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hey this is my quest of medibang gorg if

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you think my videos are a bit too

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technical sometimes then you should

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probably stop watching now because this

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is going to be a very very very

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technical video what I'm going to do is

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I'm going to explain how this refraction

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simulator that I wrote actually works

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I'm going to have a look at the source

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code and the mathematics and the

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equations behind it and I'll try to do

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it reasonably quickly so you can just

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look at the actual stuff yourself

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hopefully I should give you enough of an

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understanding so refraction simulator is

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basically simulating the effects of

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atmospheric refraction and you see here

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we have rays of light which are coming

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from the camera this is using the

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principle of reversibility of light goes

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exactly the same way through air

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regardless of whether you go this way or

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this way now of course light is actually

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going from the target to the camera but

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we can actually cast rays from the

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camera to the target and it goes along

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the same path so what we want to do in

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terms of code is that we start out from

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a position over here where the camera is

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and we give the Ray a direction which

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and it's like in tentacle now I use a

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two-dimensional vector

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I just use a two-dimensional unit vector

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which is a vector of length 1 and that

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specifies which way the line is pointing

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and I store its position as another 2d

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vector and X Y vector where X is the

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horizontal distance and Y is the

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vertical distance 0 0 in this situation

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is the ground beneath the feet of the

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where the guy with a camera is

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theoretically sending so if we move in

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way up here 0 0 in our coordinate system

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is this position right here and so this

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the nuts and bolts of the code are

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tracing a ray which means we start with

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a ray a certain position which we call P

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and we have a direction which I specify

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with a unit vector which I call you and

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I trace it forward one step at a time

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along a distance D and we move through

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this medium which is a varying

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refractive index and the actual

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refractive index affects how the ray

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actually curves so we move it a step

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forward we see as it moved up or down a

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little bit and then we move it up or

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down and then we move it again and we

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keep moving and eventually we'll hit

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something and then where we hit it

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determines what red watch line of the

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image ends up in the final result so

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probably lost you all already but I'm

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gonna plow on now if you want to see how

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this simulator actually works then you

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have to look at the source code and here

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in chrome I can just go to view

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developer view source and the source

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code will pop up blah blah blah it's all

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there but actually a couple of other

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files which you can see listed at the

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top let's see they're gonna be right

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here you can click on them to just load

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them and then you can you can see them

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here but another way of doing this is in

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chrome you do view developer and

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developer tools and it will bring you up

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this little list neat neatly shows kind

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of the project structure and where the

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other files are jQuery is a strong that

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standard library library the action most

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of the code is actually in the index

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file index.html and most of the actual

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interesting code is in there some other

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stuff is in this file here pseudo ID and

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refraction GS which has a few equations

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for calculating refractive index then

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the curve editor which is a thing I

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wrote to easily edit the actual curves

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it's not very complicated it's just

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Bezier curves which I guess is

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complicated but you know that's these

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are the only extra bits of code that are

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relevant but

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I wrote this is just an implementation

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of standard equations and this is just

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editing a curve right back to the index

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I'm actually going to show you this in

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the program I use which piece phpstorm

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and I've made the font nice and big so

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you can see you can see what what's

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actually going on so the heart of the

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program is this one function here trace

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one line and it's taking a line number

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which is actually the line number of the

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the actual target image which would be

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when I was the the resultant image the

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rendered image so the line on the screen

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basically so it's my life checking to

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see if it's being done already is it

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just things asynchronously so it will do

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like every one out of ten lines and then

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it Flags them as being done and there's

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an array called rays which contains all

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the lines and as a bunch of bunch of

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flags in but not that difficult

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not that complicated but you can look at

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the actual structure of the rays array

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and see the various of the four things

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in it

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we can I collect the angle of the Ray

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and now obviously when when array is

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leaving your eye goes out at a

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particular angle zero is going to be

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perfectly horizontal and this is going

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to be up at whatever the field of view

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is which is 1.5 degrees here remember

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this is greatly exaggerated so it looks

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like it's a fairly wide field of view

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it's only actually 1.5 degrees which is

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a very very narrow field of view it's

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taking x and y's being the star position

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there's just the the the eye x and y

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which is the basically the camera

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position and it takes this angle or

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whatever it's doing there I guess it's

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just check checking to see if you're

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going up or down coloring the Lions red

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or blue

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this is detecting which of the lines is

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eye level not very important really just

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cosmetic thing and here this is where we

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get to some of the actual mathematics

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here getting the value D to be the step

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along the the line which is how

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we're gonna go in one step in our ray

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tracing algorithm which is typically the

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distance to the target and divided by

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500 I think now we have the unit vector

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U which is the direction of the Ray and

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we just use two variables u ax and UI to

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store this and we calculating it simply

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as being the cosine and the negative

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sine of the angle and you want to think

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about how that actually works you can

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think of a actual vector d DX and dy

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which is the the length of D multiplied

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by the cosine and the negative sine of

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the angle and then you calculate the

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length of that and then you calculate

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the unit vector by dividing it by the

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length so this gives you the actual off

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certainty but for our purposes we're

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only using the unit vector so we don't

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need to have these intermediate values

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which you might if you're doing it in a

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slightly different way all right

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so for this Ray we are going to build in

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an array of line points which is just

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point XY point and we're going to check

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to see if we hit the ground and we check

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to see if we recheck later to see if we

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hit the target this is how much we're

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actually going to trace we just

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basically trace the amount that is

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visible in this diagram here so we'll

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trace all the way over to here to see if

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you know we hit something we don't we

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just basically stop if we get it to this

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point

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and if if it's a flat earth then we

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actually triple the amount that we trace

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because we want to see the horizon

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beyond things this isn't work entirely

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right and I need to improve that but

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that's why I'm doing that try to get a

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more accurate horizon beyond the actual

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target ok so this is the main loop that

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loops and calculates all of the points

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in the ray tracing this this function

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call here is the key thing and which

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will need the most explanation

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it takes us parameters x and y which is

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the current position it takes UX and uy

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which is the direction and it takes D

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which is how much of a step we're going

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to take so D times you actually uy will

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give you the new position and it calls

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this function here step forward and

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updates all the values if it hits the

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ground then it Flags it and then stops

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otherwise it adds a new point XY to the

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array of line points and just keeps

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going until this islet done or its hit

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hit the ground by done being going off

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the right hand side and that's pretty

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much it for that so let's delve into

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what step forward does this is the

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complicated bit if you don't think it's

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complicated enough already so I'm going

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to zip into that function step forward

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it's planes wait I see a given x and y

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we're going to step forward by x times d

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uy times d and blah blah blah so to

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explain how this works there's a venture

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this is what it actually does here to

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explain how this works I'm going to have

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to show you a diagram

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okay now if every country this base has

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simple integration of a point P of

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distance D in Direction u here's a point

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P the point P is the point on a ray of

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light but really you've got to think of

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the motion of light as being the motion

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of a wavefront you see this squiggly

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line here it's meant to be a wavefront

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so it's the the ray of light is actually

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just one point on a wave that's going

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along and we have this point P on that

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wave and we want to know if it's gone up

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distance D along direction vector u what

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will be the new point P - which is

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sometimes called P Prime but old British

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so I'm calling it P - and what will be

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the new direction u - so the mathematics

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we're doing here is how do we take P and

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you and turn it into P - and you - so

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the old way I did it was that I would

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calculates two points a and B on either

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side a B either side of p on this

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wavefront then I would calculate the

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points a dash and B dash and the way I

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do that is that a and B will be in

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different parts of the medium so they're

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in different refractive indices the

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refractive index will be different for a

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and B so it'll move at different speeds

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through through this the speed of light

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in a medium is inversely proportional to

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the refractive index so you have to

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divide the speed of light by the

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refractive index and that gives you the

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actual speed of light and that tell you

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how far it can go

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so this point P will move based on

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whatever the refractive index is here

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this point a will move based on whether

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the refractive index is here and end up

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in a - so you could calculate these two

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points a and B's and calculate some new

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points a dash and B dash and then that

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would give you this line here and you

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can use this line here to find out what

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the new you - would be because it's

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simply the vector which is perpendicular

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to this line

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I do us something of a an optimization

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or a simplification here instead of

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calculating these points a and B I just

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directly calculate the value of U from

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this value G where G is the unit

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00:12:37,389 --> 00:12:32,819
gradient of the refractive index field

273
00:12:41,869 --> 00:12:37,399
at P that's the UM perhaps a little bit

274
00:12:44,809 --> 00:12:41,879
hard to understand like I said it's from

275
00:12:47,569 --> 00:12:44,819
your brother hard to understand but the

276
00:12:51,519 --> 00:12:47,579
refractive index gradient that is how

277
00:12:57,079 --> 00:12:51,529
much is how much is the value of N

278
00:12:58,970 --> 00:12:57,089
changing in the X direction and in the

279
00:13:05,749 --> 00:12:58,980
in the X direction and in the Y

280
00:13:08,569 --> 00:13:05,759
direction per foot and that is a value G

281
00:13:11,239 --> 00:13:08,579
which is a vector value obviously to GX

282
00:13:12,650 --> 00:13:11,249
and Gy because it's how much is it

283
00:13:14,269 --> 00:13:12,660
changing X and what's is changing and

284
00:13:16,579 --> 00:13:14,279
why it's represented by an arrow here

285
00:13:19,929 --> 00:13:16,589
because it's going to be a direction it

286
00:13:21,980 --> 00:13:19,939
will be not necessarily a unit vector

287
00:13:24,079 --> 00:13:21,990
because it will have a certain length

288
00:13:31,189 --> 00:13:24,089
and the steeper the gradient the longer

289
00:13:33,889 --> 00:13:31,199
this this G will be so the meth is not

290
00:13:36,259 --> 00:13:33,899
too long but it's perhaps difficult to

291
00:13:37,999 --> 00:13:36,269
understand so let's go through it U is

292
00:13:39,350 --> 00:13:38,009
the unit vector or the rate erection we

293
00:13:42,199 --> 00:13:39,360
already know that it's this value here

294
00:13:45,139 --> 00:13:42,209
it's the unit vector along this this

295
00:13:47,840 --> 00:13:45,149
line of travel P is the current rate

296
00:13:49,549 --> 00:13:47,850
where are we starting a and B are two

297
00:13:54,439 --> 00:13:49,559
points on the way from perpendicular to

298
00:13:56,119 --> 00:13:54,449
u which they're going to be virtual

299
00:13:59,809 --> 00:13:56,129
points but right now imagine they are

300
00:14:02,509 --> 00:13:59,819
actually real points V now I see V as a

301
00:14:04,879 --> 00:14:02,519
- as a little underscore which means

302
00:14:08,059 --> 00:14:04,889
that it's a vector quantity V is the

303
00:14:10,340 --> 00:14:08,069
left rotated you so here's you and

304
00:14:14,600 --> 00:14:10,350
here's V it's rotated counterclockwise

305
00:14:18,110 --> 00:14:14,610
to the left by 90 degrees so it's the

306
00:14:20,480 --> 00:14:18,120
vector perpendicular to - you

307
00:14:24,079 --> 00:14:20,490
is the refractive index at P so it's the

308
00:14:29,299 --> 00:14:24,089
refractive index at this position here G

309
00:14:31,100 --> 00:14:29,309
as with it is the gradient of n at P now

310
00:14:34,009 --> 00:14:31,110
this is just explaining how you

311
00:14:36,860 --> 00:14:34,019
calculate rotated right and rotated left

312
00:14:41,210 --> 00:14:36,870
values rotate right it's just simply hue

313
00:14:43,280 --> 00:14:41,220
I minus X and rotate left is minus y

314
00:14:44,509 --> 00:14:43,290
minus X and let's explain why there's a

315
00:14:47,780 --> 00:14:44,519
little diagram here I should be fairly

316
00:14:50,629 --> 00:14:47,790
straightforward alright so as we have

317
00:14:54,410 --> 00:14:50,639
all these values we can figure out

318
00:14:58,670 --> 00:14:54,420
certain things we know that a is P plus

319
00:15:01,910 --> 00:14:58,680
B we're going to use an offset of 1 so

320
00:15:04,369 --> 00:15:01,920
we just use the unit vector to get the

321
00:15:05,989 --> 00:15:04,379
natural position of a and the B of

322
00:15:07,670 --> 00:15:05,999
course is P minus V because it's just in

323
00:15:15,170 --> 00:15:07,680
the other direction then we can

324
00:15:25,299 --> 00:15:15,180
calculate a dash as people's v8 a she's

325
00:15:29,419 --> 00:15:25,309
a plus u times D multiplied by n plus GB

326
00:15:32,169 --> 00:15:29,429
this is this is very frank it's a little

327
00:15:38,210 --> 00:15:32,179
confused but basically what that does is

328
00:15:42,259 --> 00:15:38,220
it takes the position a and he adds u

329
00:15:45,319 --> 00:15:42,269
times D scaled by a you deal with this

330
00:15:49,100 --> 00:15:45,329
lengthier from P - P - scaled by the

331
00:15:54,980 --> 00:15:49,110
refractive index at P which is n plus G

332
00:16:00,139 --> 00:15:54,990
dot V which is the component of G along

333
00:16:01,999 --> 00:16:00,149
V and which gives you how much the

334
00:16:06,079 --> 00:16:02,009
refractive index is changing in this

335
00:16:08,269 --> 00:16:06,089
direction from P to a and similarly we

336
00:16:11,629 --> 00:16:08,279
do the same thing for our B - we take B

337
00:16:15,590 --> 00:16:11,639
and we add u times D which is this

338
00:16:19,220 --> 00:16:15,600
vector scaled by n - G dot V which is

339
00:16:22,340 --> 00:16:19,230
going to be how much G varies in this

340
00:16:25,400 --> 00:16:22,350
direction so that will actually give you

341
00:16:29,030 --> 00:16:25,410
a virtual a and B so now we can

342
00:16:31,820 --> 00:16:29,040
calculate another vector B - a - which

343
00:16:35,840 --> 00:16:31,830
is this one here which is

344
00:16:37,550 --> 00:16:35,850
to be a - - B - which if you take these

345
00:16:41,840 --> 00:16:37,560
just works out to be something nice

346
00:16:46,490 --> 00:16:41,850
which doesn't involve a or b so - v + 2

347
00:16:49,250 --> 00:16:46,500
UD times govt and we can work out quite

348
00:16:51,350 --> 00:16:49,260
simply that you - which is the new unit

349
00:16:56,540 --> 00:16:51,360
vector in this direction is the right

350
00:16:58,340 --> 00:16:56,550
rotated vector B 2 a so it is B 2 a is

351
00:17:00,080 --> 00:16:58,350
that vector and then we were setting it

352
00:17:01,970 --> 00:17:00,090
right 90 degrees and then we take the

353
00:17:03,710 --> 00:17:01,980
unit vector which is what I'm indicating

354
00:17:07,100 --> 00:17:03,720
with these bars on either side and that

355
00:17:09,560 --> 00:17:07,110
gives you a new direction vector which

356
00:17:13,460 --> 00:17:09,570
gives us this nice little equation here

357
00:17:19,180 --> 00:17:13,470
you - you write rotated V + UD terms

358
00:17:24,140 --> 00:17:19,190
govt which slightly simplifies to this

359
00:17:27,740 --> 00:17:24,150
you - is U plus V times D times G dot V

360
00:17:33,560 --> 00:17:27,750
because the U and V are the right

361
00:17:36,470 --> 00:17:33,570
rotated versions of V and u then we have

362
00:17:38,630 --> 00:17:36,480
simply P - is P Plus u times D now we

363
00:17:42,080 --> 00:17:38,640
could multiply that by n times P but it

364
00:17:43,400 --> 00:17:42,090
doesn't really matter because the it

365
00:17:44,660 --> 00:17:43,410
doesn't really make much difference but

366
00:17:46,880 --> 00:17:44,670
I think I do actually do this very much

367
00:17:48,470 --> 00:17:46,890
not sure but if you can introduce this

368
00:17:51,110 --> 00:17:48,480
term into the code it doesn't really

369
00:17:52,880 --> 00:17:51,120
make very much difference all right now

370
00:17:56,030 --> 00:17:52,890
if thoroughly confuse everybody we can

371
00:17:58,660 --> 00:17:56,040
go back to the code and we can see that

372
00:18:02,810 --> 00:17:58,670
what we have is just simply an

373
00:18:07,580 --> 00:18:02,820
implementation of those equations we

374
00:18:11,930 --> 00:18:07,590
have NeoGeo v DG v and then you witness

375
00:18:16,750 --> 00:18:11,940
the new ux 1 UI 1 which is the work you

376
00:18:18,860 --> 00:18:16,760
do for you - calculated here and then

377
00:18:20,960 --> 00:18:18,870
normalizing it here by dividing by the

378
00:18:23,270 --> 00:18:20,970
length and then returning it and doing

379
00:18:26,480 --> 00:18:23,280
at x equals ya I don't actually add in

380
00:18:27,800 --> 00:18:26,490
the refractive index - you can you try

381
00:18:29,510 --> 00:18:27,810
it with them without but it doesn't

382
00:18:31,930 --> 00:18:29,520
actually make any difference so I left

383
00:18:36,550 --> 00:18:31,940
it off for speed

384
00:18:39,790 --> 00:18:36,560
ok that is how that works now

385
00:18:41,650 --> 00:18:39,800
I must admit it's very complicated

386
00:18:44,050 --> 00:18:41,660
sounding I wouldn't expect most people

387
00:18:47,170 --> 00:18:44,060
to understand it but if you want to

388
00:18:49,630 --> 00:18:47,180
understand how the code works then

389
00:18:51,360 --> 00:18:49,640
hopefully that would lead you through it

390
00:18:55,750 --> 00:18:51,370
and you can go through it step by step

391
00:18:57,870 --> 00:18:55,760
and follow along with what I just did

392
00:19:00,690 --> 00:18:57,880
and check the math and then see that

393
00:19:08,920 --> 00:19:00,700
these equations hearings step forward

394
00:19:13,060 --> 00:19:08,930
match that math okay so other things in

395
00:19:14,560 --> 00:19:13,070
the code that are math related are how

396
00:19:19,030 --> 00:19:14,570
do we actually calculate the refractive

397
00:19:24,370 --> 00:19:19,040
index the refractive index is calculated

398
00:19:26,020 --> 00:19:24,380
let's see in this function somewhere

399
00:19:27,520 --> 00:19:26,030
here it uses this function here gradient

400
00:19:29,770 --> 00:19:27,530
because we're using the gradient of the

401
00:19:32,350 --> 00:19:29,780
refractive index so how do we calculate

402
00:19:36,220 --> 00:19:32,360
the gradient of GX and Gy we go into

403
00:19:38,380 --> 00:19:36,230
that and we see gradient x and y is just

404
00:19:41,260 --> 00:19:38,390
simply taking the refractive index

405
00:19:42,880 --> 00:19:41,270
that's half a foot on either side and

406
00:19:46,030 --> 00:19:42,890
then returning the difference between

407
00:19:48,310 --> 00:19:46,040
them and that gives you the gradient for

408
00:19:51,040 --> 00:19:48,320
the X and the gradient for the Y so it's

409
00:19:53,020 --> 00:19:51,050
very very simple really so what does

410
00:19:55,990 --> 00:19:53,030
this function do refractive index let's

411
00:19:58,270 --> 00:19:56,000
pop into that one I call this function

412
00:20:02,320 --> 00:19:58,280
here which is refractive index from H

413
00:20:05,140 --> 00:20:02,330
and that function here is always a it's

414
00:20:09,280 --> 00:20:05,150
a virtual function that it actually

415
00:20:11,890 --> 00:20:09,290
comes from this or this function here

416
00:20:14,950 --> 00:20:11,900
which calculates the function which

417
00:20:18,640 --> 00:20:14,960
ultimately calls this function here

418
00:20:20,530 --> 00:20:18,650
refractive index old which the oldest

419
00:20:24,580 --> 00:20:20,540
indicators before I did all this this

420
00:20:27,580 --> 00:20:24,590
fancy optimization and and or or they're

421
00:20:31,750 --> 00:20:27,590
doing a bit of a bit of a camel ROM

422
00:20:34,890 --> 00:20:31,760
spline interpolation down here which is

423
00:20:37,570 --> 00:20:34,900
an optimization so the actual relevant

424
00:20:40,630 --> 00:20:37,580
code for how we calculate the refractive

425
00:20:48,400 --> 00:20:40,640
index is in refractive index old which

426
00:20:49,850 --> 00:20:48,410
is right here which just takes the drop

427
00:20:52,430 --> 00:20:49,860
into account to calculate the height

428
00:20:55,130 --> 00:20:52,440
above the surface it calls pressure at

429
00:20:56,780 --> 00:20:55,140
altitude to calculate what's that

430
00:21:00,770 --> 00:20:56,790
pressure would be at that altitude which

431
00:21:05,840 --> 00:21:00,780
uses this equation here which comes from

432
00:21:09,919 --> 00:21:05,850
this Wikipedia page it calculates the

433
00:21:11,630 --> 00:21:09,929
temperature from the curve from this

434
00:21:13,130 --> 00:21:11,640
curve here you have to look in the it's

435
00:21:15,140 --> 00:21:13,140
just basically interpolating along this

436
00:21:21,620 --> 00:21:15,150
curve is very simple nothing complicated

437
00:21:25,190 --> 00:21:21,630
there it takes the RH from the default

438
00:21:27,799 --> 00:21:25,200
RH if you know editing the RH which will

439
00:21:29,960 --> 00:21:27,809
be 50 generally if you are aged in the

440
00:21:31,760 --> 00:21:29,970
RH it comes from this graph here and

441
00:21:34,520 --> 00:21:31,770
similarly it's interpolated along that

442
00:21:36,560 --> 00:21:34,530
curve and then the refractive index is

443
00:21:38,810 --> 00:21:36,570
calculated from that temperature and the

444
00:21:42,289 --> 00:21:38,820
pressure and the RH by calling this

445
00:21:45,919 --> 00:21:42,299
function here air index Sidor which if

446
00:21:49,610 --> 00:21:45,929
we go into that we can see is the full

447
00:21:51,380 --> 00:21:49,620
Sidor equation from NIST website which

448
00:21:53,710 --> 00:21:51,390
is sudo is well known in the scientific

449
00:21:55,850 --> 00:21:53,720
literature as being the most accurate

450
00:21:58,159 --> 00:21:55,860
equation but of course it's incredibly

451
00:21:59,630 --> 00:21:58,169
complicated there's actually other

452
00:22:01,370 --> 00:21:59,640
equations you can use in here there's

453
00:22:03,560 --> 00:22:01,380
the application which is a bit simpler

454
00:22:05,510 --> 00:22:03,570
or there's this missed shot fourth

455
00:22:08,419 --> 00:22:05,520
equation which is relatively simple and

456
00:22:12,740 --> 00:22:08,429
straightforward but that essentially is

457
00:22:13,430 --> 00:22:12,750
all the math that is relevant in in the

458
00:22:15,860 --> 00:22:13,440
simulator

459
00:22:18,230 --> 00:22:15,870
so if hopefully that will explain to

460
00:22:21,919 --> 00:22:18,240
people who want to know who put in the

461
00:22:24,380 --> 00:22:21,929
time how it actually works and if you

462
00:22:27,500 --> 00:22:24,390
want to replicate it or if you want to

463
00:22:29,270 --> 00:22:27,510
verify to if you want to improve it then

464
00:22:32,570 --> 00:22:29,280
it should give you a good understanding

465
00:22:33,860 --> 00:22:32,580
of the technical details behind it

466
00:22:35,470 --> 00:22:33,870
there's other stuff like how it does the

467
00:22:38,930 --> 00:22:35,480
rendering but that's just you know

468
00:22:40,789 --> 00:22:38,940
straightforward programming and just

469
00:22:44,080 --> 00:22:40,799
basically data management it's not

470
00:22:47,600 --> 00:22:44,090
really pretty interesting the actual

471
00:22:49,640 --> 00:22:47,610
technical stuff is this stuff right here

472
00:22:51,530 --> 00:22:49,650
the mathematics the equations the

473
00:22:55,150 --> 00:22:51,540
implementation of the those equations

474
00:22:59,120 --> 00:22:55,160
and how we step through them with our

475
00:23:02,659 --> 00:22:59,130
gradient there's various issues with it

476
00:23:03,710 --> 00:23:02,669
like the step size the calculation of

477
00:23:07,970 --> 00:23:03,720
the gradient

478
00:23:10,610 --> 00:23:07,980
is based on a step of one and you know

479
00:23:13,909 --> 00:23:10,620
things could be improved and it's not

480
00:23:15,770 --> 00:23:13,919
the same as the the the formula that

481
00:23:17,960 --> 00:23:15,780
Andy Young and Walter bisland used which

482
00:23:22,070 --> 00:23:17,970
uses curved segments but that is based

483
00:23:23,600 --> 00:23:22,080
on a much simpler version of the

484
00:23:25,430 --> 00:23:23,610
refractive index essentially it's using

485
00:23:26,930 --> 00:23:25,440
this very simple calculation which

486
00:23:28,580 --> 00:23:26,940
doesn't really use the full suit or

487
00:23:30,560 --> 00:23:28,590
equation so even though it's more

488
00:23:32,690 --> 00:23:30,570
accurate in some ways and it's going

489
00:23:33,980 --> 00:23:32,700
through a curve it's less accurate in

490
00:23:36,020 --> 00:23:33,990
this using this very very simple

491
00:23:40,220 --> 00:23:36,030
equation which is more resembles more

492
00:23:42,950 --> 00:23:40,230
the NIST equation here alright so I'm

493
00:23:46,039 --> 00:23:42,960
going to sign off because I think I'm

494
00:23:47,600 --> 00:23:46,049
done for now so if anybody sat through