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👉 Write a new method for interact that takes a photon and a hit of type Intersection{Sphere}, that implements refraction. It returns a new Photon positioned at the hit point, with the refracted velocity and the new index of refraction.
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Homework 7: Raytracing in 2D
18.S191, fall 2020
This notebook contains built-in, live answer checks! In some exercises you will see a coloured box, which runs a test case on your code, and provides feedback based on the result. Simply edit the code, run it, and the check runs again.
For MIT students: there will also be some additional (secret) test cases that will be run as part of the grading process, and we will look at your notebook and write comments.
Feel free to ask questions!
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Exercise 3.1
Just like with the Wall, our first step is to be able to find the intersection point of a ray of light and a sphere.
This one is a bit more challenging than the intersction with the wall, in particular because there are 3 potential outcomes of a line interacting with a sphere:
You see that Wall is included in the type. This will be very useful later, when we want to do something different depending on the intersected object (wall, sphere, etc.) using multiple dispatch. We can write one method for ::Intersection{Sphere}, and one for ::Intersection{Wall}.
We are now able to find the Intersection of a single photon with a single wall (or detect a Miss). Great! To make our simulation more interesting, we will combine multiple walls into a single scene.
We will write a function that finds the location where a photon hits the wall. Instead of moving the photon forward in small timesteps until we reach the wall, we will compute the intersection directly, making use of the fact that the wall is a geometrically simple object.
Our function will return one of two possible types: a Miss or a Intersection. We define these types below, and both definitions need some elaboration.
In Exercise 3 and 4, we will add a Sphere type, and our scene will consist of Walls (mirrors) and Spheres (lenses). But before we move on, let's review what we have done so far.
Our main character is a Photon, which bounces around a scene made up of Walls.
Using intersection(photon, wall::Wall) we can find either an Intersection (containing the wall, the distance and the point) or a Miss.
Our scene is just a Vector or objects, and we compute the intersection between the photon and every object.
By adding Base.isless methods we have told Julia how to compare hit/miss results, and we get the closest one using minimum(all_intersections).
We wrote a function interact(photon, hit::Intersection{Wall}) that returns a new photon after interacting with a wall collision.
We repeat these four steps to trace a ray through the scene.
In the next two exercises we will reuse some of the functionality that we have already written, using multiple dispatch! For example, we add a method intersection(photon, sphere::Sphere), and steps 2 and 3 magically also work with spheres!
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In our simulations, we will enclose our scene in a box of four walls, to make sure that no rays can escape the scene. We have written this box (i.e. vector of walls) below, but we are still missing the roof.
For this, we will start with refraction from the surface of water and then move on to a spherical lens.
So, how does refraction work? Well, every time light enters a new medium that is more dense than air, it will bend towards the normal to the surface, like so:
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Exercise 2.1 - reflect
For this one, we need to implement a reflection function. This one is way easier than refraction. All we need to do is find how much of the light is moving in the direction of the surface's normal and subtract that twice.
$$\ell_2 = \ell_1 - 2(\ell_1\cdot \hat n)\hat n$$
Where $\ell_1$ and $\ell_2$ are the photon directions before and after the reflection off a surface with normal $\hat{n}$. Let's write that in code:
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Exercise 2.2 - step
Our event-driven simulation is a stepping method, but instead of taking small steps in time, we take large steps from one collision event to the next.
👉 Write a function interact that takes a photon and a hit::Intersection{Wall} and returns a new Photon at the next step. The new photon is located at the hit point, its direction is reflected off the wall's normal and the photon.ior is reused.
👉 Modify the definition of box_scene to be a vector of 4 walls, instead of 3. The fourth wall should be positioned at [0,10], and point downwards.
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In the next exercise, we will find the intersection of a ray of light and a wall. To represent light, we create a struct called Photon, holding the position and travel direction of a single particle of light. We also include the index of refraction of the medium it is currently traveling in, we will use this later.
To start, let's look at the intersection of a point and a sphere. So long as the relative distance between the photon and the sphere's center satisfies the sphere equation, we can be considered inside of the sphere. More specifically, we are inside the sphere if:
$$(x_s-x_p)^2+(y_s-y_p) < r^2.$$
where the $s$ and $p$ subscripts represent the sphere and photon, respectively. We know we are on the sphere if
where $\mathbf{R}$ and $\mathbf{S}$ are the $x$, $y$, and $z$ location of the photon and sphere, respectively.
Returning to the timestepping example from above, we know that our ray is moving forward with time such that $\mathbf{R} = \mathbf{R}_0 + v dt = \mathbf{R}_0 + \ell t$. We now need to ask ourselves if there is any time when our ray interacts with the sphere. Plugging this in to the dot product from above, we get
$$(\mathbf{R}_0 + \ell t - \mathbf{S})\cdot(\mathbf{R}_0 + \ell t - \mathbf{S}) = r^2$$
To solve this for $t$, we first need to reorder everything into the form of a polynomial, such that:
This can be solved with the good ol' fashioned quadratic equation:
$$\frac{-b\pm\sqrt{b^2-4ac}}{2a},$$
where $a = \ell\cdot\ell$, $b = 2\ell\cdot(\mathbf{R}_0-\mathbf{S})$, and $c=(\mathbf{R}_0-\mathbf{S})\cdot(\mathbf{R}_0-\mathbf{S}) - r^2$
If the quadratic equation returns no roots, there is no intersection. If it returns 1 root, the ray just barely hits the edge of the sphere. If it returns 2 roots, it goes right through!
The easiest way to check this is by looking at the discriminant $d = b^2-4ac$.
$$\text{Number of roots} = \left\{
\begin{align}
&0, \qquad \text{if } d < 0 \\
&1, \qquad \text{if } d = 0 \\
&2, \qquad \text{if } d > 0 \\
\end{align}
\right.$$
In the case that there are 2 roots, the second root corresponds to when the ray would interact with the far edge of the sphere if there were no refraction or reflection!; therefore, we only care about returning the closest point.
With all this said, we are ready to write some code.
👉 Write a new method intersection that takes a Photon and a Sphere, and returns either a Miss or an Intersection, using the method described above. Go back to Exercise 1.3 where we defined the first method, and see how we adapt it to a sphere.
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👉 Change the definition of test_sphere to test different situations:
Hit the circle
Miss the circle
Start inside the cricle (you should hit the exit boundary)
To test your code, modify the definition of test_lens_photon and test_lens below.
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👉 Recreate the spherical aberration figure from the lecture (around the end of the video), and make the index of refraction interactive using a Slider. Or make something else!
So which of these five results should we use to determine what the photon does next? It should be the closest intersection.
Because we used two different types for hits and misses, we can express this in a charming way. We define what it means for one to be better than the other:
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Hint
You can use ray.ior == 1.0 to check whether this is a ray entering or leaving the sphere.
Now that we know how to bounce light around mirrors, we want to simulate a spherical lens to make things more interesting. Let's define a Sphere.
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Now we're going to make a bold claim: All walls in this simulation are mirrors. This is just for simplicity so we don't need to worry about rays stopping at the boundaries.
We are already able to find the intersection of a light ray with a mirror, but we still need to tell our friendly computer what a reflection is.
Here, $n$ is the index of refraction, $v$ is the speed (not velocity (sorry for the notation!)), and $\theta$ is the angle with respect to the surface normal. Any variables with an subscript of 1 are in the outer medium (air), and any variables with a subscript 2 are in the inner medium (water).
This means that we can find the angle of the new ray of light as
The problem is that $\sin$ is slow, so we typically want to rewrite this in terms of vector operations. This means that we want to rewrite everything to be in terms of dot products, but because $A\cdot B = |A||B|cos(\theta)$, we really want to rewrite everything in terms of cosines first. So, using the fact that $\sin(\theta)^2 + \cos(\theta)^2 = 1$, we can rewrite the above equation to be:
Finally, we know that the new light direction should be the same as the old one, but shifted towards (or away) from the normal according to the new refractive index. In particular:
Now to move on to lenses. Like in lecture, we will focus exclusively on spherical lenses. Ultimately, there isn't a big difference between a lens and a spherical drop of water. It just has a slightly different refractive index and it's normal is defined slightly differently.
is a parametric type. The first field (object) is of type T, and T is a subtype of Object. Have a look at the definition above, and take note of how we write such statements in Julia syntax.
We also could have used Object directly as the type for the field object, but what's special about parametric types is that T becomes "part of the type". Let's have a look at an example:
When we shoot a photon at the scene, we compute the intersections between the photon and every object in the scene. Click on the vector below to see all elements:
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Wall geometry
So, how do we find the location where it hits the wall? Well, because our walls are infinitely long, we are essentially trying to find the point at which 2 lines intersect.
To do this, we can combine a few dot products: one to find how far away we are, and another to scale that distance. Mathematically, it would look like:
where $p$ is the position, $\hat \ell$ is the direction of the light, and $\hat n$ is the normal vector for the wall. subscripts $i$, $r$, and $w$ represent the intersection point, ray, and wall respectively. The result is $D$, the amount that the photon needs to travel until it hits the wall.
👉 Write a function intersection_distance that implements this formula, and returns $D$. You can use dot(a,b) to compute the vector dot product $a \cdot b$.
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And we can now use all of Julia's built in functions to work with a vector of hit/miss results. For example, we can sort it:
As discussed the lecture, event-driven simulations are the traditional method used for raytracing. Here, we look for any objects in our path and analytically determine how far away they are. From there, we take one big timestep all the way to the surface boundary, calculate refraction or reflection to see what direction we are moving in, and then seek out any other object we could potentially run into.
So let's start simple with determining when a ray of light could intersect with a wall.
Exercise 1.1 - what is a wall?
To start, let's create the concept of a wall. For our purposes, walls will be infinitely long, so we only need to create an object that has a position and a normal vector at that position:
We need a helper functions to find the normal of the sphere's surface at any position. Remember that the normal will always be pointing perpendicularly from the surface of the sphere. This means that no matter what point you are at, the normal will just be a normalized vector of your current location minus the sphere's position:
👉 Write a function trace that takes an initial Photon, a vector of Objects and N, the number of steps to make. Return a vector of Photons. Try to use accumulate.
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By defining a method for interact that takes a sphere intersection, we are now able to use the machinery developed in Exercise 2 to simulate a scene with both lenses and mirrors. Let's try it out!
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👉 Verify that the function reflect works by writing a simple test case:
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Exercise 1.3 - hitting the wall
👉 Write a function intersection that takes a photon and a wall, and returns either a Miss or an Intersection, based on the result of intersection_distance(photon, wall)$= D$.
If $D$ is positive, then the photon will hit the wall, and we should return an Intersection. We already have the intersected object, and we have $D$, our intersection distance. To find the intersection point, we use the photon's position and velocity.
If $D$ is negative (or zero), then the wall is behind the photon - we should return a Miss.
Floating points
We are using floating points (Float64) to store positions, distances, etc., which means that we need to account for small errors. Like in the lecture, we will not check for D > 0, but D > ϵ with ϵ = 1e-3.
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cell_id$1df82c20-19c4-11eb-0959-8543a0d5630ddownstream_cells_mapupstream_cells_map@md_strstudent$1e202680-19c4-11eb-29a7-99061b886b3cgetindex$337918f4-194f-11eb-0b45-b13fef3b23bfprecedence_heuristic cell_id$337918f4-194f-11eb-0b45-b13fef3b23bfdownstream_cells_mapupstream_cells_mapPlutoUI$c3e52bf2-ca9a-11ea-13aa-03a4335f2906PlutoUI.Resource$70b8401e-1c7e-11eb-16b2-d54d8f66d71aprecedence_heuristic cell_id$70b8401e-1c7e-11eb-16b2-d54d8f66d71adownstream_cells_mapupstream_cells_map@md_strgetindex$eb35ac4a-1acc-11eb-0729-ff85c8406c45precedence_heuristic cell_id$eb35ac4a-1acc-11eb-0729-ff85c8406c45downstream_cells_mapaberration_viz_ior$f83da7f8-1acc-11eb-02d7-f33ffe518531upstream_cells_mapCoreBase:PlutoRunner.create_bondPlutoRunnerCore.applicable@bindBase.getSlider$0b03316c-1c80-11eb-347c-1b5c9a0ae379precedence_heuristic cell_id$0b03316c-1c80-11eb-347c-1b5c9a0ae379downstream_cells_maptest_new_photon$fb70cc0c-1c7f-11eb-31b5-87b168a66e19upstream_cells_maptest_closest$b8cd4112-1c7c-11eb-3b2d-29170ad9beb5philip$55187168-1c78-11eb-1182-ab4336b577a4interact$e70b9e24-1a07-11eb-13db-b95c07880893$e1cb1622-1a0c-11eb-224c-559af7b90f49$ec3ed530-19c3-11eb-10bb-a55e77550d1fprecedence_heuristic cell_id$ec3ed530-19c3-11eb-10bb-a55e77550d1fdownstream_cells_mapalmost$038d5e88-1ac7-11eb-2020-a9d7e19feebcupstream_cells_mapMarkdown.AdmonitionMarkdown.MDMarkdown$42d65f56-1aca-11eb-1079-e32f85554349precedence_heuristic cell_id$42d65f56-1aca-11eb-1079-e32f85554349downstream_cells_mapupstream_cells_map@md_strBase.getindexBaseBase.Docs.HTML@html_str$1b0c0e4c-1c73-11eb-225d-23c731455755precedence_heuristic cell_id$1b0c0e4c-1c73-11eb-225d-23c731455755downstream_cells_mapall_intersections$e055262c-1c73-11eb-14de-2f537a19b012$c3090e4a-1a09-11eb-0f32-d3bbfd9992e0$63ef21c6-1c7a-11eb-2f3c-c5ac16bc289fupstream_cells_mapex_1_scene$2158a356-1a05-11eb-3f5b-4dfa810fc602philip$55187168-1c78-11eb-1182-ab4336b577a4intersection$aa19faa4-1941-11eb-2b61-9b78aaf42876$885ac814-1953-11eb-30d9-85dcd198a1d8$e45e1d36-1a12-11eb-2720-294c4be6e9fdprecedence_heuristic cell_id$e45e1d36-1a12-11eb-2720-294c4be6e9fddownstream_cells_mapupstream_cells_maptest_wall$0906b340-19d3-11eb-112c-e568f69deb5dplot_scene$eff9329e-1a05-11eb-261f-734127d36750$5501a700-19ec-11eb-0ded-53e41f7f821aprecedence_heuristic cell_id$5501a700-19ec-11eb-0ded-53e41f7f821adownstream_cells_mapupstream_cells_mapex_1_scene$2158a356-1a05-11eb-3f5b-4dfa810fc602plot_photon_arrow!$eabca8ce-1c73-11eb-26ad-271f6eba889bphilip$55187168-1c78-11eb-1182-ab4336b577a4plot_scene$eff9329e-1a05-11eb-261f-734127d36750$b3ab93d2-1a0b-11eb-0f5a-cdca19af3d89precedence_heuristic cell_id$b3ab93d2-1a0b-11eb-0f5a-cdca19af3d89downstream_cells_mapex_3_scene$83aa9cea-1a0c-11eb-281d-699665da2b4fupstream_cells_mapbox_scene$d257a728-1a04-11eb-281d-bde30644f5f5test_sphere$af5c6bea-1c9c-11eb-35ae-250337e4fc86$83aa9cea-1a0c-11eb-281d-699665da2b4fprecedence_heuristic cell_id$83aa9cea-1a0c-11eb-281d-699665da2b4fdownstream_cells_mapupstream_cells_mapIntersection$8018fbf0-1a05-11eb-3032-95aae07ca78f:plot_photon_arrow!$eabca8ce-1c73-11eb-26ad-271f6eba889b|>isaex_3_scene$b3ab93d2-1a0b-11eb-0f5a-cdca19af3d89as_svgphilip$55187168-1c78-11eb-1182-ab4336b577a4plot_scene$eff9329e-1a05-11eb-261f-734127d36750sphere_intersection$251f0262-1a0c-11eb-39a3-09be67091dc8scatter!$900d6622-1a08-11eb-1475-bfadc2aac749precedence_heuristic cell_id$900d6622-1a08-11eb-1475-bfadc2aac749downstream_cells_maptrace$1ee0787e-1a08-11eb-233b-43a654f70117upstream_cells_map:accumulatestep_ray$76ef6e46-1a06-11eb-03e3-9f40a86dc9aaObject$d851a202-1ca0-11eb-3da0-51fcb656783cPhoton$24b0d4ba-192c-11eb-0f66-e77b544b0510Vector$14dc73d2-1a0d-11eb-1a3c-0f793e74da9bprecedence_heuristic cell_id$14dc73d2-1a0d-11eb-1a3c-0f793e74da9bdownstream_cells_maprefract$e1cb1622-1a0c-11eb-224c-559af7b90f49upstream_cells_mapnormalizesqrtislessVector ϵ
point = photon.p + D * photon.l
Intersection(wall, D, point)
else
Miss()
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$d851a202-1ca0-11eb-3da0-51fcb656783ccell_id$d851a202-1ca0-11eb-3da0-51fcb656783ccodeabstract type Object endmetadatashow_logsèdisabled®skip_as_script«code_folded$392c25b8-1add-11eb-225d-49cfca27bef4cell_id$392c25b8-1add-11eb-225d-49cfca27bef4codemd"""
👉 Write a new method for `interact` that takes a `photon` and a `hit` of type `Intersection{Sphere}`, that implements refraction. It returns a new `Photon` positioned at the hit point, with the refracted velocity and the new index of refraction.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$5895d9ae-1c9e-11eb-2f4e-671f2a7a0150cell_id$5895d9ae-1c9e-11eb-2f4e-671f2a7a0150code,test_lens = Sphere(
[5, -1.5],
3,
1.5,
)metadatashow_logsèdisabled®skip_as_script«code_folded$83acf10e-1c9e-11eb-3426-bb28e7bc6c79cell_id$83acf10e-1c9e-11eb-3426-bb28e7bc6c79codeXlet
scene = [test_lens, box_scene...]
N = 3
p = plot_scene(scene, legend=false, xlim=(-11,11), ylim=(-11,11))
path = accumulate(1:N; init=test_lens_photon) do old_photon, i
step_ray(old_photon, scene)
end
line = [test_lens_photon.p, [r.p for r in path]...]
plot!(p, first.(line), last.(line), lw=5, color=:red)
p
end |> as_svgmetadatashow_logsèdisabled®skip_as_script«code_folded$2a2b7284-1ade-11eb-3b71-d17fe2ca638acell_id$2a2b7284-1ade-11eb-3b71-d17fe2ca638acodeXplot_scene([example_sphere], size=(400,200), legend=false, xlim=(-15,15), ylim=(-10,10))metadatashow_logsèdisabled®skip_as_script«code_folded$bc10541e-1ac3-11eb-0b5f-916922f1a8e8cell_id$bc10541e-1ac3-11eb-0b5f-916922f1a8e8codetypeof(test_intersection_1)metadatashow_logsèdisabled®skip_as_script«code_folded$f83da7f8-1acc-11eb-02d7-f33ffe518531cell_id$f83da7f8-1acc-11eb-02d7-f33ffe518531code"aberration_viz(aberration_viz_ior)metadatashow_logsèdisabled®skip_as_script«code_folded$76ef6e46-1a06-11eb-03e3-9f40a86dc9aacell_id$76ef6e46-1a06-11eb-03e3-9f40a86dc9aacode}function step_ray(photon::Photon, objects::Vector{<:Object})
hit = closest_hit(photon, objects)
interact(photon, hit)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$c492a1f8-1a0c-11eb-2c38-5921c39cf5f8cell_id$c492a1f8-1a0c-11eb-2c38-5921c39cf5f8code/@bind sphere_test_ray_N Slider(1:30; default=4)metadatashow_logsèdisabled®skip_as_script«code_folded$eabca8ce-1c73-11eb-26ad-271f6eba889bcell_id$eabca8ce-1c73-11eb-26ad-271f6eba889bcode=function plot_photon_arrow!(p, photon::Photon, length=2; kwargs...)
line = [photon.p, photon.p .+ length*photon.l]
plot!(p, first.(line), last.(line); lw=2, arrow=true, color=:darkred, kwargs..., label=nothing)
scatter!(p, photon.p[1:1], photon.p[2:2]; color=:darkred, markersize=3, label=nothing, kwargs...)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$24b0d4ba-192c-11eb-0f66-e77b544b0510cell_id$24b0d4ba-192c-11eb-0f66-e77b544b0510codeُstruct Photon
"Position vector"
p::Vector{Float64}
"Direction vector"
l::Vector{Float64}
"Current Index of Refraction"
ior::Real
endmetadatashow_logsèdisabled®skip_as_script«code_folded$1e109620-19c4-11eb-013e-1bc95c14c2bacell_id$1e109620-19c4-11eb-013e-1bc95c14c2bacodemd"""
# **Homework 7**: _Raytracing in 2D_
`18.S191`, fall 2020
This notebook contains _built-in, live answer checks_! In some exercises you will see a coloured box, which runs a test case on your code, and provides feedback based on the result. Simply edit the code, run it, and the check runs again.
_For MIT students:_ there will also be some additional (secret) test cases that will be run as part of the grading process, and we will look at your notebook and write comments.
Feel free to ask questions!
"""metadatashow_logsèdisabled®skip_as_script«code_folded$af5c6bea-1c9c-11eb-35ae-250337e4fc86cell_id$af5c6bea-1c9c-11eb-35ae-250337e4fc86code+test_sphere = Sphere(
[7, -6],
2,
1.5,
)metadatashow_logsèdisabled®skip_as_script«code_folded$6fdf613c-193f-11eb-0029-957541d2ed4dcell_id$6fdf613c-193f-11eb-0029-957541d2ed4dcodeUfunction sphere_normal_at(p::Vector{Float64}, s::Sphere)
normalize(p - s.center)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$ec57b460-19c3-11eb-2142-07cf28dcf02bcell_id$ec57b460-19c3-11eb-2142-07cf28dcf02bcodeفkeep_working(text=md"The answer is not quite right.") = Markdown.MD(Markdown.Admonition("danger", "Keep working on it!", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$e2a8d1d6-1add-11eb-0da1-cda1492a950ccell_id$e2a8d1d6-1add-11eb-0da1-cda1492a950ccodevmd"
#### Exercise 3.1
Just like with the `Wall`, our first step is to be able to find the intersection point of a ray of light and a sphere.
This one is a bit more challenging than the intersction with the wall, in particular because there are 3 potential outcomes of a line interacting with a sphere:
- No intersection
- 1 intersection
- 2 intersections
As shown below:
"metadatashow_logsèdisabled®skip_as_script«code_folded$2c6defd0-1ca1-11eb-17db-d5cb498f3265cell_id$2c6defd0-1ca1-11eb-17db-d5cb498f3265codeV# function interact(photon::Photon, hit::Intersection{Wall})
# return missing
# endmetadatashow_logsèdisabled®skip_as_script«code_folded$3f727a2c-1c80-11eb-3608-e55ccb9786d9cell_id$3f727a2c-1c80-11eb-3608-e55ccb9786d9codeْmd"""
For convenience, we define a function `step_ray` that combines these two actions: it finds the closest hit and computes the interaction.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$d39f149e-1ac3-11eb-39a2-41c2030d7d49cell_id$d39f149e-1ac3-11eb-39a2-41c2030d7d49code/md"""
You see that `Wall` is **included in the type**. This will be very useful later, when we want to do something different _depending on the intersected object_ (wall, sphere, etc.) using multiple dispatch. We can write one method for `::Intersection{Sphere}`, and one for `::Intersection{Wall}`.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$d70380a4-1ad0-11eb-1184-f7e9b84a83adcell_id$d70380a4-1ad0-11eb-1184-f7e9b84a83adcodemd"""
$(html"
")
#### Exercise 1.4 - _which wall?_
We are now able to find the `Intersection` of a single photon with a single wall (or detect a `Miss`). Great! To make our simulation more interesting, we will combine **multiple walls** into a single scene.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$aa43ef1c-1941-11eb-04de-552719a08da0cell_id$aa43ef1c-1941-11eb-04de-552719a08da0codemd"""
$(html"
")
#### Exercise 1.2 - _how far is the wall?_
We will write a function that finds the location where a photon hits the wall. Instead of moving the photon forward in small timesteps until we reach the wall, we will compute the intersection directly, making use of the fact that the wall is a geometrically simple object.
Our function will return one of two possible types: a `Miss` or a `Intersection`. We define these types below, and both definitions need some elaboration.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$7478330a-1c81-11eb-2f9f-099f1111032ccell_id$7478330a-1c81-11eb-2f9f-099f1111032ccodemd"""
#### Recap
In Exercise 3 and 4, we will add a `Sphere` type, and our scene will consist of `Wall`s (mirrors) and `Sphere`s (lenses). But before we move on, let's review what we have done so far.
Our main character is a `Photon`, which bounces around a scene made up of `Wall`s.
1. Using `intersection(photon, wall::Wall)` we can find either an `Intersection` (containing the `wall`, the `distance` and the `point`) or a `Miss`.
2. Our scene is just a `Vector` or objects, and we compute the intersection between the photon and every object.
3. By adding `Base.isless` methods we have told Julia how to compare hit/miss results, and we get the closest one using `minimum(all_intersections)`.
4. We wrote a function `interact(photon, hit::Intersection{Wall})` that returns a new photon after interacting with a wall collision.
We repeat these four steps to trace a ray through the scene.
---
In the next two exercises we will reuse some of the functionality that we have already written, using multiple dispatch! For example, we add a method `intersection(photon, sphere::Sphere)`, and steps 2 and 3 magically also work with spheres!
"""
# We have a type `Photon` and a type `Wall`, and umetadatashow_logsèdisabled®skip_as_script«code_folded$bff04784-1acc-11eb-36c2-9335a58be23acell_id$bff04784-1acc-11eb-36c2-9335a58be23acodefunction aberration_viz(ior)
s = Sphere([5,0], 1, ior)
aberration_scene = [s, box_scene...]
p = plot_scene(aberration_scene, legend=false, xlim=(3,8), ylim=(-1.5,1.5))
for y in LinRange(-0.9, 0.9, 8)
start = Photon([0,y], [1,0], 1.0)
path = accumulate(1:3; init=start) do old_photon, i
step_ray(old_photon, aberration_scene)
end
line = [start.p, [r.p for r in path]...]
plot!(p, first.(line), last.(line), lw=2, color=:darkred)
end
p |> as_svg
endmetadatashow_logsèdisabled®skip_as_script«code_folded$c3090e4a-1a09-11eb-0f32-d3bbfd9992e0cell_id$c3090e4a-1a09-11eb-0f32-d3bbfd9992e0codesort(all_intersections)metadatashow_logsèdisabled®skip_as_script«code_folded$5f551588-1ac4-11eb-1f86-197442f1ef1dcell_id$5f551588-1ac4-11eb-1f86-197442f1ef1dcodemd"""
In our simulations, we will enclose our scene in a box of **four walls**, to make sure that no rays can escape the scene. We have written this box (i.e. vector of walls) below, but we are still missing the roof.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$584ce620-1935-11eb-177a-f75d9ad8a399cell_id$584ce620-1935-11eb-177a-f75d9ad8a399codemd"""
$(html"
")
## **Exercise 4:** _Lenses_
For this, we will start with refraction from the surface of water and then move on to a spherical lens.
So, how does refraction work? Well, every time light enters a new medium that is more dense than air, it will bend towards the normal to the surface, like so:
$(RemoteResource("https://upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Snells_law2.svg/800px-Snells_law2.svg.png", :width=>200, :style=>"display: block; margin: auto;"))
"""metadatashow_logsèdisabled®skip_as_script«code_folded$c00eb0a6-cab2-11ea-3887-070ebd8d56e2cell_id$c00eb0a6-cab2-11ea-3887-070ebd8d56e2codeemd"
#### Spherical aberration
Now we can put it all together into an image of spherical aberration!
"metadatashow_logsèdisabled®skip_as_script«code_folded$7f286ccc-1c75-11eb-1270-95a87840b300cell_id$7f286ccc-1c75-11eb-1270-95a87840b300code3@bind dizzy_angle Slider(0:0.0001:2π, default=2.2)metadatashow_logsèdisabled®skip_as_script«code_folded$0787f130-1aca-11eb-24b4-2ff2ddd0bc48cell_id$0787f130-1aca-11eb-24b4-2ff2ddd0bc48codelet
p = Photon([5,0], [1,0], 1.0)
w = Wall([10,10], normalize([-1,-1]))
result = intersection_distance(p, w)
if result isa Missing
still_missing()
elseif !(result isa Real)
keep_working(md"You need to return a number.")
else
if abs(result - (20 - 5)) > 0.1
if abs(-result - (20 - 5)) > 0.1
keep_working(md"The returned distance is not correct.")
else
keep_working(md"Did you forget the minus sign?")
end
else
correct()
end
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$ec5d59b0-19c3-11eb-0206-cbd1a5415c28cell_id$ec5d59b0-19c3-11eb-0206-cbd1a5415c28codeyays = [md"Fantastic!", md"Splendid!", md"Great!", md"Yay ❤", md"Great! 🎉", md"Well done!", md"Keep it up!", md"Good job!", md"Awesome!", md"You got the right answer!", md"Let's move on to the next section."]metadatashow_logsèdisabled®skip_as_script«code_folded$f76ab794-1ac9-11eb-26e3-b9d0baa05d49cell_id$f76ab794-1ac9-11eb-26e3-b9d0baa05d49codeوfunction intersection_distance(photon::Photon, wall::Wall)
-dot(photon.p - wall.position, wall.normal) / dot(photon.l, wall.normal)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$e1cb1622-1a0c-11eb-224c-559af7b90f49cell_id$e1cb1622-1a0c-11eb-224c-559af7b90f49codefunction interact(photon::Photon, hit::Intersection{Sphere})
old_ior = photon.ior
new_ior = if photon.ior == 1.0
hit.object.ior
else
1.0
end
normal = sphere_normal_at(hit.point, hit.object)
Photon(hit.point, refract(photon.l, normal, old_ior, new_ior), new_ior)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$dad5acfa-194c-11eb-27f9-01f40342a681cell_id$dad5acfa-194c-11eb-27f9-01f40342a681codemd"
#### Exercise 2.1 - _reflect_
For this one, we need to implement a reflection function. This one is way easier than refraction. All we need to do is find how much of the light is moving in the direction of the surface's normal and subtract that twice.
$\ell_2 = \ell_1 - 2(\ell_1\cdot \hat n)\hat n$
Where $\ell_1$ and $\ell_2$ are the photon directions before and after the reflection off a surface with normal ``\hat{n}``. Let's write that in code:
"metadatashow_logsèdisabled®skip_as_script«code_folded$3cd36ac0-1a09-11eb-1818-75b36e67594acell_id$3cd36ac0-1a09-11eb-1818-75b36e67594acode/@bind mirror_test_ray_N Slider(1:30; default=4)metadatashow_logsèdisabled®skip_as_script«code_folded$55187168-1c78-11eb-1182-ab4336b577a4cell_id$55187168-1c78-11eb-1182-ab4336b577a4code0philip = Photon([3, 0], normalize([.5,-1]), 1.0)metadatashow_logsèdisabled®skip_as_script«code_folded$b6614d80-194b-11eb-1edb-dba3c29672f8cell_id$b6614d80-194b-11eb-1edb-dba3c29672f8codemd"""
#### Exercise 2.2 - _step_
Our event-driven simulation is a stepping method, but instead of taking small steps in time, we take large steps from one collision event to the next.
👉 Write a function `interact` that takes a photon and a `hit::Intersection{Wall}` and returns a new `Photon` at the next step. The new photon is located at the hit point, its direction is reflected off the wall's normal and the `photon.ior` is reused.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$e055262c-1c73-11eb-14de-2f537a19b012cell_id$e055262c-1c73-11eb-14de-2f537a19b012codelet
p = plot_scene(ex_1_scene)
plot_photon_arrow!(p, philip, 4; label="Philip")
for (i,hit) in enumerate(all_intersections)
if hit isa Intersection
scatter!(p, hit.point[1:1], hit.point[2:2], label="intersection $i")
end
end
p |> as_svg
endmetadatashow_logsèdisabled®skip_as_script«code_folded$038d5e88-1ac7-11eb-2020-a9d7e19feebccell_id$038d5e88-1ac7-11eb-2020-a9d7e19feebccodelet
p = Photon([5,0], [1,0], 1.0)
w = Wall([10,10], normalize([-1,-1]))
result = intersection(p, w)
if result isa Missing
still_missing()
elseif !(result isa Miss || result isa Intersection)
keep_working(md"You need to return a `Miss` or a `Intersection`.")
else
if result isa Miss
keep_working(md"You returned a `Miss` for a photon that hit the wall.")
else
if abs(result.distance - (20 - 5)) > 0.1
keep_working(md"The returned distance is not correct.")
else
p = Photon([5,0], [-1,0], 1.0)
result = intersection(p, w)
if !(result isa Miss)
almost(md"What should happen when ``D < 0``?")
else
p = Photon([10,10], [1,0], 1.0)
w = Wall([10,10], normalize([-1,-1]))
result = intersection(p, w)
if result isa Miss
correct()
else
almost(md"Remember to use ``\epsilon``.")
end
end
end
end
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$1e202680-19c4-11eb-29a7-99061b886b3ccell_id$1e202680-19c4-11eb-29a7-99061b886b3ccode# edit the code below to set your name and kerberos ID (i.e. email without @mit.edu)
student = (name = "Jazzy Doe", kerberos_id = "jazz")
# you might need to wait until all other cells in this notebook have completed running.
# scroll around the page to see what's upmetadatashow_logsèdisabled®skip_as_script«code_folded$293776f8-1ac4-11eb-21db-9d023c09e89fcell_id$293776f8-1ac4-11eb-21db-9d023c09e89fcode٣md"""
👉 Modify the definition of `box_scene` to be a vector of 4 walls, instead of 3. The fourth wall should be positioned at `[0,10]`, and point downwards.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$6de1bafc-1a01-11eb-3d67-c9d9b6c3cea8cell_id$6de1bafc-1a01-11eb-3d67-c9d9b6c3cea8codeFfunction plot_object!(p, wall::Wall)
# old_xlims = xlims(p)
# old_ylims = ylims(p)
adjacent = [wall.normal[2], -wall.normal[1]]
a = wall.position + adjacent * 20
b = wall.position - adjacent * 20
line = [a, b]
plot!(p, first.(line), last.(line), label="Wall")
# xlims!(p, old_xlims)
# xlims!(p, old_xlims)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$e5ed6098-1c70-11eb-0b58-31d1830b9a10cell_id$e5ed6098-1c70-11eb-0b58-31d1830b9a10codePmd"""
In the next exercise, we will find the intersection of a ray of light and a wall. To represent light, we create a `struct` called **`Photon`**, holding the position and travel direction of a single particle of light. We also include the _index of refraction_ of the medium it is currently traveling in, we will use this later.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$e9c5d68c-1ac2-11eb-04ec-3b72eb133239cell_id$e9c5d68c-1ac2-11eb-04ec-3b72eb133239code٧md"""
##### `Miss`
is a struct with _no fields_. It does not contain any information, except the fact that it is a `Miss`. You create a new `Miss` object like so:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$492b257a-194f-11eb-17fb-f770b4d3da2ecell_id$492b257a-194f-11eb-17fb-f770b4d3da2ecode
/md"""
So we need a way of finding all of these.
To start, let's look at the intersection of a **point** and a sphere. So long as the relative distance between the photon and the sphere's center satisfies the sphere equation, we can be considered inside of the sphere. More specifically, we are inside the sphere if:
$(x_s-x_p)^2+(y_s-y_p) < r^2.$
where the $s$ and $p$ subscripts represent the sphere and photon, respectively. We know we are *on* the sphere if
$(x_s-x_p)^2+(y_s-y_p) = r^2.$
Let's rewrite this in vector notation as:
$(\mathbf{R} - \mathbf{S})\cdot(\mathbf{R} - \mathbf{S}) = r^2,$
where $\mathbf{R}$ and $\mathbf{S}$ are the $x$, $y$, and $z$ location of the photon and sphere, respectively.
Returning to the timestepping example from above, we know that our ray is moving forward with time such that $\mathbf{R} = \mathbf{R}_0 + v dt = \mathbf{R}_0 + \ell t$. We now need to ask ourselves if there is any time when our ray interacts with the sphere. Plugging this in to the dot product from above, we get
$(\mathbf{R}_0 + \ell t - \mathbf{S})\cdot(\mathbf{R}_0 + \ell t - \mathbf{S}) = r^2$
To solve this for $t$, we first need to reorder everything into the form of a polynomial, such that:
$t^2(\ell\cdot\ell)+2t\ell\cdot(\mathbf{R_0}-\mathbf{S})+(\mathbf{R}_0-\mathbf{S})\cdot(\mathbf{R}_0-\mathbf{S}) - r^2=0.$
This can be solved with the good ol' fashioned quadratic equation:
$\frac{-b\pm\sqrt{b^2-4ac}}{2a},$
where $a = \ell\cdot\ell$, $b = 2\ell\cdot(\mathbf{R}_0-\mathbf{S})$, and $c=(\mathbf{R}_0-\mathbf{S})\cdot(\mathbf{R}_0-\mathbf{S}) - r^2$
If the quadratic equation returns no roots, there is no intersection. If it returns 1 root, the ray just barely hits the edge of the sphere. If it returns 2 roots, it goes right through!
The easiest way to check this is by looking at the discriminant $d = b^2-4ac$.
```math
\text{Number of roots} = \left\{
\begin{align}
&0, \qquad \text{if } d < 0 \\
&1, \qquad \text{if } d = 0 \\
&2, \qquad \text{if } d > 0 \\
\end{align}
\right.
```
In the case that there are 2 roots, the second root corresponds to when the ray would interact with the far edge of the sphere *if there were no refraction or reflection!*; therefore, we only care about returning the closest point.
With all this said, we are ready to write some code.
👉 Write a new method `intersection` that takes a `Photon` and a `Sphere`, and returns either a `Miss` or an `Intersection`, using the method described above. Go back to Exercise 1.3 where we defined the first method, and see how we adapt it to a sphere.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$0393dd3a-1a06-11eb-18a9-494ae7a26bc0cell_id$0393dd3a-1a06-11eb-18a9-494ae7a26bc0code3plot_scene(box_scene, legend=false, size=(400,200))metadatashow_logsèdisabled®skip_as_script«code_folded$ec698eb0-19c3-11eb-340a-e319abb8ebb5cell_id$ec698eb0-19c3-11eb-340a-e319abb8ebb5codeYcorrect(text=rand(yays)) = Markdown.MD(Markdown.Admonition("correct", "Got it!", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$65aec4fc-1c9e-11eb-1c5a-6dd7c533d3b8cell_id$65aec4fc-1c9e-11eb-1c5a-6dd7c533d3b8code,test_lens_photon = Photon([0,0], [1,0], 1.0)metadatashow_logsèdisabled®skip_as_script«code_folded$19cf420e-1c7c-11eb-1cb8-dd939fee1276cell_id$19cf420e-1c7c-11eb-1cb8-dd939fee1276code[# function closest_hit(photon::Photon, objects::Vector{<:Object})
# return missing
# endmetadatashow_logsèdisabled®skip_as_script«code_folded$71dc652e-1c9c-11eb-396c-cfd9ee2261fecell_id$71dc652e-1c9c-11eb-396c-cfd9ee2261fecodeٳmd"""
👉 Change the definition of `test_sphere` to test different situations:
- Hit the circle
- Miss the circle
- Start inside the cricle (you should hit the exit boundary)
"""metadatashow_logsèdisabled®skip_as_script«code_folded$ac9bafaa-1ac4-11eb-16c4-0df8133f9c98cell_id$ac9bafaa-1ac4-11eb-16c4-0df8133f9c98codeٝ# box_scene = [
# Wall(
# [10,0],
# [-1,0]
# ),
# Wall(
# [-10,0],
# [1,0]
# ),
# Wall(
# [0,-10],
# [0,1]
# ),
# # your code here
# ]metadatashow_logsèdisabled®skip_as_script«code_folded$dced1fd0-1c9e-11eb-3226-17dc1e09e018cell_id$dced1fd0-1c9e-11eb-3226-17dc1e09e018code_md"""
To test your code, modify the definition of `test_lens_photon` and `test_lens` below.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$a306e880-19eb-11eb-0ff1-d7ef49777f63cell_id$a306e880-19eb-11eb-0ff1-d7ef49777f63code2test_intersection = intersection(dizzy, test_wall)metadatashow_logsèdisabled®skip_as_script«code_folded$b65d9a0c-1a0c-11eb-3cd5-e5a2c4302c7ecell_id$b65d9a0c-1a0c-11eb-3cd5-e5a2c4302c7ecode_let
scene = [test_lens, box_scene...]
p = plot_scene(scene, legend=false, xlim=(-11,11), ylim=(-11,11))
path = accumulate(1:sphere_test_ray_N; init=test_lens_photon) do old_photon, i
step_ray(old_photon, scene)
end
line = [test_lens_photon.p, [r.p for r in path]...]
plot!(p, first.(line), last.(line), lw=5, color=:red)
p
end |> as_svgmetadatashow_logsèdisabled®skip_as_script«code_folded$ec31dce0-19c3-11eb-1487-23cc20cd5277cell_id$ec31dce0-19c3-11eb-1487-23cc20cd5277codeEhint(text) = Markdown.MD(Markdown.Admonition("hint", "Hint", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$9df1d0f2-1ac3-11eb-0eac-d90eccca669ccell_id$9df1d0f2-1ac3-11eb-0eac-d90eccca669ccode=test_intersection_1 = Intersection(test_wall, 3.0, [1.0,2.0])metadatashow_logsèdisabled®skip_as_script«code_folded$fb70cc0c-1c7f-11eb-31b5-87b168a66e19cell_id$fb70cc0c-1c7f-11eb-31b5-87b168a66e19codeٴlet
p = plot_scene(ex_1_scene)
plot_photon_arrow!(p, philip, 4; label="Philip")
plot_photon_arrow!(p, test_new_photon, 4; label="Philip after interaction")
p |> as_svg
endmetadatashow_logsèdisabled®skip_as_script«code_folded$ec7638e0-19c3-11eb-1ca1-0b3aa3b40240cell_id$ec7638e0-19c3-11eb-1ca1-0b3aa3b40240codeٱnot_defined(variable_name) = Markdown.MD(Markdown.Admonition("danger", "Oopsie!", [md"Make sure that you define a variable called **$(Markdown.Code(string(variable_name)))**"]))metadatashow_logsèdisabled®skip_as_script«code_folded$885ac814-1953-11eb-30d9-85dcd198a1d8cell_id$885ac814-1953-11eb-30d9-85dcd198a1d8codefunction intersection(photon::Photon, sphere::Sphere; ϵ=1e-3)
a = dot(photon.l, photon.l)
b = 2 * dot(photon.l, photon.p - sphere.center)
c = dot(photon.p - sphere.center, photon.p - sphere.center) - sphere.radius^2
d = b^2 - 4*a*c
if d <= 0
Miss()
else
t1 = (-b-sqrt(d))/2a
t2 = (-b+sqrt(d))/2a
t = if t1 > ϵ
t1
elseif t2 > ϵ
t2
else
return Miss()
end
point = photon.p + t * photon.l
Intersection(sphere, t, point)
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$3dd0a48c-1ca3-11eb-1127-e7c43b5d1666cell_id$3dd0a48c-1ca3-11eb-1127-e7c43b5d1666codemd"""
👉 Recreate the spherical aberration figure from [the lecture](https://www.youtube.com/watch?v=MkkZb5V6HqM) (around the end of the video), and make the index of refraction interactive using a `Slider`. _Or make something else!_
"""metadatashow_logsèdisabled®skip_as_script«code_folded$63ef21c6-1c7a-11eb-2f3c-c5ac16bc289fcell_id$63ef21c6-1c7a-11eb-2f3c-c5ac16bc289fcodeminimum(all_intersections)metadatashow_logsèdisabled®skip_as_script«code_folded$c3e52bf2-ca9a-11ea-13aa-03a4335f2906cell_id$c3e52bf2-ca9a-11ea-13aa-03a4335f2906codebegin
import Pkg
Pkg.activate(mktempdir())
Pkg.add([
Pkg.PackageSpec(name="Plots", version="1.6-1"),
Pkg.PackageSpec(name="PlutoUI", version="0.6.8-0.6"),
])
using Plots
using PlutoUI
using LinearAlgebra
endmetadatashow_logsèdisabled®skip_as_script«code_folded$a5847264-1ca0-11eb-0b45-eb5388f6e688cell_id$a5847264-1ca0-11eb-0b45-eb5388f6e688codeV# function intersection(photon::Photon, wall::Wall; ϵ=1e-3)
# return missing
# endmetadatashow_logsèdisabled®skip_as_script«code_folded$5342430e-1c79-11eb-261c-15abd0f8cfc1cell_id$5342430e-1c79-11eb-261c-15abd0f8cfc1code)md"""
So which of these **five** results should we use to determine what the photon does next? It should be the _closest intersection_.
Because we used two different types for hits and misses, we can express this in a charming way. We define what it means for one to be better than the other:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$392fe192-1ca1-11eb-36c4-f9bd2b01a5e5cell_id$392fe192-1ca1-11eb-36c4-f9bd2b01a5e5codeZ# function intersection(photon::Photon, sphere::Sphere; ϵ=1e-3)
# return missing
# endmetadatashow_logsèdisabled®skip_as_script«code_folded$1df32310-19c4-11eb-0824-6766cd21aaf4cell_id$1df32310-19c4-11eb-0824-6766cd21aaf4codemd"_homework 7, version 1_"metadatashow_logsèdisabled®skip_as_script«code_folded$e70b9e24-1a07-11eb-13db-b95c07880893cell_id$e70b9e24-1a07-11eb-13db-b95c07880893codeمfunction interact(photon::Photon, hit::Intersection{Wall})
Photon(hit.point, reflect(photon.l, hit.object.normal), photon.ior)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$c25caf08-1a13-11eb-3c4d-0567faf4e662cell_id$c25caf08-1a13-11eb-3c4d-0567faf4e662codeqmd"""
You can use `ray.ior == 1.0` to check whether this is a ray _entering_ or _leaving_ the sphere.
""" |> hintmetadatashow_logsèdisabled®skip_as_script«code_folded$b8cd4112-1c7c-11eb-3b2d-29170ad9beb5cell_id$b8cd4112-1c7c-11eb-3b2d-29170ad9beb5code.test_closest = closest_hit(philip, ex_1_scene)metadatashow_logsèdisabled®skip_as_script«code_folded$ba0a869a-1ad1-11eb-091f-916e9151f052cell_id$ba0a869a-1ad1-11eb-091f-916e9151f052codemd"""
$(html"
")
## **Exercise 3:** _Spheres_
Now that we know how to bounce light around mirrors, we want to simulate a _spherical lens_ to make things more interesting. Let's define a `Sphere`.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$925e98d4-1c78-11eb-230d-994518f0060ecell_id$925e98d4-1c78-11eb-230d-994518f0060ecode6test_photon = Photon([-1, 2], normalize([1,-.8]), 1.0)metadatashow_logsèdisabled®skip_as_script«code_folded$43306bd4-194d-11eb-2e30-07eabb8b29efcell_id$43306bd4-194d-11eb-2e30-07eabb8b29efcodeRreflect(ℓ₁::Vector, n̂::Vector)::Vector = ℓ₁ - 2 * dot(ℓ₁, n̂) * n̂metadatashow_logsèdisabled®skip_as_script«code_folded$6544be90-19d3-11eb-153c-218025f738c6cell_id$6544be90-19d3-11eb-153c-218025f738c6codeUdizzy = Photon([0, 1], normalize([cos(dizzy_angle + π),sin(dizzy_angle + π)]), 1.0)metadatashow_logsèdisabled®skip_as_script«code_folded$522e6b22-194d-11eb-167c-052e65f6b703cell_id$522e6b22-194d-11eb-167c-052e65f6b703codeJmd"""
Now we're going to make a bold claim: All walls in this simulation are mirrors. This is just for simplicity so we don't need to worry about rays stopping at the boundaries.
We are already able to find the intersection of a light ray with a mirror, but we still need to tell our friendly computer what a _reflection_ is.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$251f0262-1a0c-11eb-39a3-09be67091dc8cell_id$251f0262-1a0c-11eb-39a3-09be67091dc8code7sphere_intersection = intersection(philip, test_sphere)metadatashow_logsèdisabled®skip_as_script«code_folded$8acef4b0-1a09-11eb-068d-79a259244ed1cell_id$8acef4b0-1a09-11eb-068d-79a259244ed1codestruct Miss endmetadatashow_logsèdisabled®skip_as_script«code_folded$8018fbf0-1a05-11eb-3032-95aae07ca78fcell_id$8018fbf0-1a05-11eb-3032-95aae07ca78fcodeXstruct Intersection{T<:Object}
object::T
distance::Float64
point::Vector{Float64}
endmetadatashow_logsèdisabled®skip_as_script«code_folded$2158a356-1a05-11eb-3f5b-4dfa810fc602cell_id$2158a356-1a05-11eb-3f5b-4dfa810fc602code&ex_1_scene = [box_scene..., test_wall]metadatashow_logsèdisabled®skip_as_script«code_folded$99c61b74-1941-11eb-2323-2bdb7c120a28cell_id$99c61b74-1941-11eb-2323-2bdb7c120a28codekstruct Wall <: Object
"Position"
position::Vector{Float64}
"Normal vector"
normal::Vector{Float64}
endmetadatashow_logsèdisabled®skip_as_script«code_folded$78915326-1937-11eb-014f-fff29b3660a0cell_id$78915326-1937-11eb-014f-fff29b3660a0code
md"""
This can be described by Snell's law:
$\frac{n_1}{n_2} = \frac{v_2}{v_1} = \frac{\sin(\theta_2)}{\sin(\theta_1)}$
Here, $n$ is the index of refraction, $v$ is the speed (not velocity (sorry for the notation!)), and $\theta$ is the angle with respect to the surface normal. Any variables with an subscript of 1 are in the outer medium (air), and any variables with a subscript 2 are in the inner medium (water).
This means that we can find the angle of the new ray of light as
$\sin(\theta_2) = \frac{n_1}{n_2}\sin(\theta_1)$
The problem is that $\sin$ is slow, so we typically want to rewrite this in terms of vector operations. This means that we want to rewrite everything to be in terms of dot products, but because $A\cdot B = |A||B|cos(\theta)$, we really want to rewrite everything in terms of cosines first. So, using the fact that $\sin(\theta)^2 + \cos(\theta)^2 = 1$, we can rewrite the above equation to be:
$\sin(\theta_2) = \frac{n_1}{n_2}\sqrt{1-\cos(\theta_1)^2}$
We also know that
$\cos(\theta_2) = \sqrt{1-\sin(\theta_2)^2} = \sqrt{1-\left(\frac{n_1}{n_2}\right)^2\left(1-\cos(\theta_1)^2\right)}.$
Finally, we know that the new light direction should be the same as the old one, but shifted towards (or away) from the normal according to the new refractive index.
In particular:
$n_2 \ell _2 = {n_1} \ell _1 + (n_1\cos(\theta_1)-n_2\cos(\theta_2))\hat n,$
where $\hat n$ is the normal from the water's surface. Rewriting this, we find:
$\ell _2 = \left(\frac{n_1}{n_2}\right) \ell _1 + \left(\left(\frac{n_1}{n_2}\right)\cos(\theta_1)-\cos(\theta_2)\right)\hat n.$
Now, we already know $\cos(\theta_2)$ in terms of $\cos(\theta_1)$, so we can just plug that in... But first, let's do some simplifications, such that
$r = \frac{n_1}{n_2}$
and
$c = -\hat n \cdot \ell_1.$
Now, we can rewrite everything such that
$\ell_2 = r\ell_1 + \left(rc-\sqrt{1-r^2(1-c^2)}\right)\hat n.$
The last step is to write this in code with a function that takes the light direction, the normal, and old and new indices of refraction:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$71b70da6-193e-11eb-0bc4-f309d24fd4efcell_id$71b70da6-193e-11eb-0bc4-f309d24fd4efcodemd"
Now to move on to lenses. Like in lecture, we will focus exclusively on spherical lenses. Ultimately, there isn't a big difference between a lens and a spherical drop of water. It just has a slightly different refractive index and it's normal is defined slightly differently.
"metadatashow_logsèdisabled®skip_as_script«code_folded$ad5a7420-1c7f-11eb-042f-115a9ef4c676cell_id$ad5a7420-1c7f-11eb-042f-115a9ef4c676codeٔhint(md"`Intersection` contains the intersected object, so you can retrieve the wall using `hit.object`, and the normal using `hit.object.normal`.")metadatashow_logsèdisabled®skip_as_script«code_folded$4d69c36a-1c73-11eb-3ae3-23900db09c27cell_id$4d69c36a-1c73-11eb-3ae3-23900db09c27codeKmd"""
There are two misses and three intersections. Just what we hoped!
"""metadatashow_logsèdisabled®skip_as_script«code_folded$55f475a8-1c7a-11eb-377e-91d07fa0bdb6cell_id$55f475a8-1c7a-11eb-377e-91d07fa0bdb6code*md"""
And we can take the **minimum**:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$ebd05bf0-19c3-11eb-2559-7d0745a84025cell_id$ebd05bf0-19c3-11eb-2559-7d0745a84025codeif student.name == "Jazzy Doe" || student.kerberos_id == "jazz"
md"""
!!! danger "Before you submit"
Remember to fill in your **name** and **Kerberos ID** at the top of this notebook.
"""
endmetadatashow_logsèdisabled®skip_as_script«code_folded$5aa7c4e8-1ac3-11eb-23f3-03bd58e75c4bcell_id$5aa7c4e8-1ac3-11eb-23f3-03bd58e75c4bcodemd"""
##### `Intersection`
is a **parametric type**. The first field (`object`) is of type `T`, and `T` is a subtype of `Object`. Have a look at the definition above, and take note of how we write such statements in Julia syntax.
We also could have used `Object` directly as the type for the field `object`, but what's special about parametric types is that `T` becomes "part of the type". Let's have a look at an example:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$87a8e280-1c7c-11eb-2bb0-034011f6c10fcell_id$87a8e280-1c7c-11eb-2bb0-034011f6c10fcode٬md"""
When we shoot a photon at the scene, we compute the intersections between the photon and every object in the scene. Click on the vector below to see all elements:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$754eeec4-1a07-11eb-1329-8d9ae0948613cell_id$754eeec4-1a07-11eb-1329-8d9ae0948613code}function closest_hit(photon::Photon, objects::Vector{<:Object})
hits = intersection.([photon], objects)
minimum(hits)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$e135d490-1ac2-11eb-053e-914051f16e31cell_id$e135d490-1ac2-11eb-053e-914051f16e31codemd"""
$(html"
")
##### Wall geometry
So, how do we find the location where it hits the wall? Well, because our walls are infinitely long, we are essentially trying to find the point at which 2 lines intersect.
To do this, we can combine a few dot products: one to find how far away we are, and another to scale that distance. Mathematically, it would look like:
$D = -\frac{(p_{\text{ray}} - p_{\text{wall}})\cdot \hat n}{\hat \ell \cdot \hat n},$
where $p$ is the position, $\hat \ell$ is the direction of the light, and $\hat n$ is the normal vector for the wall. subscripts $i$, $r$, and $w$ represent the intersection point, ray, and wall respectively. The result is $D$, the amount that the photon needs to travel until it hits the wall.
👉 Write a function `intersection_distance` that implements this formula, and returns $D$. You can use `dot(a,b)` to compute the vector dot product ``a \cdot b``.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$d257a728-1a04-11eb-281d-bde30644f5f5cell_id$d257a728-1a04-11eb-281d-bde30644f5f5codeٌbox_scene = [
Wall(
[10,0],
[-1,0]
),
Wall(
[-10,0],
[1,0]
),
Wall(
[0,-10],
[0,1]
),
Wall(
[0,10],
[0,-1]
),
]metadatashow_logsèdisabled®skip_as_script«code_folded$8cfa4902-1ad3-11eb-03a1-736898ff9cefcell_id$8cfa4902-1ad3-11eb-03a1-736898ff9cefcodeRTODO_note(text) = Markdown.MD(Markdown.Admonition("warning", "TODO note", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$ec85c940-19c3-11eb-3375-a90735beaec1cell_id$ec85c940-19c3-11eb-3375-a90735beaec1codehTODO = html"TODO"metadatashow_logsèdisabled®skip_as_script«code_folded$6c37c5f4-1a09-11eb-08ae-9dce752f29cbcell_id$6c37c5f4-1a09-11eb-08ae-9dce752f29cbcodebegin
Base.isless(a::Miss, b::Miss) = false
Base.isless(a::Miss, b::Intersection) = false
Base.isless(a::Intersection, b::Miss) = true
Base.isless(a::Intersection, b::Intersection) = a.distance < b.distance
endmetadatashow_logsèdisabled®skip_as_script«code_folded$79532662-1c7e-11eb-2edf-57e7cfbc1edacell_id$79532662-1c7e-11eb-2edf-57e7cfbc1edacodemetadatashow_logsèdisabled®skip_as_script«code_folded$76d4351c-1c78-11eb-243f-5f6f5e485d5dcell_id$76d4351c-1c78-11eb-243f-5f6f5e485d5dcodejlet
p = plot_scene(box_scene, legend=false, size=(400,200))
plot_photon_arrow!(p, test_photon, 7)
p
endmetadatashow_logsèdisabled®skip_as_script«code_folded$3663bf80-1a06-11eb-3596-8fbbed28cc38cell_id$3663bf80-1a06-11eb-3596-8fbbed28cc38codelet
p = plot_scene([test_wall])
plot_photon_arrow!(p, dizzy, 4; label="Philip")
try
scatter!(p, test_intersection.point[1:1], test_intersection.point[2:2], label="Intersection point")
catch
end
p
endmetadatashow_logsèdisabled®skip_as_script«code_folded$caa98732-19cd-11eb-04ce-2f018275cf01cell_id$caa98732-19cd-11eb-04ce-2f018275cf01codefunction plot_object!(p::Plots.Plot, sphere::Sphere)
points = [
sphere.center .+ sphere.radius .* [cos(ϕ), sin(ϕ)]
for ϕ in LinRange(0,2π,50)
]
plot!(p, points .|> first, points .|> last, seriestype=:shape, label="Sphere", fillopacity=0.2)
p
endmetadatashow_logsèdisabled®skip_as_script«code_folded$427747d6-1ca1-11eb-28ae-ff50728c84fecell_id$427747d6-1ca1-11eb-28ae-ff50728c84fecodeX# function interact(photon::Photon, hit::Intersection{Sphere})
# return missing
# endmetadatashow_logsèdisabled®skip_as_script«code_folded$052dc502-1c7a-11eb-2316-d3a1eef2af94cell_id$052dc502-1c7a-11eb-2316-d3a1eef2af94codeيmd"""
And we can now use all of Julia's built in functions to work with a vector of hit/miss results. For example, we can **sort** it:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$3aa539ce-193f-11eb-2a0f-bbc6b83528b7cell_id$3aa539ce-193f-11eb-2a0f-bbc6b83528b7code~struct Sphere <: Object
# Position
center::Vector{Float64}
# Radius
radius::Real
# Index of refraction
ior::Real
endmetadatashow_logsèdisabled®skip_as_script«code_folded$a45e1012-194d-11eb-3252-bb89daed3c8dcell_id$a45e1012-194d-11eb-3252-bb89daed3c8dcodeMmd"
Great! Next, we will repeat this action to trace the path of a photon.
"metadatashow_logsèdisabled®skip_as_script«code_folded$e5c0e960-19cc-11eb-107d-39b397a783abcell_id$e5c0e960-19cc-11eb-107d-39b397a783abcode.example_sphere = Sphere(
[7, -6],
2,
1.5,
)metadatashow_logsèdisabled®skip_as_script«code_folded$92290e54-1940-11eb-1a24-5d1eaee9f6cacell_id$92290e54-1940-11eb-1a24-5d1eaee9f6cacodemd"""
## **Exercise 1:** _Walls_
As discussed the lecture, event-driven simulations are the traditional method used for raytracing. Here, we look for any objects in our path and _analytically_ determine how far away they are. From there, we take one big timestep all the way to the surface boundary, calculate refraction or reflection to see what direction we are moving in, and then seek out any other object we could potentially run into.
So let's start simple with determining when a ray of light could intersect with a wall.
#### Exercise 1.1 - _what is a wall?_
To start, let's create the concept of a wall. For our purposes, walls will be infinitely long, so we only need to create an object that has a position and a normal vector at that position:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1e01c912-19c4-11eb-269a-9796cccdf274cell_id$1e01c912-19c4-11eb-269a-9796cccdf274code# WARNING FOR OLD PLUTO VERSIONS, DONT DELETE ME
html"""
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1e2cd0b0-19c4-11eb-3583-0b82092139aacell_id$1e2cd0b0-19c4-11eb-3583-0b82092139aacode)md"_Let's create a package environment:_"metadatashow_logsèdisabled®skip_as_script«code_folded$6cf7df1a-1c7a-11eb-230b-df1333f191c7cell_id$6cf7df1a-1c7a-11eb-230b-df1333f191c7codeumd"""
> Note that we did not define the `sort` and `minimum` methods ourselves! We only added methods for `Base.isless`.
By taking the minimum, we have found our closest hit! Let's turn this into a function.
👉 Write a function `closest_hit` that takes a `photon` and a vector of objects. Calculate the vector of `Intersection`s/`Miss`es, and return the `minimum`.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$5a9d00f6-1ac3-11eb-01fb-53c35796e766cell_id$5a9d00f6-1ac3-11eb-01fb-53c35796e766codea_miss = Miss()metadatashow_logsèdisabled®skip_as_script«code_folded$ec275590-19c3-11eb-23d0-cb3d9f62ba92cell_id$ec275590-19c3-11eb-23d0-cb3d9f62ba92codeImd"## Function library
Just some helper functions used in the notebook."metadatashow_logsèdisabled®skip_as_script«code_folded$0e9a240c-1ac5-11eb-1a7e-b3c43c459484cell_id$0e9a240c-1ac5-11eb-1a7e-b3c43c459484codelet
if length(box_scene) != 4
keep_working()
elseif !(box_scene isa Vector{Wall})
keep_working(md"`box_scene` should be a Vector of `Wall` objects.")
else
w = last(box_scene)
if w.position != [0,10]
keep_working(md"The wall's position is not correct.")
elseif w.normal != [0,-1]
keep_working(md"The wall's direction is not correct.")
else
correct()
end
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$e9c6a0b8-1ad0-11eb-1606-0319caf0948acell_id$e9c6a0b8-1ad0-11eb-1606-0319caf0948acodeAmd"""
$(html"
")
## **Exercise 2:** _Mirrors_
"""metadatashow_logsèdisabled®skip_as_script«code_folded$54b81de0-193f-11eb-004d-f90ec43588f8cell_id$54b81de0-193f-11eb-004d-f90ec43588f8codeTmd"""
We need a helper functions to find the normal of the sphere's surface at any position. Remember that the normal will always be pointing perpendicularly from the surface of the sphere. This means that no matter what point you are at, the normal will just be a normalized vector of your current location minus the sphere's position:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$7ba5dda0-1ad1-11eb-1c4e-2391c11f54b3cell_id$7ba5dda0-1ad1-11eb-1c4e-2391c11f54b3codemd"""
#### Exercise 2.3 - _accumulate_
👉 Write a function `trace` that takes an initial `Photon`, a vector of `Object`s and `N`, the number of steps to make. Return a vector of `Photon`s. Try to use `accumulate`.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1ee0787e-1a08-11eb-233b-43a654f70117cell_id$1ee0787e-1a08-11eb-233b-43a654f70117code=let
p = plot_scene(ex_1_scene, legend=false, xlim=(-11,11), ylim=(-11,11))
path = trace(philip, ex_1_scene, mirror_test_ray_N)
line = [philip.p, [r.p for r in path]...]
plot!(p, first.(line), last.(line), lw=5, color=:pink)
plot_photon_arrow!(p, philip)
plot_photon_arrow!.([p], path)
p
end |> as_svgmetadatashow_logsèdisabled®skip_as_script«code_folded$ec4abc12-19c3-11eb-1ca4-b5e9d3cd100bcell_id$ec4abc12-19c3-11eb-1ca4-b5e9d3cd100bcodeـstill_missing(text=md"Replace `missing` with your answer.") = Markdown.MD(Markdown.Admonition("warning", "Here we go!", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$eff9329e-1a05-11eb-261f-734127d36750cell_id$eff9329e-1a05-11eb-261f-734127d36750code٘function plot_scene(objects::Vector{<:Object}; kwargs...)
p = plot(aspect_ratio=:equal; kwargs...)
for o in objects
plot_object!(p, o)
end
p
endmetadatashow_logsèdisabled®skip_as_script«code_folded$0906b340-19d3-11eb-112c-e568f69deb5dcell_id$0906b340-19d3-11eb-112c-e568f69deb5dcode0test_wall = Wall(
[8,-1],
normalize([-3,1]),
)metadatashow_logsèdisabled®skip_as_script«code_folded$a99c40bc-1c7c-11eb-036b-7fe6e9b937e5cell_id$a99c40bc-1c7c-11eb-036b-7fe6e9b937e5codeټlet
p = plot_scene(ex_1_scene)
plot_photon_arrow!(p, philip, 4; label="Philip")
scatter!(p, test_closest.point[1:1], test_closest.point[2:2], label="Closest hit")
p |> as_svg
endmetadatashow_logsèdisabled®skip_as_script«code_folded$13fef49c-1c9e-11eb-2aa3-d3aa2bfd0d57cell_id$13fef49c-1c9e-11eb-2aa3-d3aa2bfd0d57codemd"""
By defining a method for `interact` that takes a sphere intersection, we are now able to use the machinery developed in Exercise 2 to simulate a scene with both lenses and mirrors. Let's try it out!
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1df82c20-19c4-11eb-0959-8543a0d5630dcell_id$1df82c20-19c4-11eb-0959-8543a0d5630dcodePmd"""
Submission by: **_$(student.name)_** ($(student.kerberos_id)@mit.edu)
"""metadatashow_logsèdisabled®skip_as_script«code_folded$337918f4-194f-11eb-0b45-b13fef3b23bfcell_id$337918f4-194f-11eb-0b45-b13fef3b23bfcodelPlutoUI.Resource("https://upload.wikimedia.org/wikipedia/commons/6/67/Line-Sphere_Intersection_Cropped.png")metadatashow_logsèdisabled®skip_as_script«code_folded$70b8401e-1c7e-11eb-16b2-d54d8f66d71acell_id$70b8401e-1c7e-11eb-16b2-d54d8f66d71acodeVmd"""
👉 Verify that the function `reflect` works by writing a simple test case:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$eb35ac4a-1acc-11eb-0729-ff85c8406c45cell_id$eb35ac4a-1acc-11eb-0729-ff85c8406c45code@@bind aberration_viz_ior Slider(1.0:0.0001:2.0, show_value=true)metadatashow_logsèdisabled®skip_as_script«code_folded$0b03316c-1c80-11eb-347c-1b5c9a0ae379cell_id$0b03316c-1c80-11eb-347c-1b5c9a0ae379code0test_new_photon = interact(philip, test_closest)metadatashow_logsèdisabled®skip_as_script«code_folded$ec3ed530-19c3-11eb-10bb-a55e77550d1fcell_id$ec3ed530-19c3-11eb-10bb-a55e77550d1fcodeSalmost(text) = Markdown.MD(Markdown.Admonition("warning", "Almost there!", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$42d65f56-1aca-11eb-1079-e32f85554349cell_id$42d65f56-1aca-11eb-1079-e32f85554349codemd"""
$(html"
")
#### Exercise 1.3 - _hitting the wall_
👉 Write a function `intersection` that takes a `photon` and a `wall`, and returns either a `Miss` or an `Intersection`, based on the result of `intersection_distance(photon, wall)` ``= D``.
If $D$ is _positive_, then the photon will hit the wall, and we should return an `Intersection`. We already have the intersected object, and we have $D$, our intersection distance. To find the intersection _point_, we use the photon's position and velocity.
$p_{\text{intersection}} = p_{\text{ray}} + D\hat \ell$
If $D$ is _negative_ (or zero), then the wall is _behind_ the photon - we should return a `Miss`.
##### Floating points
We are using _floating points_ (`Float64`) to store positions, distances, etc., which means that we need to account for small errors. Like in the lecture, we will not check for `D > 0`, but `D > ϵ` with `ϵ = 1e-3`.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1b0c0e4c-1c73-11eb-225d-23c731455755cell_id$1b0c0e4c-1c73-11eb-225d-23c731455755codeAall_intersections = [intersection(philip, o) for o in ex_1_scene]metadatashow_logsèdisabled®skip_as_script«code_folded$e45e1d36-1a12-11eb-2720-294c4be6e9fdcell_id$e45e1d36-1a12-11eb-2720-294c4be6e9fdcode'plot_scene([test_wall], size=(400,200))metadatashow_logsèdisabled®skip_as_script«code_folded$5501a700-19ec-11eb-0ded-53e41f7f821acell_id$5501a700-19ec-11eb-0ded-53e41f7f821acodeclet
p = plot_scene(ex_1_scene, legend=false, size=(400,200))
plot_photon_arrow!(p, philip, 5)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$b3ab93d2-1a0b-11eb-0f5a-cdca19af3d89cell_id$b3ab93d2-1a0b-11eb-0f5a-cdca19af3d89code(ex_3_scene = [test_sphere, box_scene...]metadatashow_logsèdisabled®skip_as_script«code_folded$83aa9cea-1a0c-11eb-281d-699665da2b4fcell_id$83aa9cea-1a0c-11eb-281d-699665da2b4fcodelet
p = plot_scene(ex_3_scene)
plot_photon_arrow!(p, philip, 4; label="Philip")
if sphere_intersection isa Intersection
scatter!(p, sphere_intersection.point[1:1], sphere_intersection.point[2:2], label="Intersection point")
end
p |> as_svg
endmetadatashow_logsèdisabled®skip_as_script«code_folded$900d6622-1a08-11eb-1475-bfadc2aac749cell_id$900d6622-1a08-11eb-1475-bfadc2aac749codeِfunction trace(photon::Photon, scene::Vector{<:Object}, N)
accumulate(1:N; init=photon) do old_photon, i
step_ray(old_photon, scene)
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$14dc73d2-1a0d-11eb-1a3c-0f793e74da9bcell_id$14dc73d2-1a0d-11eb-1a3c-0f793e74da9bcodefunction refract(
ℓ₁::Vector, n̂::Vector,
old_ior, new_ior
)
r = old_ior / new_ior
n̂_oriented = if -dot(ℓ₁, n̂) < 0
-n̂
else
n̂
end
c = -dot(ℓ₁, n̂_oriented)
normalize(r * ℓ₁ + (r*c - sqrt(1 - r^2 * (1 - c^2))) * n̂_oriented)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$1a43b70c-1ca3-11eb-12a5-a94ebbba0e86cell_id$1a43b70c-1ca3-11eb-12a5-a94ebbba0e86codeV# function trace(photon::Photon, scene::Vector{<:Object}, N)
# return missing
# endmetadatashow_logsèdisabled®skip_as_script«code_folded$abe3de54-1ca0-11eb-01cd-11fe798bfb97cell_id$abe3de54-1ca0-11eb-01cd-11fe798bfb97codeV# function intersection_distance(photon::Photon, wall::Wall)
# return missing
# endmetadatashow_logsèdisabled®skip_as_script«code_folded«notebook_id$118fc668-4aa6-11f0-2ffc-8f791164447bin_temp_dir¨metadata
mimetext/htmlrootassigneelast_run_timestampA p ypersist_js_state·has_pluto_hook_features§cell_id$337918f4-194f-11eb-0b45-b13fef3b23bfdepends_on_disabled_cells§runtimeڎ^published_object_keysdepends_on_skipped_cells§errored$70b8401e-1c7e-11eb-16b2-d54d8f66d71aqueued¤logsrunning¦outputbody{