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Density of Intersecting Integer Circles and Bisection of Integer Angles
1
Introduction
2
Notions and Definitions
3
Density of the Intersection of Integer Circles
4
Spectra of Circumscribed Centres
5
Inscribed Circles in Integer Geometry
6
Conclusion
A
Python Program for Example
3.1.4
B
Python Program for Example
3.1.6
C
Python for Example
3.1.15
D
Python for Example
3.1.16
E
Python for Example
3.1.20
F
Python for Example
3.2.2
Density of Intersecting Integer Circles and Bisection of Integer Angles
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Density of Intersecting Integer Circles and Bisection of Integer Angles
Rebecca Sheppard
July 4, 2025
Contents
1
Introduction
2
Notions and Definitions
2.1
Integer Length and Integer Distance
2.2
Continued Fractions
2.3
Integer Angles
2.4
Integer Circles
2.5
Density of sets in the lattice
ℤ
2
\mathbb{Z}^{2}
2.6
Multiplicative Functions
3
Density of the Intersection of Integer Circles
3.1
Intersection of Unit Integer Circles
3.1.1
Density of Two Integer Circles with respect to
ℐ
\mathcal{I}
3.1.2
Density of Intersection of Three Integer Circles
3.1.3
Density of Intersection with respect to
ℬ
\mathcal{B}
3.1.4
Comparison between
#
ℐ
\#_{\mathcal{I}}
and
#
ℬ
(
1
,
1
)
\#_{\mathcal{B}(1,1)}
3.2
Intersection of Two Integer Circles with different radii
4
Spectra of Circumscribed Centres
4.1
Set of circumscribed circles
5
Inscribed Circles in Integer Geometry
5.1
Inscribed Circles of Integer Angles
5.1.1
Bisectors of Integer Angles
5.1.2
k
k
-secting integer angles
5.2
Inscribed Circles of Integer Polygons
6
Conclusion
A
Python Program for Example
3.1.4
A.1
Python Code:
A.2
Python Output
B
Python Program for Example
3.1.6
B.1
Python Code:
B.2
Program Output:
B.3
Result Table for Example
3.1.6
C
Python for Example
3.1.15
C.1
Python Code:
C.2
Python Output:
D
Python for Example
3.1.16
D.1
Python Code:
D.2
Python Output:
E
Python for Example
3.1.20
E.1
Python Code:
E.2
Python Output:
F
Python for Example
3.2.2
F.1
Discussion:
F.2
Python Code: