File:FS HC dia.png

Summary[edit]

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The semicircle as base element. And the largest inscribed circle.

General case[edit]

Segments in the general case[edit]

0) The radius of the semicircle: ${\displaystyle r_{0}}$
1) The radius of the inscribed circle: ${\displaystyle r_{1}={\frac {r_{0}}{2}}}$

Perimeters in the general case[edit]

0) Perimeter of base semicircle: ${\displaystyle P_{0}=2\cdot r_{0}+\pi \cdot r_{0}=(2+\pi )\cdot r_{0}\quad \approx 5.142\cdot r_{0}}$
1) Perimeter of inscribed circle: ${\displaystyle P_{1}=2\cdot \pi \cdot r_{1}=\pi \cdot r_{0}\quad \approx 3.142\cdot r_{0}}$

Areas in the general case[edit]

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

Centroids in the general case[edit]

0) By definition the centroid point of a base shape is ${\displaystyle S_{0}=0+0i}$
1) The centroid of the inscribed circle relative to the base centroid is: ${\displaystyle S_{1}=0+{\frac {3\pi -8}{6\pi }}\cdot r_{0}\cdot i}$, see Calculation 1

Normalised case[edit]

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

Segments in the normalised case[edit]

0) Radius of the base semicircle ${\displaystyle r_{0}={\sqrt {\frac {2}{\pi }}}\quad \approx 0.798}$
1) Radius of the inscribed circle ${\displaystyle r_{1}={\frac {1}{2}}{\sqrt {\frac {2}{\pi }}}\quad \approx 0.399}$

Perimeter in the normalised case[edit]

0) Perimeter of base semicircle: ${\displaystyle P_{0}=(2+\pi )\cdot r_{0}=(2+\pi )\cdot {\sqrt {\frac {2}{\pi }}}=4.1023974}$
1) Perimeter of inscribed circle: ${\displaystyle P_{1}=\pi \cdot {\sqrt {\frac {2}{\pi }}}\quad \approx 2.5066283}$
S) Sum of perimeters: ${\displaystyle P_{s}=P_{0}+P_{1}=6.6090257}$

Area in the normalised case[edit]

0) Area of the base semicircle is by definition ${\displaystyle A_{0}=1}$
1) Area of the base semicircle ${\displaystyle A_{1}={\frac {A_{0}}{2}}=0.500}$

Centroids in the normalised case[edit]

0) ${\displaystyle S_{0}=0+0i}$
1) ${\displaystyle S_{1}=0+{\frac {3\pi -8}{6\pi }}\cdot {\sqrt {\frac {2}{\pi }}}\cdot i\quad \approx 0+0.060i}$

Distances of centroids[edit]

The distance between the centroid of the base semicircle and the centroid of the circle is:
${\displaystyle d_{01}={\sqrt {(x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}}}\quad \approx 0.0603096}$
Sum of distances: ${\displaystyle d_{s}=d_{01}\quad \approx 0.0603096}$

Identifying number[edit]

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.

${\displaystyle decimalpart(6.6090257+0.0603096)=decimalpart(6.6693352)=.6693352}$

So the identifying number is: ${\displaystyle 1.6693352}$

Calculations[edit]

Calculation 1[edit]

${\displaystyle S_{1}=S_{0}+{\overrightarrow {S_{0}D}}+{\overrightarrow {DS_{1}}}}$
${\displaystyle \Leftrightarrow S_{1}=(0+0i)-\left(0+\left({\frac {4\cdot r_{0}}{3\cdot \pi }}\right)\cdot i\right)+(0+r_{1}\cdot i)}$
${\displaystyle \Leftrightarrow S_{1}=0-\left({\frac {4}{3\cdot \pi }}\right)\cdot r_{0}\cdot i+r_{1}\cdot i}$
${\displaystyle \Leftrightarrow S_{1}=0+\left({\frac {1}{2}}-{\frac {4}{3\cdot \pi }}\right)\cdot r_{0}\cdot i}$
${\displaystyle \Leftrightarrow S_{1}=0+\left({\frac {3\pi }{6\pi }}-{\frac {8}{6\pi }}\right)\cdot r_{0}\cdot i}$
${\displaystyle \Leftrightarrow S_{1}=0+{\frac {3\pi -8}{6\pi }}\cdot r_{0}\cdot i}$