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The semicircle as base element.
Segments in the general case [ edit ]
The radius of the semicircle
=
r
0
{\displaystyle =r_{0}}
Perimeters in the general case [ edit ]
Perimeter of base semicircle
P
0
=
2
⋅
r
0
+
π
⋅
r
0
=
(
2
+
π
)
⋅
r
0
{\displaystyle P_{0}=2\cdot r_{0}+\pi \cdot r_{0}=(2+\pi )\cdot r_{0}}
Areas in the general case [ edit ]
Area of the base semicircle
A
0
=
π
⋅
r
0
2
2
{\displaystyle A_{0}={\frac {\pi \cdot r_{0}^{2}}{2}}}
Centroids in the general case [ edit ]
By definition the centroid points of a base shape are x_0=0 and y_0=0. Relatively is the lower left point of the of the base of the semicircle at
x
L
=
−
r
0
y
L
=
−
4
⋅
r
0
3
⋅
π
{\displaystyle \quad x_{L}=-r_{0}\quad y_{L}=-{\frac {4\cdot r_{0}}{3\cdot \pi }}}
In the normalised case the area of the base semicircle is set to 1.
So
A
0
=
π
⋅
r
0
2
2
=
1
⇒
r
0
2
=
2
π
⇒
r
0
=
2
π
{\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 ]
Segment of the base semicircle
r
0
=
2
π
=
0.79788...
{\displaystyle r_{0}={\sqrt {\frac {2}{\pi }}}=0.79788...}
Perimeter in the normalised case [ edit ]
Perimeter of base semicircle:
P
0
=
(
2
+
π
)
⋅
r
0
=
(
2
+
π
)
⋅
2
π
=
4.102397...
{\displaystyle P_{0}=(2+\pi )\cdot r_{0}=(2+\pi )\cdot {\sqrt {\frac {2}{\pi }}}=4.102397...}
Area in the normalised case [ edit ]
Area of the base semicircle is by definition
A
0
=
1
{\displaystyle A_{0}=1}
Centroids in the normalised case [ edit ]
Positions of the lower left point of the semicircle:
x
L
=
−
r
0
=
−
2
π
=
−
0.79788...
y
L
=
−
4
⋅
r
0
3
⋅
π
=
−
4
3
⋅
π
⋅
2
π
=
−
32
9
⋅
π
3
=
−
0.33863...
{\displaystyle x_{L}=-r_{0}=-{\sqrt {\frac {2}{\pi }}}=-0.79788...\quad y_{L}=-{\frac {4\cdot r_{0}}{3\cdot \pi }}=-{\frac {4}{3\cdot \pi }}\cdot {\sqrt {\frac {2}{\pi }}}=-{\sqrt {\frac {32}{9\cdot \pi ^{3}}}}=-0.33863...}
Apart of the base element there is no other shape allocated. Therefore the integer part of the identifying number is 0.
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.
d
e
c
i
m
a
l
p
a
r
t
(
4.1023974...
+
0
)
=
d
e
c
i
m
a
l
p
a
r
t
(
4.1023974...
)
=
.1023974
.
.
.
{\displaystyle decimalpart(4.1023974...+0)=decimalpart(4.1023974...)=.1023974...}
So the identifying number is:
0.1023974
{\displaystyle 0.1023974}