# ShinyTouch Progress Update + Fresnel’s Equations

So I was looking through wikipedia to find out if there were some magical equations to govern how it should mix the color of the background screen contents and the reflection and make the application work better. I think Fresnel’s equations fit that description. It basically gives the reflectiveness of the substance from information about the substance, the surrounding substance (air) and the angle of incidence.

Well, this image really is quite intimidating (maybe i'm not not elitist enough). I won't even pretend to understand it, but it looks like Fresnel's equations with different values of n1 and n2 (some ratio for different temperatures). And is the plot on the right the same Total Internal Reflection in FTIR?

It’s quite interesting, partly because the shinyness (and thus the ratios used to combine the background color with the finger to compare) depends on the angle of the webcam to the finger, which depends on the distance (yay for trigonometry?). So the value used isn’t the 50-50 ration that it currently uses in the algorithm universally, but it’s dependent on variables to Fresnel’s Equations and the distance of the finger.

I forgot what this was supposed to describe

Anyway, time for a graphic that doesn’t really explain anything because I lost my train of thought while trying to understand how to use Inkscape!

So here’s something more descriptive. The two hands (at least it’s not 3, and why they’re just lines with no fingers isn’t my fault) and they’re positioned at different locations, one (hand 1) is close to the camera while the second (hand 2) is quite far away. Because of magic and trigonometry, the angle of the hand is greater when it’s further away. Also, this plugs into Fresnel’s Equations which mean the surface is shinier for where hand 2 is touching while it’s less shiny for hand 1. So the algorithm has to adjust for the variation (and if this works, then it might not need the complex region specific range values).

Notice how Angle 2 for hand 2 is much larger than angle 1 because of how it's farther from the camera than hand 1

So here it’s pretty ideal to have the angle be pretty extreme right? Those graphs sure seem imply that extreme angles are a good thing. But no, because quite interestingly, the more extreme the angle is, then the less accurate the measures from the x axis become. So in the image below, you can see that cam b is farther from the monitor (and thus has a greater angle from the monitor) and it can discern far more accurately depth than than cam a. The field of view for cam a is squished down to that very thin angle whereas the cam b viewing area is far larger. Imagine if there was a cam c which was mounted directly in front of the monitor, it would suffer from no compression of the x axis like a or b but instead it has full possible depth.

the more extreme the angle is, then the lower the resolution of the usable x axis becomes. So while you get better accuracy (shinier = easier to detect) the accuracy point you can reduce it to declines proportionally.

So for the math portion, interestingly, the plot for the decline in % of the total possible width is equivalent to the 1-sin() (I think, but if I’m wrong then it could be cos() and i suck at math anyway).

So since it's a given that angle C is *always* 90 degrees, you can think of precision to be 2a (or at least that's what I think, I'm probably wrong though, but if you're measuring A from the center of the monitor, then it's no longer 2a but just a instead). So you can figure that the % of precision decreases as the 2*sin(A)

So if you graph out 1-sin(n) then you get a curve where it starts at 100% when the angle the camera is positioned at is 0 degrees from an imaginary line perpendicular to the center of the surface, and it approaches 0% as the degree measure reaches 90 deg.

So interestingly when you plot it, basically what happens is a trade-off between the angle the camera is at and the precision (% of ideal maximum horizontal resolution) and accuracy (shinyness of reflection). I had the same theory a few days ago, even before I discovered Fresnel’s Equations, though mine was more linear. I thought that it was just a point in which the values dropped for the shinyness. I thought that the reason the monitor was shinier from the side is that it was beyond the intended viewing angle, so since there is less light at the direction, the innate shinyness is more potent.

So what does this mean for the project? Well, it confirms my initial thoughts that this is far too complicated for me to do alone, and makes me quite sad (partly because of the post titled Fail that was published in january). It’s really far too complicated for me. Right now the algorithm I use is very approximate (and noticably so). The formula improperly adjusts for perception and so if you try to draw a straight line across the monitor, you end up with a curved section of a sinusoidal-wave.

Trying to draw a straight line across the screen ends up looking curved because it uses a linear approximate distortion adjustment algorithm. Note that the spaces between the bars is because of the limited horizontal resolution, partly due to the angle, mostly due to hacks for how slow python is.

So it’s far more complicated than I could have imagined at first, and I imagined it as far too complicated for me to venture in this alone. But I’m trying even with this sub-ideal situation. So the rest of the algorithm for now will also remain with more linear approximations. I’m going to experiment in making more linear approximations of the plot of Fresnel’s equations. And hopefully it’ll work this time.

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