File:FS CV.1C dia.png

Shows the largest circle within a circle that contains the largest quarter circle.

Base is the circle of given radius ${\displaystyle r_{0}}$ around point ${\displaystyle S_{0}}$ and the resulting quarter circle ${\displaystyle [ABC]}$ of radius ${\displaystyle r_{1}}$ around point ${\displaystyle A}$. From the developing of FS_CV we know that ${\displaystyle r_{1}={\sqrt {2}}\cdot r_{0}}$.
Inscribed is the largest possible circle in the shape ${\displaystyle [ABC]}$ having radius ${\displaystyle r_{2}}$ around point ${\displaystyle S_{2}}$. This is also the largest circle in a CV type sangaku.

In order to find radius ${\displaystyle r_{2}}$ of the circle, the following reasoning is used:
The line segment ${\displaystyle [AH]}$ corresponds to the radius ${\displaystyle r_{1}}$ of the quarter circle around ${\displaystyle A}$. So we have:
(1)${\displaystyle \quad |AH|=|AS_{2}|+|S_{2}H|=r_{1}={\sqrt {2}}\cdot r_{0}}$

The line segments ${\displaystyle [S_{2}F],[S_{2}G],[S_{2}H]}$ correspond to the radius ${\displaystyle r_{2}}$ of the circle around ${\displaystyle S_{2}}$. This means that:
(2)${\displaystyle \quad |S_{2}F|=|S_{2}G|=|S_{2}H|=r_{2}}$

It also means that the points ${\displaystyle [AGEF]}$ form a square with side length ${\displaystyle r_{2}}$. So applying the Pythagorean theorem we get:
(3)${\displaystyle \quad |AS_{2}|^{2}=|S_{2}F|^{2}+|AF|^{2}\quad \Leftrightarrow |AS_{2}|^{2}=2\cdot r_{2}^{2}\quad \Leftrightarrow |AS_{2}|={\sqrt {2}}\cdot r_{2}}$

Using (2) and (3) on (1) gives:
${\displaystyle \quad |AH|={\sqrt {2}}\cdot r_{2}+r_{2}={\sqrt {2}}\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow ({\sqrt {2}}+1)\cdot r_{2}={\sqrt {2}}\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow r_{2}={\frac {\sqrt {2}}{{\sqrt {2}}+1}}\cdot r_{0}={\frac {\sqrt {2}}{{\sqrt {2}}+1}}\cdot {\frac {{\sqrt {2}}-1}{{\sqrt {2}}-1}}\cdot r_{0}=(2-{\sqrt {2}})\cdot r_{0}}$

0) The radius of the base circle ${\displaystyle =r_{0}}$
1) Radius of the quarter circle ${\displaystyle r_{1}={\sqrt {2}}\cdot r_{0}\quad \approx 1.414\cdot r_{0}}$
2) Radius of the additional circle ${\displaystyle r_{2}=(2-{\sqrt {2}})\cdot r_{0}\quad \approx 0.586\cdot r_{0}}$

0) Perimeter of base circle ${\displaystyle P_{0}=2\cdot \pi \cdot r_{0}}$
1) Perimeter of the quarter circle ${\displaystyle P_{1}={\frac {4+\pi }{\sqrt {2}}}\cdot r_{0}}$
2) Perimeter of additional circle ${\displaystyle P_{2}=2\cdot \pi \cdot r_{2}=2\cdot \pi \cdot (2-{\sqrt {2}})\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow P_{2}=(2-{\sqrt {2}})\cdot P_{0}}$

0) Area of the base circle ${\displaystyle A_{0}=\pi \cdot r_{0}^{2}}$
1) Area of the inscribed quarter circle ${\displaystyle A_{1}={\frac {1}{2}}\cdot A_{0}}$
2) Area of the additional circle
${\displaystyle \quad A_{2}=\pi \cdot r_{2}^{2}=\pi \cdot \left((2-{\sqrt {2}})\cdot r_{0}\right)^{2}=\pi \cdot (2-{\sqrt {2}})^{2}\cdot r_{0}^{2}=\pi \cdot (4-2\cdot 2\cdot {\sqrt {2}}+2)\cdot r_{0}^{2}}$
${\displaystyle \quad \Leftrightarrow A_{2}=(6-4{\sqrt {2}})\cdot A_{0}}$

Covered surface of the base shape: ${\displaystyle C_{b}={\frac {A_{1}}{A_{0}}}=0.500=50\%}$

Centroid positions are measured from the centroid point of the base shape
0) Centroid positions of the base square: ${\displaystyle S_{0}=0+0i}$
1) Centroid positions of the inscribed quarter circle: ${\displaystyle S_{1}=\left({\frac {8-3\pi }{3{\sqrt {2}}\cdot \pi }}\right)\cdot (1+i)\cdot r_{0}\quad \approx (-0.107-0.107i)\cdot r_{0}}$
2) Centroids of the additional circle: ${\displaystyle S_{2}=\left({\frac {4-3{\sqrt {2}}}{2}}\right)\cdot (1+i)\cdot r_{0}\quad \approx (-0.121-0.121i)\cdot r_{0}\quad }$, see calculation (1)

The centroid positions of the following shapes will be expressed orientated so that the first shape n with ${\displaystyle S_{n}\neq S_{0}}$ will be of type ${\displaystyle S_{n}=0-ki}$ with ${\displaystyle k>0}$. This means that the graphical representation will not correspond to the mathematical expression.
0) Orientated centroid position of the base circle: ${\displaystyle S_{0}=0+0i}$
1) Orientated centroid positions of the inscribed quarter circle: ${\displaystyle S_{1}=\left({\frac {8-3\pi }{3{\sqrt {2}}\cdot \pi }}\right)\cdot {\sqrt {2}}\cdot (0+i)\cdot r_{0}\quad \approx (0-0.151i)\cdot r_{0}}$
2) Orientazed centroid of the additional circle: ${\displaystyle S_{2}=\left({\frac {4-3{\sqrt {2}}}{2}}\right)\cdot {\sqrt {2}}\cdot (0+i)\cdot r_{0}=\left(2{\sqrt {2}}-3\right)\cdot (0+i)\cdot r_{0}\quad \approx (0-0.172i)\cdot r_{0}}$

In the normalised case the area of the base is set to 1.
${\displaystyle A_{0}=1\quad \Rightarrow r_{0}={\frac {1}{\sqrt {\pi }}}}$

0) Radius of the base circle ${\displaystyle r_{0}={\frac {1}{\sqrt {\pi }}}=0.56...}$
1) Radius of the inscribed quarter circle ${\displaystyle r_{1}={\sqrt {\frac {2}{\pi }}}=0.79...}$
2) Radius of the additional circle
${\displaystyle \quad r_{2}=(2-{\sqrt {2}})\cdot r_{0}={\frac {2-{\sqrt {2}}}{\sqrt {\pi }}}=0.33...}$

0) Perimeter of base square ${\displaystyle P_{0}=2\cdot {\sqrt {\pi }}=3.5449077...}$
1) Perimeter of the inscribed quarter circle ${\displaystyle P_{1}={\frac {4+\pi }{\sqrt {2\cdot \pi }}}=2.8490832...}$
2) Perimeter of additional circle ${\displaystyle P_{2}=2\pi \cdot (2-{\sqrt {2}})\cdot {\frac {1}{\sqrt {\pi }}}=2{\sqrt {\pi }}\cdot (2-{\sqrt {2}})=2.07655885...}$
S) Sum of perimeters ${\displaystyle P_{s}=P_{0}+P_{1}+P_{2}=8.4705498...}$

0) Area of the base square ${\displaystyle A_{0}=1}$
1) Area of the inscribed quarter circle ${\displaystyle A_{1}=0.5}$
2) Area of the additional circle ${\displaystyle A_{2}=\pi \cdot \left({\frac {2-{\sqrt {2}}}{\sqrt {\pi }}}\right)^{2}=\pi \cdot {\frac {(2-{\sqrt {2}})^{2}}{\pi }}=6-4{\sqrt {2}}=0.34...}$

Centroid positions are measured from the centroid point of the base shape
0) Centroid positions of the base square: ${\displaystyle S_{0}=0+0i}$
1) Centroid positions of the inscribed quarter circle:
${\displaystyle S_{1}=\left({\frac {8-3\pi }{3{\sqrt {2}}\cdot \pi }}\right)\cdot (1+i)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx (-0.06-0.06i)}$
2) Centroid of the additional circle: ${\displaystyle S_{2}=\left({\frac {4-3{\sqrt {2}}}{2}}\right)\cdot (1+i)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx (-0.068-0.068i)\quad }$

The centroid positions of the following shapes will be expressed orientated so that the first shape n with ${\displaystyle S_{n}\neq S_{0}}$ will be of type ${\displaystyle S_{n}=0-ki}$ with ${\displaystyle k>0}$. This means that the graphical representation will not correspond to the mathematical expression.
0) Orientated centroid position of the base circle: ${\displaystyle S_{0}=0+0i}$
1) Orientated centroid positions of the inscribed quarter circle: ${\displaystyle S_{1}=\left({\frac {8-3\pi }{3{\sqrt {2}}\cdot \pi }}\right)\cdot {\sqrt {2}}\cdot (0+i)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx (0-0.085i)}$
2) Orientazed centroid of the additional circle: ${\displaystyle S_{2}=\left(2{\sqrt {2}}-3\right)\cdot (0+i){\frac {1}{\sqrt {\pi }}}\quad \approx (0-0.097i)}$

Calculating ${\displaystyle S_{2}}$ ${\displaystyle \quad S_{2}=S_{0}+{\overrightarrow {S_{0}A}}+{\overrightarrow {AS_{2}}}}$
${\displaystyle \quad \Leftrightarrow S_{2}=0-|S_{0}A|\cdot {\frac {(1+i)}{\sqrt {2}}}+{\overrightarrow {AS_{2}}}}$
${\displaystyle \quad \Leftrightarrow S_{2}=-|S_{0}A|\cdot {\frac {(1+i)}{\sqrt {2}}}+|AS_{2}|\cdot {\frac {(1+i)}{\sqrt {2}}}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left(-|S_{0}A|+|AS_{2}|\right)\cdot {\frac {(1+i)}{\sqrt {2}}}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left(|AS_{2}|-|S_{0}A|\right)\cdot {\frac {(1+i)}{\sqrt {2}}}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left({\sqrt {2}}\cdot r_{2}-|S_{0}A|\right)\cdot {\frac {(1+i)}{\sqrt {2}}}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left({\sqrt {2}}\cdot r_{2}-r_{0}\right)\cdot {\frac {(1+i)}{\sqrt {2}}}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left({\sqrt {2}}\cdot (2-{\sqrt {2}})\cdot r_{0}-r_{0}\right)\cdot {\frac {(1+i)}{\sqrt {2}}}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left({\sqrt {2}}\cdot (2-{\sqrt {2}})-1\right)\cdot {\frac {(1+i)}{\sqrt {2}}}\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left(2{\sqrt {2}}-2-1\right)\cdot {\frac {(1+i)}{\sqrt {2}}}\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left(2{\sqrt {2}}-3\right)\cdot {\frac {(1+i)}{\sqrt {2}}}\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left(2-{\frac {3}{\sqrt {2}}}\right)\cdot (1+i)\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left(2-{\frac {3{\sqrt {2}}}{2}}\right)\cdot (1+i)\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left({\frac {4-3{\sqrt {2}}}{2}}\right)\cdot (1+i)\cdot r_{0}\quad \approx (-0.121-0.121i)\cdot r_{0}}$

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