 You probably know how to find the refracted ray using Snell's law. When you plug in your index of refraction, you measure your first angle, then use your calculator to measure or to find the second angle, and then you just draw it. But there is a way to play it much faster and without even using a calculator. So in this method what you have to do first is you have to draw two circles with the radius proportional to the index of refraction. So here I had an index of refraction and one, and about three times bigger an index of refraction and two. Now all you have to do is continue your incoming ray until it hits the angle of refraction or the circle of the index of refraction that you are coming from, in this case N1, and then make a parallel line and connect this to the circle of the second index of refraction and you have found your refracted ray. Now you might wonder why is this working or how comes this is working and the answer is the following. If I have my angle T1 here, then I have my angle T1 here. The angle of T1 sign of it times N1 is this line here. So I have N1 times T1 is my green line. Now what we have here, here I have my second radius which is my N2 times sign of T2 gives me this distance here and those two distances I made them equal so basically I have just genetically solved. I can also use this if I want to go from the higher index to the lower index. So same thing, let's say I have a ray coming up like this. All I need to do is continue until I hit the index of refraction coming from making a parallel line to the normal down. Look at the intersection here and here is my refracted ray. Also I can very easily find the critical angle. The critical angle happens when this parallel line comes down it's not hitting my lower index of refraction anymore. So to find the critical incoming angle all I go to the most extreme end here my lower index of refraction. I connect like this and here the red one was my critical angle which will be refracted along the surface. Here I have my critical angle and that's how we do this without even opening up a calculator and I'm sure it's much faster than doing it with a calculator.