File:FS C(2C) dia.png

Summary[edit]

Task[edit]

The largest pair of circles inscribed in a circle of radius ${\displaystyle r_{0}}$

General case[edit]

Segments in the general case[edit]

0) The radius of the base circle: ${\displaystyle r_{0}}$
1) The radii of the inscribed circles: ${\displaystyle r_{1}=r_{2}={\frac {r_{0}}{2}}\quad }$

Perimeters in the general case[edit]

0) Perimeter of base circle: ${\displaystyle P_{0}=2\pi r_{0}}$
1) Perimeter of the inscribed circles: ${\displaystyle P_{1}+P_{2}=2\cdot 2\pi r_{1}=2\cdot 2\pi {\frac {r_{0}}{2}}=2\pi r_{0}=P_{0}\quad }$
S) Sum of perimeters: ${\displaystyle P_{s}=P_{0}+P_{0}=2\cdot (2\pi r_{0})=4\pi r_{0}}$

Areas in the general case[edit]

0) Area of the base circle: ${\displaystyle A_{0}=\pi \cdot r_{0}^{2}}$
1) Area of the inscribed circles: ${\displaystyle A_{1}+A_{2}=2\cdot \pi \cdot r_{1}^{2}=2\cdot \pi \cdot \left({\frac {r_{0}}{2}}\right)^{2}=2\cdot \pi \cdot \left({\frac {r_{0}^{2}}{4}}\right)={\frac {1}{2}}\cdot \pi \cdot r_{0}^{2}={\frac {1}{2}}\cdot A_{0}}$

Centroids in the general case[edit]

1) Centroids as graphically displayed[edit]

0) Centroid position of the base circle: ${\displaystyle S_{0}=0+0i}$
1) Centroid position of the first inscribed circle measured from the centroid of the base shape: ${\displaystyle S_{1}=(0+0i)+({\frac {r_{1}}{\sqrt {2}}}+{\frac {r_{1}}{\sqrt {2}}}i)={\frac {r_{0}}{2{\sqrt {2}}}}+{\frac {r_{0}}{2{\sqrt {2}}}}i\quad \approx (0.354+0.354i)\cdot r_{0}}$
2) Centroid position of the second inscribed circle measured from the centroid of the base shape: ${\displaystyle S_{2}=(0+0i)+(-{\frac {r_{1}}{\sqrt {2}}}-{\frac {r_{1}}{\sqrt {2}}}i)=-{\frac {r_{0}}{2{\sqrt {2}}}}-{\frac {r_{0}}{2{\sqrt {2}}}}i\quad \approx (-0.354-0.354i)\cdot r_{0}}$
W) Weighted centroid: ${\displaystyle S_{w}=A_{1}\cdot S_{1}+A_{2}\cdot S_{2}=A_{1}\cdot S_{1}-A_{1}\cdot S_{1}=0+0i}$

2) Orientated centroids[edit]

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 position of the first inscribed circle measured from the centroid of the base shape: ${\displaystyle S_{1}=(0-r_{1}i)=0-{\frac {r_{0}}{2}}i\quad =(0-0.500i)\cdot r_{0}}$
2) Orientated centroid position of the second inscribed circle measured from the centroid of the base shape: ${\displaystyle S_{2}=(0+r_{1}i)=0+{\frac {r_{0}}{2}}i\quad =(0+0.500i)\cdot r_{0}}$

Normalised case[edit]

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

Segments in the normalised case[edit]

0) Radius of the base circle ${\displaystyle r_{0}={\frac {1}{\sqrt {\pi }}}\quad \approx 0.564}$
1) Radii of the inscribed circles: ${\displaystyle r_{1}={\frac {\frac {1}{\sqrt {\pi }}}{2}}={\frac {1}{2{\sqrt {\pi }}}}\quad \approx 0.282}$

Perimeters in the normalised case[edit]

0) Perimeter of base circle: ${\displaystyle P_{0}=2\pi \cdot {\frac {1}{\sqrt {\pi }}}=2{\sqrt {\pi }}\quad \approx 3.545}$
1) Perimeter of the inscribed circles: ${\displaystyle P_{1}+P_{2}=P_{0}=2{\sqrt {\pi }}\quad \approx 3.545}$
S) Sum of perimeters: ${\displaystyle P_{s}=2\cdot P_{0}=4\cdot {\sqrt {\pi }}\quad \approx 7.09}$

Areas in the normalised case[edit]

0) Area of the base square is by definition ${\displaystyle A_{0}=1}$
1) Area of the inscribed circles: ${\displaystyle A_{1}+A_{2}={\frac {1}{2}}\cdot A_{0}={\frac {1}{2}}=0.500}$

Covered surface of the base shape: ${\displaystyle {\frac {A_{1}+A_{2}}{A_{0}}}=50\%}$

Centroids in the normalised case[edit]

1) Centroids as graphically displayed[edit]

0) Centroid position of the base square: ${\displaystyle S_{0}=0+0i}$
1) Centroid position of the first inscribed circle measured from the centroid of the base shape: ${\displaystyle S_{1}={\frac {1}{2{\sqrt {2}}\cdot {\sqrt {\pi }}}}+{\frac {1}{2{\sqrt {2}}\cdot {\sqrt {\pi }}}}i\quad \approx 0.199+0.199i}$
2) Centroid position of the second inscribed circle measured from the centroid of the base shape: ${\displaystyle S_{1}=-{\frac {1}{2{\sqrt {2}}\cdot {\sqrt {\pi }}}}-{\frac {1}{2{\sqrt {2}}\cdot {\sqrt {\pi }}}}i\quad \approx -0.199-0.199i}$
W) Weighted centroid: ${\displaystyle S_{w}=0+0i}$

2) Orientated centroids[edit]

0) Orientated centroid position of the base square: ${\displaystyle S_{0}=0+0i}$
1) Orientated centroid position of the first inscribed circle measured from the centroid of the base shape: ${\displaystyle S_{1}=0-{\frac {1}{2{\sqrt {\pi }}}}i\quad \approx 0-0.282i}$
2) Centroid position of the second inscribed circle measured from the centroid of the base shape: ${\displaystyle S_{1}=0+{\frac {1}{2{\sqrt {\pi }}}}i\quad \approx 0+0.282i}$

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