Nickboey22305

Nick Boey

The Boey Equation

The Boey Equation was made to parabolically rate something on an arbitrary scale. It is as thus:

$S={\frac {Bl-Bh}{(Rb-Rp)^{2}}}(Rr-Rp)^{2}+Bh$

This is if:

Bl = The lowest possible score (i.e. 1)

Bh = The highest possible score (i.e. 10)

Rl = The score that would be given a score of Bl

Rp = The score that would be given a score of Bh

Rr = The score to be evaluated

S = The result of Rr being rated on Rp on a scale of Bl to Bh

$\theta =90-{\frac {(n-2)90}{n}}$

$w=r\left(1-\cos \left({\frac {180}{n}}\right)\right)$

$l={\sqrt {2(r-t)^{2}\left(1-\cos \left({\frac {360}{n}}\right)\right)}}+2w\tan \theta$

$l={\sqrt {2(r-t)^{2}\left(1-\cos \left({\frac {360}{n}}\right)\right)}}+2r\left(1-\cos \left({\frac {180}{n}}\right)\right)\tan \left(90-{\frac {(n-2)90}{n}}\right)$

In example, presume that one would want to rate the 6.5 parabolically on a scale of 1 to 10, with a 5 being given the perfect score of 10 and 3 given the lowest score of 1. Thus:

Bl = 1

Bh = 10

Rl = 3

Rp = 5

Rr = 6.5

Sub these into the equation:

$S={\frac {1-10}{(3-5)^{2}}}(6.5-5)^{2}+10$

Which solves to give S = 4.9375, which is the answer of rating 6.5 on 5 on a scale of 1 to 10 with 3 receiving 1. If S is less than Bl, then S = Bl

Non-symmetrical Boey Equation

In the case above, the value of Rr that recieves Bl is actually both Rl and Rp + (Rp - Rl). This makes the curve symmetrical, though the equation gains more functionality when the curve is non-symmetrical. To do this, a different value of Rl must be used if Rr is less than or more than Rp. Thus the gradient of the curve on either side of Rp is different.

In the symmetrical equation, if there is a constant c, then a line S = c, if it intersects the curve (i.e. c is less than Rp), will have a gradient m at the first intersection and a gradient -m at the second intersection. However, in the non-symmetrical equation, a line S = c, if it intersects the curve, will have a gradient m at the first intersection but a different gradient at the second intersection.