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Warframe And Curve Approximations (With Loki's Famous 'grin') (Part Eight)


Renegade343
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So, from my main hub thread, I received the following request: 

 

How about approximating Lokis famous 'grin'? 

And since I have a freshly built Loki, I decided to do it. 

 

This time, however, the Arsenal screen is not sufficient enough to clearly see the curve, so for the first time, I have to break the rule. I have to set Loki to have a Noble animation set (so he will have a straight back and head), then turned the third person camera very carefully to make the camera be aimed approximately straight at the back of his helmet. Thus, I obtained this image (dashed box is... I think you will know by now): 

 

DEOMLQQ.png

 

Then, doing all the stuff with Geogebra and such (no rotations needed) (110.27 PPI, image size: 0.03732m x 0.01612m), I then (being a lazy person), decided to use polynomial interpolation (Monomial basis yet again [that tool is so easy and so addictive to use]) for three points (meaning a parabola). And thus, I obtained points D, E, and F (scanner is still broken [i think one factor for the long wait might be because I am too lazy to bring the scanner to be fixed]): 

 

M1pbIlb.png

 

The result of the Monomial basis based on these three points is below: 

 

XDLshGm.png

 

Afterwards, noticing that the other side is also approximately the same curve as the curve I approximated the function of, I then plotted point G: 

 

n0AJo3o.png

 

Doing a few graph transformation formula and techniques (y-intercept shift, x-coordinate shift), I obtained the approximate function for the other side, along with the point of intersection for the two curves (Point H): 

 

kFCHYlE.png

 

Finally, I obtained the following piecewise function (since my calculator is not cooperating today and dies when I attempt to try solving any simultaneous equations with ≥4 equations, so I have to make do with three simultaneous equations [not to mention the fact that I feel under the weather, so I am not motivated to solve any simultaneous equations with ≥4 equations by hand]): 

 

QZotOsB.png

 

But anyways, the piecewise function does fit the 'grin', although I suspect (with good reason) that a quartic function will work almost, if not just, as well.

 

Next time, I will do just that (once I get a new calculator and I feel better).

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