# TU Wien Rendering #6 - Snell's Law and Total Internal Reflection

## Метаданные

- **Канал:** Two Minute Papers
- **YouTube:** https://www.youtube.com/watch?v=LD6xRkCJ6ek
- **Дата:** 25.03.2015
- **Длительность:** 15:32
- **Просмотры:** 12,057
- **Источник:** https://ekstraktznaniy.ru/video/15006

## Описание

Why does the straw look bent in a glass of water? Why is the world distorted when looking through a lens or a glass marble ball? How can light get trapped in transparent surfaces? We formulate Snell's law, an incredibly simple equation to answer these questions.

About the course:
This course aims to give an overview of basic and state-of-the-art methods of rendering. Offline methods such as ray and path tracing, photon mapping and many other algorithms are introduced and various refinement are explained. 

The basics of the involved physics, such as geometric optics, surface and media interaction with light and camera models are outlined. 

The apparatus of Monte Carlo methods is introduced which is heavily used in several algorithms and its refinement in the form of stratified sampling and the Metropolis-Hastings method is explained. 

At the end of the course students should be familiar with common techniques in rendering and find their way around the current state-of-the-art of the

## Транскрипт

### Segment 1 (00:00 - 05:00) []

there is still one thing that we don't know and it's about angles so we know about probabilities whatever you give me I can tell you what is the probability that light gets reflected or refracted I know the probabilities but I don't know about angles and what we need to know is that light rays of light slow down they travel with the speed of light but they slow down as they enter mediate because there's uh atoms particles in there and it's more difficult to get so light slows down the index of refraction tells you by exactly how much so the index of refraction of a medium is given by this fraction the speed of light in vacuum that we know over this medium so uh this is how we can write it up let's look up an example the index of refraction of glass is 1. 5 so we know exactly what the speed of light inside glass is well it's 300 million m/s in vacuum and what we know is this equation so this is the index of refraction so I can just reorder this thing and conclude that uh light if it travels in vacuum 200 300 million uh m/s but in glass it loses the third of its velocity and it's only 200 million m/s so that's a pretty easy calculation and it's pretty neat and another absolutely beautiful thing hopefully you have uh studied the maxal equations and pointing vectors in physics uh light is ultimately a wave so here above and below you can see some wave Behavior and the ray is essentially the wavefronts of these waves so light Can Be Imagined as Rays if you uh take into consideration that I would need to compute many of these wave fronts in order to take account for the wave behavior and don't look at the red only look at the Blue uh this is above this is uh vacuum and below this could be for instance glass and you can see that the waves slow down in this medium and uh what this means if we go back to the definition of the wavefronts the red lines then they are essentially bending because the wave fronts are going to look like this so it's very interesting because if you imagine light as a wave it only slows down but if you imagine light as a ray then it bends it changes Direction so uh I think that's absolutely beautiful and the question is uh why is the light refracting inwards because what I would imagine is that it continues normally it continues its way uh with this Theta T equals Theta I because it would just continue its way and it doesn't continue its way but it it fends a bit and the question is why and now we have Can Academy time raise your hand if you know Can Academy okay awesome well educated people uh so this is shamelessly stolen from Can Academy because this is the best way to describe how refraction works so basically uh you imagine that you have a large car and the air vacuum interface is now Road and mud I mean the road is the air and mud is glass for instance and imagine as you are approaching this boundary line between the two then the first wheel of the car like the lower left on this image is entering the mud but the other but on the other side the wheels are still on the road so therefore this wheel will slow down in the mud but this is still going as fast as it used to be so what will the car do if this happens it will start to turn and you know exactly where it would turn because this is going slow this is going faster so therefore it would turn inwards so this is I think an amazing interpretation of the whole thing I think also it's easy to explain with the waves because when the waves slow down then the dire um then we can see that the circles will get bigger radius and then if you go perpendicular to the waves exactly then we will go down exactly like in the previous figure little that's another intuition I just wanted to ask if this is actually physically happens because I don't think so no this is intuition that's why it's a bit misleading and then the it's a nice this is strictly intuition okay so if you would start to model uh raise of light

### Segment 2 (05:00 - 10:00) [5:00]

uh like trucks on going Road and not then you're going to encounter problems you could technically interpret it as part of the wave hits the medium first and by that the entire wave gets rotated a bit that's like how I would interpret no is it I tend to give multiple ways to interpret things because you know different minds work differently some graphical ways are working for different people better okay so SN law and uh we're almost done for today SN law tells you in what angle refracted rays are going to continue their path and this is given by this expression signs of the angles uh against velocities against the reciprocal of indices of refraction okay so let's do the a class example of the previous image let's State our expectations before we go so I'm interested in the relation of theta I versus Theta T so uh I know these Expressions exactly how much is Theta I how in degrees 60 it's 60 okay excellent how much is Theta T in degrees 3 it's it's something around 35 exactly okay so the light is reflected inwards therefore the Theta T must be less than the Theta I so this this uh let's compute the equation and see if this works and if it doesn't work we're going to call out the physicist on so let's just reorder some things and let's put there the indices of refractions uh and the incoming uh light angle that we know and just some very simple reordering we are almost there and if we actually compute the sign of 60° we get this well uh this is we can also uh carry out the division but at this point I'm not interested in the sign of theta T I'm interested in Theta T so I would multiply both sides invert the equation by multiplying with the inverse of the sign so this Theta T should be uh the arc sign of this and if I compute everything back to degrees then I will get this Theta T which is 34. 75 so whoever said the 35 was very close to the actual result but also not to forget that there's different kinds of glasses I mean uh there's multiple ways of creating and Manufacturing glasses and they have different indices of reflection more or less the same but it's still different but we can see that this is in really good agreement with what we see in real life well what did we say Theta T should be less than Theta I but 35 is definitely less than 60 so again physics works and physicists are uh smart people and uh just another example if you think about the car example or whichever example you like better you will uh hopefully immediately see that if we would be going with the yellow uh Arrow this is going to bend inwards after going back from the water to the air now whoa holding right there uh what is happening I don't see any reflection whatsoever right so it seems to me that if I go back at around how much is this in degrees that's it 50° exactly so Something Fishy happens at 50° well I don't know what is happening I'll tell you the name and we going to compute whether this is possible or not well if it's not possible then our math sucks so uh but let's see so what we uh what we call this is total internal reflection there is a critical angle and after this critical angle there is no more uh refraction there's only uh there's only reflection happens and there is many examples of that and there's many applications of that uh this is one of the more beautiful examples so let's compute what's going on here uh what I know is that I have the indices of refractions I know this degree uh that we uh just dealt with and something interesting should happen here uh and something interesting already happened so I just plugged in everything what I have seen on this image and I get

### Segment 3 (10:00 - 15:00) [10:00]

this and this is awfully horribly terribly wrong someone please help me out why is that it's okay yes sign can be bigger than one exactly so the support of the sign is between 1 and minus one at least according to my experiences uh so it's saying that the sign of an angle is more than one is mathematically not possible so it says that there's no such angle what would be the angle of refraction if I would use be using 50° then it says something that mathematically doesn't make any sense so math actually suggests to you if you use the right numbers it suggests to you already that this total internal reflection would happen uh let's try to compute the critical angle and this I just reordered things this is hopefully the critical angle that I will be uh trying to compute well if I have this Theta one this is a relatively small number then there is going to be refraction there is this critical angle on the second figure at which I have this 90° uh refraction so it says that at the critical angle this thing is going to be 90° and after that so this is smaller than this after this critical angle there's only going to be reflection now let's try to compute this uh note uh notice the 90° here so what I put here is this is what I'm interested in and this happens when this refraction is at 90° so I put there this 90° explicitly and I want to know this uh Theta one it's going to be the uh that's critical angle well if I actually uh do the computation with the 90° then I'm going to get one for the sign so this is N2 over N1 well I'm still not interested in the sign of this angle I'm interested in the angle so I have to invert the other side both sides of the equation and this is the definition of the critical angle and if you write it in Wikipedia critical Ang you are going to get the very same formula but the most interesting thing is that you can actually derive this yourself and this is not a huge derivation this is very simple this is where this 90° refraction happens so what is our expectation for this critical angle let's look at the reality again on the right side what so this is I'm just trying to hint without saying telling you the solution so let's try it without hints uh what could be the critical angle here raise your hand if you know the answer okay sorry I will for pedagogical reason I will ask someone who I haven't asked before okay let's see if it's correct I have asked you have I asked you before nothing important okay now you got an important question it should be something smaller than 50° because now is only exactly so so the the usual answer what I get is that 50° because I see uh total internal reflection but total internal reflection means that after some at some point after that there's only going to be reflection so it doesn't mean that if this is that point because if I would be trying 60 I would also see reflection but this doesn't mean that 60° is the critical angle it's before that at some point okay was this your answer too as well yes but uh I was thinking that if it's critical angle for example let's say 50 is critical angle then we would have also reflection and then one exactly yes what you have seen on the figure so at the critical angle you see this instead of this so this is over the critical angle so it has to be less than 50° less than 6 50° okay so this is very simple from here let's just substitute the indices of refraction 4181 wonderful physics works and we are still alive so that's basically it for today uh we have used reality to

### Segment 4 (15:00 - 15:00) [15:00]

be our judge we are not just writing formula on paper and then be happy about how much we can understand or memorize of them we put everything to use and you will see all of this in C++ code not so long uh from now so that would be the introductory course and I'll see you next week thank you