# 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!

# The Chemical Brothers: “Wide Open”

Earlier this year at Tokyo Demo Fest 2016, we were honored with the presence of Solid Angle‘s founder, Marcos Fajardo, who did a long presentation of the Arnold renderer. Various examples of productions that involved Arnold were included, like captures from Iron Man or Elysium, but between those blockbusters one production in particular held my attention.

It was a long shot of a dancer turning limb after limb into a lattice body while performing. That film is in fact the music video of the song “Wide Open”, by the Chemical Brothers, and directed by Dom&Nic. It is a brilliant piece of technical and artistic work, that I can only recommend to watch.

Some of the details of the creation are shared in an article by the excellent fxguide, as well as in this interview for Solid Angle, like for example how they dealt with the challenges posed by a single long shot under varying natural light. Finally, there is this behind the scenes video from the studio, The Mill:

# Coordinate mapping between sphere, disc and square

This paper, published by Martin Lambers in the Journal of Computer Graphics Techniques, compares different mappings between sphere and disc, and between disc and square. It is worth noting that the source code is available on the publication page.

Abstract:
A variety of mappings between a sphere and a disc and between a disc and a square, as well as combinations of both, are used in computer graphics applications, resulting in mappings between spheres and squares. Many options exist for each type of mapping; to pick the right methods for a given application requires knowledge about the nature and magnitude of mapping distortions.

This paper provides an overview of forward and inverse mappings between a unit sphere, a unit disc, and a unit square. Quality measurements relevant for computer graphics applications are derived from tools used in the field of map projection, and a comparative analysis of the mapping methods is given.

# Series of articles on noise generation

I discovered last year these tutorials by Jasper Flick on how to make and use noise in Unity, and a couple of terrain and particle use examples. They present the difference between value noise and gradient noise, how Perlin noise and simplex noise work, and among others how to use curl noise to control the flow of particles.

The order information is presented is well thought, although the intention might not be clear at first. Don’t let the beginner’s tutorial tone (“You’ll learn to: create and fill a texture;”, etc.) turn you away, as the series do a great job at detailing the concepts and algorithms in a simple manner yet without cutting corners like so many articles on the topic do (when they’re not blatantly wrong and go ahead calling a blurred noise “Perlin noise”). I thought I had a pretty good grasp of gradient noise already, but reading it gave me an even better understanding.

Pushing particles around with Simplex curl noise.

While at it, other resources on the topic include Ken Perlin’s GDC 1999 talk and his two pages paper Improving noise explaining why use a 5th order polynomial for interpolation (a function I’ve sometimes seen called “smootherStep”).

# PVRTC2 compression quality

Christophe Riccio posted on his twitter feed some pictures comparing the quality of different texture compression formats, including the PowerVR’s native compression format, PVRTC2. In the light of his tests, it seems to me the new compression is a lot better than before (unfortunately they are not compared).

Last year at my work, in a context of trying to reduce loading time, memory consumption and application size, we gave a try at PVRTC and in our use case it was a clear no go. The quality was so badly impacted that the texture size we’d need for the artists to be happy was well beyond the weight of a PNG of equivalent quality. In the end we settled with WebP.

Here it is interesting to see that even at 2bpp, PVRTC2 seems to retain a lot of detail and texture. The edge tend to be muddy but this is still very good for the price.

# Various links on ray tracing

Here are some links related to ray tracing, and more specifically, path tracing.

## Some major publications:

• The rendering equation, SIGGRAPH 1986, James T. Kajiya. From the paper:

We present an integral equation which generalizes a variety of known rendering algorithms.
[…]
We mention that the idea behind the rendering equation is hardly new.
[…]
However, the form in which we present this equation is well suited for computer graphics, and we believe that this form has not appeared before.

• Bi-directional path tracing, Compugraphics 1993, Eric P. Lafortune and Yves D. Willems. From the paper:

The basic idea is that particles are shot at the same time from a selected light source and from the viewing point, in much the same way. All hit points on respective particle paths are then connected using shadow rays and the appropriate contributions are added to the flux of pixel  in question.

• Optimally Combining Sampling Techniques for Monte Carlo Rendering, SIGGRAPH 1995, Eric Veach and Leonidas J. Guibas. From the abstract:

We present a powerful alternative for constructing robust Monte Carlo estimators, by combining samples from several distributions in a way that is provably good.

• Metropolis Light Transport, SIGGRAPH 1997, Eric Veach and Leonidas J. Guibas. From the abstract:

To render an image, we generate a sequence of light transport paths by randomly mutating a single current path (e.g. adding a new vertex to the path).

• Robust Monte Carlo methods for light transport simulation, 1998, Erich Veach PhD thesis (432 pages pdf): it presents bidirectional path tracing, and introduces Metropolis Light Transport and Multiple Importance Sampling. From the abstract:

Our statistical contributions include a new technique called multiple importance sampling, which can greatly increase the robustness of Monte Carlo integration. It uses more than one sampling technique to evaluate an integral, and then combines these samples in a way that is provably close to optimal. This leads to estimators that have low variance for a broad class of integrands. We also describe a new variance reduction technique called efficiency-optimized Russian roulette.

[…]

The second algorithm we describe is Metropolis light transport, inspired by the Metropolis sampling method from computational physics. Paths are generated by following a random walk through path space, such that the probability density of visiting each path is proportional to the contribution it makes to the ideal image.

## Other:

On a slightly different topic, fxguide had a great series of articles on the state of rendering in the film industry, which I previously mentioned.