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Warframe And Curve Approximations (With Saryn's Breast Area) (Part Four)


Renegade343
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(As of 2014-09-08, the series Warframe and Taylor's Expansion will be renamed to Warframe and Curve Approximations, since I am starting to learn other methods of curve approximations, in which I may use them if applicable.)

 

(And I still need to link the series to my profile page. So much for reminding myself.)

 

From Part Three of the series, I have received a request from a member about approximating a curve: 

 



Saryn's underboob or Rhino's codpiece

Now, since I do not have Rhino, I have decided to choose the first option, since I do have Saryn. 

 

So, going to the Arsenal, setting Saryn to have no animation set, and turning her π/4 radians to her right, I obtained the following picture (dashed box indicates the area we will be doing a curve approximation [and since I do not know what the hell is an 'underboob', I just went and decided to crop out the breast area]): 

 

PRaib1u.png

 

Then, turning the image π/4 radians clockwise in order to obtain a function (since one-to-many cannot be called a function [see the formula of a circle]), I then placed it into Geogebra, scaled the axis to be in meters, and manipulated it so the image would scale with the axis when I zoom in or out (110.27 PPI, image size: 0.01428m x 0.01658m). Afterwards, suspecting that the curve could be a cubic function, given the rather linear slope at the right side of the maximum point of Saryn's breast (which a parabola would not really have [probably unless its axis of symmetry is not parallel to the y-axis or x-axis, but that is out of my knowledge for now]), I plotted four points: One at the maximum point, two at the limits, and one random point, to obtain this result: 

 

6bsQQTX.png

 

Then, I used the technique of polynomial interpolation (in layman's terms, finding a polynomial, given some points, that would go through said points) to find the curve approximation of Saryn's breast (since polynomial interpolation does have some margin of errors, as with almost all function approximations). The particular technique I will be using is called the Monomial basis, since this is the only one I understand the theory behind it and know how to do it (for a reference, see this website: http://www.math.uakron.edu/~kreider/num1/Lecture_14_poly_interp.pdf). 

 

And so, taking the coordinate of the points E, G, H, I, I started to do the Monomial basis calculations, and obtained these results: 

 

fV38QsM.jpg

 

As there were two answers I could obtain from this by substituting the values into a different equation (as far as I have tested, and for some reason which I still have to find out [the two answers]), I took the more accurate answer, and wrote it down in the above image. And so, I obtained the function, and then plotted it into Geogebra: 

 

kIgykUG.png

 

As it can be seen, the function does stick rather closely to the curvature of Saryn's breasts, but deviates starting from x = 0.01m. What I suspect are two things that could cause this: 

 

1. I may have make an error in my calculations, causing the inaccuracy at the end portion. 

2. The approximation may be too inaccurate, and may need a leading polynomial of a higher power (say a quartic or quintic polynomial) to make the approximation more accurate. 

 

And so there is my piece of art (although the accuracy does have room for improvement). And just to say that I will continue learning the other methods doing curve approximations, so that I will not be restricted by weird problems when I am trying to approximate the function of a curve in Warfarme. 

 

But the technique is worth learning (and I need to get Matlab soon if I am going to continue on with this). 

 

EDIT: And I forgot to ask this (since I do not have Vauban yet, so I cannot attend to Letter13's request): What curve do you want me to approximate in Warframe next time?

Edited by Renegade343
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Then, turning the image π/4 radians clockwise in order to obtain a function (since one-to-many cannot be called a function [see the formula of a circle]), I then placed it into Geogebra, scaled the axis to be in meters, and manipulated it so the image would scale with the axis when I zoom in or out (110.27 PPI, image size: 0.01428m x 0.01658m). Afterwards, suspecting that the curve could be a cubic function, given the rather linear slope at the right side of the maximum point of Saryn's breast (which a parabola would not really have [probably unless its axis of symmetry is not parallel to the y-axis or x-axis, but that is out of my knowledge for now]), I plotted four points: One at the maximum point, two at the limits, and one random point, to obtain this result: 

Can't you transform the image by roughly 45 degrees clockwise with respect to the origin? That would allow for the image to be a parabolic curve which would be a quadratic function instead of a cube function. Also if this was a cubic function shouldn't the point after the boob connects with the body be a stationary or inflexion point rather than a non-stationary point? Just wondering.

 

Actually taking more time to think this through a quadric function might also be applicable, where it would be a inflexion point after the maximum point followed by another maximum point. It just feels like the inflexion point is missing. I don't know how to represent the inflexion point on the graph though so sorry.

Edited by Jacate
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Can't you transform the image by roughly 45 degrees clockwise with respect to the origin? That would allow for the image to be a parabolic curve which would be a quadratic function instead of a cube function. Also if this was a cubic function shouldn't the point after the boob connects with the body be a stationary or inflexion point rather than a non-stationary point? Just wondering.

I do not know how to calculate parabola conics through rotation yet, so I am not dealing with this one until I get my hands on revising conics again. 

 

And the suspicion is from the fact that the left side of the maximum point looks like a linear line, which does not fit with a parabola with axis of symmetry parallel to the y axis (and as I said, I do not know how to do parabola conics yet). Also, from the Arsenal screen, it can be seen that the point after the breast connects with the body is an abrupt change in the curve (i.e.: dy/dx vs. x graph will be discontinuous), which indicates that if I am to include that area as well, the function obtained will be a piecewise function, and not a full cubic. 

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Actually taking more time to think this through a quadric function might also be applicable, where it would be a inflexion point after the maximum point followed by another maximum point. It just feels like the inflexion point is missing. I don't know how to represent the inflexion point on the graph though so sorry.

Inflexion point is when dy/dx vs. x graph is at a maximum point or at a minimum point (try plotting dy/dx of (x-1)(x-2)(x-3), then find the minimum (or maximum, could never remember for that graph) point of the dy/dx function. That will be the inflexion point of the original function. 

 

And besides, polynomial interpolation does take care of some of those problems rather nicely (then again, the ones I done all have nice, round values, so there is that to consider [since this time, I am dealing with a lot of decimals and such]). 

Edited by Renegade343
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Inflexion point is when dy/dx vs. x graph is at a maximum point or at a minimum point (try plotting dy/dx of (x-1)(x-2)(x-3), then find the minimum (or maximum, could never remember for that graph) point of the dy/dx function. That will be the inflexion point of the original function. 

 

And besides, polynomial interpolation does take care of some of those problems rather nicely (then again, the ones I done all have nice, round values, so there is that to consider [since this time, I am dealing with a lot of decimals and such]). 

Thanks for the explanation =D

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  • 3 months later...

(As of 2014-09-08, the series Warframe and Taylor's Expansion will be renamed to Warframe and Curve Approximations, since I am starting to learn other methods of curve approximations, in which I may use them if applicable.)

 

(And I still need to link the series to my profile page. So much for reminding myself.)

 

From Part Three of the series, I have received a request from a member about approximating a curve: 

 

Now, since I do not have Rhino, I have decided to choose the first option, since I do have Saryn. 

 

So, going to the Arsenal, setting Saryn to have no animation set, and turning her π/4 radians to her right, I obtained the following picture (dashed box indicates the area we will be doing a curve approximation [and since I do not know what the hell is an 'underboob', I just went and decided to crop out the breast area]): 

 

PRaib1u.png

 

Then, turning the image π/4 radians clockwise in order to obtain a function (since one-to-many cannot be called a function [see the formula of a circle]), I then placed it into Geogebra, scaled the axis to be in meters, and manipulated it so the image would scale with the axis when I zoom in or out (110.27 PPI, image size: 0.01428m x 0.01658m). Afterwards, suspecting that the curve could be a cubic function, given the rather linear slope at the right side of the maximum point of Saryn's breast (which a parabola would not really have [probably unless its axis of symmetry is not parallel to the y-axis or x-axis, but that is out of my knowledge for now]), I plotted four points: One at the maximum point, two at the limits, and one random point, to obtain this result: 

 

6bsQQTX.png

 

Then, I used the technique of polynomial interpolation (in layman's terms, finding a polynomial, given some points, that would go through said points) to find the curve approximation of Saryn's breast (since polynomial interpolation does have some margin of errors, as with almost all function approximations). The particular technique I will be using is called the Monomial basis, since this is the only one I understand the theory behind it and know how to do it (for a reference, see this website: http://www.math.uakron.edu/~kreider/num1/Lecture_14_poly_interp.pdf). 

 

And so, taking the coordinate of the points E, G, H, I, I started to do the Monomial basis calculations, and obtained these results: 

 

fV38QsM.jpg

 

As there were two answers I could obtain from this by substituting the values into a different equation (as far as I have tested, and for some reason which I still have to find out [the two answers]), I took the more accurate answer, and wrote it down in the above image. And so, I obtained the function, and then plotted it into Geogebra: 

 

kIgykUG.png

 

As it can be seen, the function does stick rather closely to the curvature of Saryn's breasts, but deviates starting from x = 0.01m. What I suspect are two things that could cause this: 

 

1. I may have make an error in my calculations, causing the inaccuracy at the end portion. 

2. The approximation may be too inaccurate, and may need a leading polynomial of a higher power (say a quartic or quintic polynomial) to make the approximation more accurate. 

 

And so there is my piece of art (although the accuracy does have room for improvement). And just to say that I will continue learning the other methods doing curve approximations, so that I will not be restricted by weird problems when I am trying to approximate the function of a curve in Warfarme. 

 

But the technique is worth learning (and I need to get Matlab soon if I am going to continue on with this). 

 

EDIT: And I forgot to ask this (since I do not have Vauban yet, so I cannot attend to Letter13's request): What curve do you want me to approximate in Warframe next time?

I don't.....understand anything you just said. Ok i understand some of the stuff are you a math teacher?

Edited by RAVEN-kipper
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  • 3 weeks later...

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