File:FS CV dia.png

Shows the largest quarter circle within a circle.

Base is the circle of given radius ${\displaystyle r_{0}}$ around point ${\displaystyle S_{0}}$
Inscribed is the largest possible quarter circle.

In order to find radius ${\displaystyle r_{1}}$ of the quarter circle, the following reasoning is used: Since point ${\displaystyle D}$ is the center of the circle we have:
${\displaystyle |AS_{0}|=|BS_{0}|=|CS_{0}|=r_{0}}$

The points ${\displaystyle A}$,${\displaystyle B}$ and ${\displaystyle C}$ form a rectangular, isosceles triangle with:
${\displaystyle |AB|=|AC|=r_{1}}$ and ${\displaystyle |BC|=2\cdot r_{0}}$

Applying the Pythagorean theorem on ${\displaystyle \triangle ABC}$ gives:
${\displaystyle |AB|^{2}+|AC|^{2}=|CB|^{2}}$
${\displaystyle \Rightarrow r_{1}^{2}+r_{1}^{2}=(2\cdot r_{0})^{2}}$
${\displaystyle \Rightarrow 2\cdot r_{1}^{2}=4\cdot r_{0}^{2}}$
${\displaystyle \Rightarrow r_{1}^{2}=2\cdot r_{0}^{2}}$
${\displaystyle \Rightarrow r_{1}={\sqrt {2\cdot r_{0}^{2}}}}$
${\displaystyle \Rightarrow r_{1}={\sqrt {2}}\cdot r_{0}}$

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

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

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

Covered surface of base shape: ${\displaystyle C_{b}={\frac {A_{1}}{A_{0}}}={\frac {1}{2}}\cdot A_{0}=50\%}$

Centroid positions are measured from the centroid point of the base shape
0) Centroid position of the base square: ${\displaystyle S_{0}=0+0i}$
1) Centroid position of the inscribed quarter circle:
${\displaystyle S_{1}=S_{0}+{\overrightarrow {S_{0}A}}+{\overrightarrow {AS_{1}}}=-r_{0}\cdot cos{45^{\circ }}(1+i)+{\frac {4\cdot r_{1}}{3\cdot \pi }}\cdot (1+i)=\left({\frac {4\cdot r_{1}}{3\cdot \pi }}-{\frac {r_{0}}{\sqrt {2}}}\right)\cdot (1+i)}$
${\displaystyle \Leftrightarrow S_{1}=\left({\frac {4\cdot r_{0}{\sqrt {2}}}{3\cdot \pi }}-{\frac {r_{0}}{\sqrt {2}}}\right)\cdot (1+i)=\left({\frac {4{\sqrt {2}}}{3\cdot \pi }}-{\frac {1}{\sqrt {2}}}\right)\cdot (1+i)\cdot r_{0}=\left({\frac {4\cdot 2-3\pi }{3{\sqrt {2}}\cdot \pi }}\right)\cdot (1+i)\cdot r_{0}=\left({\frac {8-3\pi }{3\pi }}\right)\cdot {\frac {1}{\sqrt {2}}}\cdot (1+i)\cdot r_{0}\quad \approx (-0.107-0.107i)\cdot r_{0}}$

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: ${\displaystyle S_{1}=\left({\frac {8-3\pi }{3{\sqrt {2}}\cdot \pi }}\right)\cdot {\sqrt {2}}\cdot (0+i)\cdot r_{0}\quad \approx (0-0.151i)\cdot r_{0}}$

In the normalised case the area of the base is set to 1.
${\displaystyle A_{0}=1\quad \Rightarrow \pi \cdot r_{0}^{2}=1\quad \Rightarrow r_{0}^{2}={\frac {1}{\pi }}\quad \Rightarrow r_{0}={\frac {1}{\sqrt {\pi }}}}$

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

0) Perimeter of base square ${\displaystyle P_{0}=2\cdot \pi \cdot r_{0}=2\cdot \pi \cdot {\frac {1}{\sqrt {\pi }}}=2\cdot {\sqrt {\pi }}=3.5449077...}$
1) Perimeter of the inscribed quarter circle ${\displaystyle P_{1}={\frac {4+\pi }{\sqrt {2}}}\cdot r_{0}={\frac {4+\pi }{\sqrt {2}}}\cdot {\frac {1}{\sqrt {\pi }}}={\frac {4+\pi }{\sqrt {2\cdot \pi }}}=2.8490832...}$
S) Sum of perimeters ${\displaystyle P_{s}=P_{0}+P_{1}=6.3939909...}$

0) Area of the base square ${\displaystyle A_{0}=1}$
1) Area of the inscribed quarter circle ${\displaystyle A_{1}={\frac {\pi }{2}}\cdot r_{0}^{2}={\frac {\pi }{2}}\cdot ({\frac {1}{\sqrt {\pi }}})^{2}={\frac {\pi }{2}}\cdot {\frac {1}{\pi }}=0.5}$

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:
${\displaystyle S_{1}=\left({\frac {8-3\pi }{3{\sqrt {2}}\cdot \pi }}\right)\cdot (1+i)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx (-0.06-0.06i)}$

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: ${\displaystyle S_{1}=\left({\frac {8-3\pi }{3{\sqrt {2}}\cdot \pi }}\right)\cdot {\sqrt {2}}\cdot (0+i){\frac {1}{\sqrt {\pi }}}\quad \approx (0-0.085i)}$

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