# TU Wien Rendering #5 - The Fresnel Equation and Schlick's Approximation

## Метаданные

- **Канал:** Two Minute Papers
- **YouTube:** https://www.youtube.com/watch?v=iKNSPETJNgo
- **Дата:** 25.03.2015
- **Длительность:** 14:57
- **Просмотры:** 18,011
- **Источник:** https://ekstraktznaniy.ru/video/15007

## Описание

What portion of light is reflected and refracted by glass-like surfaces? The Fresnel equation shows us why we see such strong reflections in windows from grazing angles, and why it's so simple to look through them. Schlick's approximation of the original equation provides simple and powerful means that can be computed rapidly. However, it has its own limitations. I wonder what they are? In this segment, we try to find out!

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

## Транскрипт

### <Untitled Chapter 1> []

now we have some image from physical reality we have an interface that is air and glass uh what I see here is that there is reflection and there is refraction so in case not everyone understands the term uh reflection is a lift reflexion and refraction is bra right if with my horribly broken German but thank God this course is not in German everyone would cry but you guys and girls have it easy because uh in Hungarian you would say fish and fish so I think reflection and refraction is much better than that at least much more convenient question which is stronger which effect is stronger raise your hand if you think that reflection that this is this effect is strong is a question about this example or in general in this example course yes so I can see that the refraction has a more pronounced effect than reflection here okay because we will deal with this in a second and uh what we can do is we can actually write up the vectors that we have been talking about for a case like that this is towards the direction of the light source there is a Surface normal it looks like this so the normal looks upwards this is where it's reflected transmitted we don't have a vector for that and we have the different angles for this uh wonderful so let's take a look at the simplified version of fral equation at

### Simplified Version of Reynolds Equation [1:45]

the

### Slinks Approximation [1:48]

the simplified version of fral equation this one at least is called schlick approximation we have no idea what this is about this is a not so complicated thing what this gives me is the probability of reflection so R of theta as I have seen the image I am interested in what is the probability of reflection and refraction because I imagine that the probability of refraction is higher in this case and I would like to compute it in some way in my computer program well let's take a quick look at this R of theta is the probability of reflection and this is important to remember because during our calculations I will you forget this approximately 15 times so uh may I ask your name Lisa okay uh well if I ask you uh what is r Theta you will tell

### Probability of Reflection [2:42]

me that this is the probability of reflection yes and exactly because I seriously I will be forgetting it all the time r0 is

### Probability of Reflection on Normal Incidence [2:52]

the probability of reflection on normal incident so this means that light is coming from if it would be coming from above what are the chances that it gets reflected and r0 can be given by this expression we'll be talking about this uh N1 and N2 are basically indices of refraction uh we will have examples with that too but let's quickly go through this and see if physics make any sense makes any sense well for an air

### Air Vacuum [3:23]

vacuum medium we have uh let's say the index of refraction of air is one and this N1 is the medium that we go into so for instance here glass and let's see what happens there uh but before we do that t is the probability of transmission uh obviously if the light is not we forget absorption for now uh if light is not reflected then there is refraction it's simple as that so if I add up these two probabilities I get one so let's play with it uh r at 0° is r0 why because cosine of theta so the cosine of 0 degrees is one so on the right side I have 1 minus one so the second term is killed by this zero therefore I will have r0 and R at 0° this is the Theta is the degrees the angle of the incoming light 0° so it means that it comes from upwards what is it this is the prob

### Probability of Refraction of Normal Incidence [4:28]

probability of reflection of normal incidence so this is basically the very same thing so if it comes like this what is the probability of it coming back bouncing off of the glass uh what's up with 90° well 90° the sign of theta is uh zero therefore I will have both of these terms very simple and this is going to

### The Probability of Reflection at 90 Degrees [4:58]

be one so the probability of reflection at 90° is one because imagine that I'm coming from above then this means that there is a very likely chance that I'm going to get through so imagine a super crowded bus in the morning and you just cannot fit in there how would you like to go in if you don't care about the health and the intactness of the other people well you will just run in there and you know hopefully they will make some space I have the best probability if I run towards them if I would be running from the side it would be very likely that they would just push me back there's a high chance that I would be reflected so I want refraction I want to get on the bus okay so as I raise this angle from normal incidence the more probability there is for the r to bounce back so this for now seems to make some sense yes but is it still reflection at 90° and not just the L itself uh you have to think in terms of limits so what is the probability at 89° it's going to get reflected if you just raise that you are going to approach the probability of one so this is a bit bugling the mind but you can see that there's like a continuous transition from here there is high probability for refraction as I go to towards 90° there's more probability for reflection and we Define that at 90° we say that it is reflection because I am moving along the boundary I'm not entering the class by the way that's a great question I was thinking about this too before so let's say that uh index of refraction of glass is 1. 5 let's compute this thing quickly uh it's 0. 5 over 2. 5 S and I do the very same substitution for the rest of the equation uh but before I get this what do I expect from the plot that's the another important mathematical principle do this all the time before you compute the result State what you would expect from the result because this gives you a much higher level of understanding well I'm interesting in R of theta what

### R of Theta Probability of Reflection [7:26]

does r of theta mean probability of reflection excellent okay Lia Nails it again so the probability of reflection at 0° I would say is something very low right okay so I have written this here that R of Z is less than 0. 1 because it's the probability of I mean r0 is the probability of oh sorry reflection yes so the probability of reflection is low I would say less than 10% okay so if I come from upwards uh reflection is likely I'm very likely to get on the bus if I Ram into the people from the front uh okay uh what's up with for instance uh 60° well we know exactly what happens at 60° because how many degrees do we have here what is the incidence now it's 60 and we can see that at 60° there is a chance for refraction and reflection and the refraction chance is higher exactly so what I would imagine is that at 60° there is a higher chance of reflection there is than the previous one I mean but refraction is still stronger as you can see on the image and we are going to compute this be let we're going to let nature be our judge whether the calculation is correct or not so 60° I quickly uh convert it to radians that's more or less one and so R of 1 is 0. 2 this means 20% chance of reflection 80% of reflection this seems to be in line of what I see here but this is just my expectation and what we have been talking about P / 2 this means 90° angle I would expect it to be one so if I convert it to radians then this is 1. 7 so R of 1. 7 I expect it to be one let's put all of these together and let's do what Engineers do all the time open wall from Alpha and try to plot this so I imagine that R of0 is less than 0. 1 so getting on the bus easily well r at zero is less than 0. 1 so far so good r at 1 is less than 0. 2 this is the 60° mostly refraction well r at one is less than 0. 2 so check mark and R at around 1. 7 is indeed around one so apparently physicists are small people and physics makes sense so prediced but there is something fishy about this so uh this this is correct I mean what we see here is in line with our expectations but there's something fishy about this plot raise your hand if you know what it is okay I'll give you a hint this plots are of Thea which is probability of reflection this is the probability of reflection okay uh what happens after I don't know if I would just extrapolate what would happen at not 1. 5 but two what would I meas I think it would be something like the very least H minus but it's going upwards but it would be at least three I don't know about you but I don't know prob about probabilities that can be larger than one right so this would give me some fishy results if I would substitute something like that uh so let's try to shed some more light on it what if I have a vacuum interaction So Below I don't have glass anymore I have vacuum well the index of refraction of

### Index of Refraction of Vacuum [11:36]

vacuum is one so let's just substitute one here so this is going to be zero and I'm going to have the second term 1 - cosine of theta to the 5th why because this is zero and this is 1 - 0 so I will keep this turn Okay engineering mode what do we expect from this spot I have vacuum or air if you wish if you will and vacuum again I start a ray of light what will happen with probability of one reflection or refraction raise your hand if you know should no reflection there should be no reflection exactly why because there is no inaction of the medium exactly so the definition of vacuum is again nothing there's nothing in there there's nothing that could reflect this light this ray of light back if there's vacuum we expect rays of light to travel in vacuum indefinitely there's no way that it could be reflected so since r0 is the probability of reflection then this R sorry R Theta what do I think this should be if there's only refraction then this R of theta will be zero fantastic okay well let's plot this looks like 1us cosine Theta to the 5th this is already fishy but let's take a look uh blah blah this is what I've been talking about so I expect it to be zero uh let's plot it and this looks like this which is uh not constant Zero by any stretch so the question is you know what went terribly wrong here and it's very easy the schek approximation is an approximation it's good for interfaces which are vacuum or air and something and not something okay so A and B it works well but it's a quick approximation because this is the original fral equation and this is much more expensive to compute and this other one was much quicker but it's a bit Limited in use so let's give it a crack so uh what would this say about the vacuum interaction well uh I would substitute N1 and ns2 equals 1 this is the index of refraction of vacuum I get the very same thing back just the n1's and n2s are gone I'm going to use a trigonometric identity which says that theare root of 1- the sin of square something is a cosine okay so let's substitute these for the cosin got it so what I see here is the cosine of theta minus cosine Theta so what is this expression exactly how much is it it's zero exactly and this is what I was expecting so apparently physicists again are smart people