec{D} \\
\frac{ \vec{P} – \vec{C} }{\lVert \vec{P} – \vec{C} \rVert} \vec{V} = \cos\theta
\end{array}
\right.
$$

We can multiply the second equation by itself to work with it, then reorder things a bit.

$$
\left\{
\begin{array}{l}
\vec{P}=\vec{O} + t\vec{D} \\
\frac{ ((\vec{P} – \vec{C}) \cdot \vec{V})^2 }{ (\vec{P} – \vec{C}) \cdot (\vec{P} – \vec{C}) } = \cos^2\theta
\end{array}
\right.
$$

$$
\left\{
\begin{array}{l}
\vec{P}=\vec{O} + t\vec{D} \\
((\vec{P} – \vec{C}) \cdot \vec{V})^2 – (\vec{P} – \vec{C}) \cdot (\vec{P} – \vec{C}) \cos^2\theta = 0
\end{array}
\right.
$$

Remember the mouthful earlier about $\hat{V}$ being in the direction of increasing radius? By elevating $\cos\theta$ to square, we’re making negative values of $\cos$ positive: values of $\theta$ beyond 90° become indistinguishable from values below 90°. This has the side effect of turning it into the equation of not one, but two cones sharing the same axis, tip and angle, but in opposite directions. We’ll fix that later.

We replace $\vec{P}$ with $\vec{O} + t\vec{D}$ and work the equation until we get a good old quadratic function that we can solve.

$$
\require{cancel}
((\vec{O} + t\vec{D} – \vec{C})\cdot\vec{V})^2 – (\vec{O} + t\vec{D} – \vec{C}) \cdot (\vec{O} + t\vec{D} – \vec{C}) \cos^2\theta = 0
$$

$$
((t\vec{D} + \vec{CO})\cdot\vec{V})^2 – (t\vec{D} + \vec{CO}) \cdot (t\vec{D} + \vec{CO}) \cos^2\theta = 0
$$

$$
(t\vec{D}\cdot\vec{V} + \vec{CO}\cdot\vec{V})^2 – (t^2\cancel{\vec{D}\cdot\vec{D}} + 2t\vec{D}\cdot\vec{CO} + \vec{CO}\cdot\vec{CO}) \cos^2\theta = 0
$$

$$
(t^2(\vec{D}\cdot\vec{V})^2 + 2t(\vec{D}\cdot\vec{V})(\vec{CO}\cdot\vec{V}) + (\vec{CO}\cdot\vec{V})^2) – (t^2 + 2t\vec{D}\cdot\vec{CO} + \vec{CO}\cdot\vec{CO}) \cos^2\theta = 0
$$

$$
t^2(\vec{D}\cdot\vec{V})^2
+ 2t(\vec{D}\cdot\vec{V})(\vec{CO}\cdot\vec{V})
+ (\vec{CO}\cdot\vec{V})^2
– t^2\cos^2\theta
– 2t\vec{D}\cdot\vec{CO}\cos^2\theta
– \vec{CO}\cdot\vec{CO}\cos^2\theta
= 0
$$

Reorder a bit:

$$
t^2((\vec{D}\cdot\vec{V})^2 – \cos^2\theta)
+ 2t((\vec{D}\cdot\vec{V})(\vec{CO}\cdot\vec{V}) – \vec{D}\cdot\vec{CO}\cos^2\theta)
+ (\vec{CO}\cdot\vec{V})^2 – \vec{CO}\cdot\vec{CO}\cos^2\theta
= 0
$$

There we go, we have our $at^2 + bt + c = 0$ equation, with:

$$
\left\{
\begin{array}{l}
a = (\vec{D}\cdot\vec{V})^2 – \cos^2\theta \\
b = 2\Big((\vec{D}\cdot\vec{V})(\vec{CO}\cdot\vec{V}) – \vec{D}\cdot\vec{CO}\cos^2\theta\Big) \\
c = (\vec{CO}\cdot\vec{V})^2 – \vec{CO}\cdot\vec{CO}\cos^2\theta
\end{array}
\right.
$$

From there, you know the drill: calculate the determinant $\Delta = b^2 – 4ac$ then depending on its value:

  • If $\Delta < 0$, the ray is not intersecting the cone.
  • If $\Delta = 0$, the ray is intersecting the cone once at $t = \frac{-b}{2a}$.
  • If $\Delta > 0$, the ray is intersecting the cone twice, at $t_1 = \frac{-b – \sqrt{\Delta}}{2a}$ and $t_2 = \frac{-b + \sqrt{\Delta}}{2a}$.

But wait! We don’t have one cone but two, so we have to reject solutions that intersect with the shadow cone. $P$ must still verify $\frac{ \vec{P} – \vec{C} }{\lVert \vec{P} – \vec{C} \rVert} \vec{V} = \cos\theta$, or simply, if $\theta < 90°$: $(\vec{P} – \vec{C})\cdot\vec{V} > 0$.

Note that there is also the corner case of the ray tangent to the cone and having an infinity of solutions to consider. I’ve completely swept it under the rug since it doesn’t matter in the context I was, but if it does to you, you’ve been warned about it. Also remember to check the sign of $t$ to know whether $P$ is in the direction of the ray. You may need to determine which of $t_1$ or $t_2$ you want to use, which depends on your use case. For example is your ray origin inside or outside of the cone?


Now for a little sanity test, let’s consider the corner case $C=O$, where the ray origin is the tip of the cone (thanks Rubix for the suggestion!). We have $b=0$ and $c=0$ thus $\Delta=0$ and $t=\frac{-b}{2a}=0$ which is the expected result.

I also tried the cases $\theta=0$ and $\theta=\pi/2$, but expanding $\Delta$ proved too tedious to proceed to the end. So this is left as an exercise, as they say. :)


Finally, to demonstrate that the result is indeed correct, here is a glorious ray traced cone scene on ShaderToy:

I hope this can prove useful to others too.

Oh, and Happy New Year by the way!