File:FS QV dia.png

Shows the largest quarter circle within a square.

Base is the square ${\displaystyle [ABCD]}$ of side length s.
Inscribed is the largest possible quarter circle of radius ${\displaystyle r_{1}}$
.

0) The side length of the base square: ${\displaystyle |AB|=|BC|=|CD|=|AD|=s_{0}}$
1) Radius of the quarter circle: ${\displaystyle r_{1}=|AB|=s_{0}}$

0) Perimeter of base square: ${\displaystyle P_{0}=|AB|+|BC|+|CD|+|AD|=4\cdot s_{0}}$
1) Perimeter of the quarter circle: ${\displaystyle P_{1}=|AB|+|BD|+|AD|=r_{1}+{\frac {\pi \cdot r_{1}}{2}}+r_{1}=(2+{\frac {\pi }{2}})\cdot r_{1}={\frac {4+\pi }{2}}\cdot r_{1}={\frac {4+\pi }{2}}\cdot s_{0}}$

0) Area of the base square: ${\displaystyle A_{0}=s_{0}^{2}}$
1) Area of the inscribed quarter circle: ${\displaystyle A_{1}={\frac {\pi \cdot r_{1}^{2}}{4}}={\frac {\pi }{4}}\cdot s_{0}^{2}}$

Centroid positions are measured from the lower left point of the square.
0) Centroid positions of the base square: ${\displaystyle x_{0}=0\quad y_{0}=0}$
1) Centroid positions of the inscribed quarter circle: ${\displaystyle x_{1}=|S_{0}S_{1}|}$
${\displaystyle \quad \Leftrightarrow x_{1}=|AS_{1}|-|AS_{0}|}$

${\displaystyle \quad \Leftrightarrow x_{1}={\frac {4\cdot r_{1}}{3\cdot \pi }}-{\frac {s_{0}}{2}}}$

${\displaystyle \quad \Leftrightarrow x_{1}={\frac {4\cdot s_{0}}{3\cdot \pi }}-{\frac {s_{0}}{2}}}$

${\displaystyle \quad \Leftrightarrow x_{1}=\left({\frac {4}{3\cdot \pi }}-{\frac {1}{2}}\right)\cdot s_{0}}$

${\displaystyle \quad \Leftrightarrow x_{1}=\left({\frac {2\cdot 4-3\pi }{2\cdot 3\cdot \pi }}\right)\cdot s_{0}}$

${\displaystyle \quad \Leftrightarrow x_{1}={\frac {8-3\pi }{6\pi }}\cdot s_{0}}$

${\displaystyle \quad \Leftrightarrow y_{1}=x_{1}={\frac {8-3\pi }{6\pi }}\cdot s_{0}}$

In the normalised case the area of the base is set to 1.
${\displaystyle ||ABCD||=1\Rightarrow s_{0}^{2}=1\Leftrightarrow s_{0}=1}$

0) Segment of the base square: ${\displaystyle s_{0}=1}$
1) Segment of the inscribed quarter circle: ${\displaystyle r_{1}=s_{0}=1}$

0) Perimeter of base square: ${\displaystyle P_{0}=4\cdot s_{0}=4}$
1) Perimeter of the inscribed quarter circle: ${\displaystyle P_{1}={\frac {4+\pi }{2}}\cdot 1=3.570796...}$
S) Sum of perimeters ${\displaystyle P_{s}=(6+{\frac {\pi }{2}})=7.570796...}$

0) Area of the base square: ${\displaystyle A_{0}=1}$
1) Area of the inscribed quarter circle: ${\displaystyle A_{1}={\frac {\pi \cdot r_{1}^{2}}{4}}={\frac {\pi }{4}}\cdot s_{0}={\frac {\pi }{4}}\approx 0.785}$

Centroid positions are measured from the centroid of the base square

0) Centroid positions of the base square: ${\displaystyle x_{0}=0\quad y_{0}=0}$
1) Centroid positions of the inscribed quarter circle: ${\displaystyle x_{1}={\frac {8-3\pi }{6\pi }}=-0.0755868...\quad y_{1}=x_{1}=-0.0755868...}$

The distance between the centroid of the base element and the centroid of the quarter circle is:
${\displaystyle d={\sqrt {(x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}}}=0,1068959...}$

Apart of the base element there is only one shape allocated. Therefore the integer part of the identifying number is 1.
The decimal part of the identifying number is the decimal part of the sum of the perimeters and the distances of the centroids in the normalised case.

${\displaystyle decimalpart(7.5707963...+0.1068959...)=decimalpart(7.6776922...)=.6776922}$

So the identifying number is: ${\displaystyle 1.6776922}$

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