File:FS CVC2 dia.png

Shows the second largest circle within a forced sudoku of CV-type.

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 ${\displaystyle A}$

Inscribed is the largest possible circle in the shape ${\displaystyle [CBG]}$ having radius ${\displaystyle r_{2}}$ around point ${\displaystyle S_{2}}$. This is the second 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 ${\displaystyle [AG]}$ is the diameter of the circle around ${\displaystyle S_{0}}$ with radius ${\displaystyle r_{0}}$. The line segment ${\displaystyle [AE]}$ is radius ${\displaystyle r_{1}}$ of the quarter circle around ${\displaystyle A}$. The line segment ${\displaystyle [EG]}$ is the diameter of the circle around ${\displaystyle S_{2}}$ with radius ${\displaystyle r_{2}}$.

${\displaystyle |AG|=|AE|+|EG|=2\cdot r_{0}}$
${\displaystyle \Rightarrow r_{1}+2\cdot r_{2}=2\cdot r_{0}}$
${\displaystyle \Leftrightarrow 2\cdot r_{2}=2\cdot r_{0}-r_{1}}$

From the construction of the quarter circle (see FS_CV) we know that ${\displaystyle r_{1}={\sqrt {2}}\cdot r_{0}}$. So:
${\displaystyle \quad 2\cdot r_{2}=2\cdot r_{0}-{\sqrt {2}}\cdot r_{0}}$
${\displaystyle \Leftrightarrow 2\cdot r_{2}=(2-{\sqrt {2}})\cdot r_{0}}$
${\displaystyle \Leftrightarrow r_{2}={\frac {2-{\sqrt {2}}}{2}}\cdot r_{0}}$

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: ${\displaystyle r_{2}={\frac {2-{\sqrt {2}}}{2}}\cdot r_{0}\quad \approx 0.293\cdot r_{0}}$

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 ${\displaystyle P_{2}=2\cdot \pi \cdot r_{2}=2\cdot \pi \cdot {\frac {2-{\sqrt {2}}}{2}}\cdot r_{0}=(2-{\sqrt {2}})\cdot \pi \cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow P_{2}={\frac {2-{\sqrt {2}}}{2}}\cdot P_{0}}$

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:
${\displaystyle \quad A_{2}=\pi \cdot r_{2}^{2}=\pi \cdot \left({\frac {2-{\sqrt {2}}}{2}}\cdot r_{0}\right)^{2}=\pi \cdot \left({\frac {2-{\sqrt {2}}}{2}}\right)^{2}\cdot r_{0}^{2}={\frac {(2-{\sqrt {2}})^{2}}{4}}\cdot \pi \cdot r_{0}^{2}={\frac {4-4{\sqrt {2}}+2}{4}}\cdot \pi \cdot r_{0}^{2}}$
${\displaystyle \quad \Leftrightarrow A_{2}={\frac {6-4{\sqrt {2}}}{4}}\cdot A_{0}}$
${\displaystyle \quad \Leftrightarrow A_{2}={\frac {3-2{\sqrt {2}}}{2}}\cdot A_{0}}$

Covered surface of base shape: ${\displaystyle C_{b}={\frac {A_{1}+A_{2}}{A_{0}}}={\frac {{\frac {1}{2}}\cdot A_{0}+{\frac {3-2{\sqrt {2}}}{2}}\cdot A_{0}}{A_{0}}}={\frac {1+3-2{\sqrt {2}}}{2}}={\frac {4-2{\sqrt {2}}}{2}}=2-{\sqrt {2}}\quad \approx 58.6\%}$

Centroid positions are measured from the centroid point of the base shape
0) Centroid of the base square: ${\displaystyle S_{0}=0+0i}$
1) Centroid of the inscribed quarter circle:${\displaystyle S_{1}=\left({\frac {8-3\pi }{3\pi }}\right)\cdot {\frac {1}{\sqrt {2}}}\cdot (1+i)\cdot r_{0}\quad \approx (-0.107-0.107i)\cdot r_{0}}$
2) Centroid of the additional circle: ${\displaystyle \quad S_{2}={\frac {1}{2}}\cdot (1+i)\cdot r_{0}=(0.5+0.5i)\cdot r_{0}}$, 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: ${\displaystyle S_{1}=\left({\frac {8-3\pi }{3\pi }}\right)\cdot (0+i)\cdot r_{0}\quad \approx (0-0.151i)\cdot r_{0}}$
2) Centroid of the additional circle: ${\displaystyle \quad S_{2}={\frac {\sqrt {2}}{2}}\cdot (0+i)\cdot r_{0}={\frac {1}{\sqrt {2}}}\cdot (0+i)\cdot r_{0}\quad \approx (0+0.707i)\cdot r_{0}}$

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
${\displaystyle \quad r_{2}={\frac {2-{\sqrt {2}}}{2}}\cdot r_{0}={\frac {2-{\sqrt {2}}}{2}}\cdot {\frac {1}{\sqrt {\pi }}}={\frac {2-{\sqrt {2}}}{2\cdot {\sqrt {\pi }}}}=0.165...}$

0) Perimeter of base circle ${\displaystyle P_{0}=2\cdot {\sqrt {\pi }}=3.5449077...}$
1) Perimeter of the inscribed quarter circle ${\displaystyle P_{1}={\frac {4+\pi }{\sqrt {2\cdot \pi }}}=2.8490832...}$
2) Perimeter of additional circle ${\displaystyle P_{2}=(2-{\sqrt {2}})\cdot \pi \cdot {\frac {1}{\sqrt {\pi }}}=(2-{\sqrt {2}})\cdot {\sqrt {\pi }}=1.0382794...}$
S) Sum of perimeters ${\displaystyle P_{s}=P_{0}+P_{1}+P_{2}=7.4322703...}$

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 ${\displaystyle A_{2}=({\frac {3}{2}}-{\sqrt {2}})\cdot A_{0}=({\frac {3}{2}}-{\sqrt {2}})=0.086...}$

Centroid positions are measured from the centroid point of the base shape
0) Centroid of the base square: ${\displaystyle S_{0}=0+0i}$
1) Centroid of the inscribed quarter circle:${\displaystyle S_{1}=\left({\frac {8-3\pi }{3\pi }}\right)\cdot {\frac {1}{\sqrt {2}}}\cdot (1+i)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx (-0.06-0.06i)}$
2) Centroid of the additional circle: ${\displaystyle \quad S_{2}={\frac {1}{2}}\cdot (1+i)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx (0.282+0.282i)}$, 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: ${\displaystyle S_{1}=\left({\frac {8-3\pi }{3\pi }}\right)\cdot (0+i)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx (0-0.085i)}$
2) Orientated centroid of the additional circle: ${\displaystyle \quad S_{2}={\frac {\sqrt {2}}{2}}\cdot (0+i)\cdot r_{0}={\frac {1}{\sqrt {2}}}\cdot (0+i)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx (0+0.399i)}$

${\displaystyle \quad S_{2}=S_{0}+{\overrightarrow {S_{0}S_{2}}}}$
${\displaystyle \quad \Leftrightarrow S_{2}=0+|S_{0}S_{2}|\cdot \cos 45^{\circ }\cdot (1+i)}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left(|AS_{2}|-|AS_{0}|\right)\cdot \cos 45^{\circ }\cdot (1+i)}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left((|AE|+|ES_{2}|)-|AS_{0}|\right)\cdot \cos 45^{\circ }\cdot (1+i)}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left((r_{1}+r_{2})-|AS_{0}|\right)\cdot \cos 45^{\circ }\cdot (1+i)}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left((r_{1}+r_{2})-r_{0}\right)\cdot \cos 45^{\circ }\cdot (1+i)}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left({\sqrt {2}}\cdot r_{0}+{\frac {2-{\sqrt {2}}}{2}}\cdot r_{0}-r_{0}\right)\cdot \cos 45^{\circ }\cdot (1+i)}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left({\sqrt {2}}+{\frac {2-{\sqrt {2}}}{2}}-1\right)\cdot \cos 45^{\circ }\cdot (1+i)\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left({\frac {2{\sqrt {2}}+2-{\sqrt {2}}-2}{2}}\right)\cdot \cos 45^{\circ }\cdot (1+i)\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left({\frac {\sqrt {2}}{2}}\right)\cdot \cos 45^{\circ }\cdot (1+i)\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow S_{2}=\left({\frac {\sqrt {2}}{2}}\right)\cdot {\frac {1}{\sqrt {2}}}\cdot (1+i)\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow S_{2}={\frac {1}{2}}\cdot (1+i)\cdot r_{0}=(0.500+0.500i)\cdot r_{0}}$

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