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r
g
=
f
⋅
g
{\displaystyle r_{g}=f\cdot g}
g
=
tan
θ
{\displaystyle g=\tan \theta }
r
s
=
f
⋅
s
{\displaystyle r_{s}=f\cdot s}
s
=
2
tan
θ
2
{\displaystyle s=2\tan {\frac {\theta }{2}}}
r
l
=
f
⋅
l
{\displaystyle r_{l}=f\cdot l}
l
=
θ
{\displaystyle l=\theta }
r
a
=
f
⋅
a
{\displaystyle r_{a}=f\cdot a}
a
=
2
sin
θ
2
{\displaystyle a=2\sin {\frac {\theta }{2}}}
r
o
=
f
⋅
o
{\displaystyle r_{o}=f\cdot o}
o
=
sin
θ
{\displaystyle o=\sin \theta }
f
{\displaystyle f}
f
=
1
{\displaystyle f=1}
r
=
r
x
y
2
+
z
2
=
x
2
+
y
2
+
z
2
{\displaystyle r={\sqrt {r_{xy}^{2}+z^{2}}}={\sqrt {x^{2}+y^{2}+z^{2}}}}
r
x
y
=
x
2
+
y
2
{\displaystyle r_{xy}={\sqrt {x^{2}+y^{2}}}}
r
=
r
(
θ
,
f
)
=
f
⋅
R
(
θ
)
{\displaystyle r=r(\theta ,f)=f\cdot R(\theta )}
M
m
=
d
r
d
θ
=
d
r
(
θ
,
f
)
d
θ
{\displaystyle M_{m}={\frac {\mathrm {d} r}{\mathrm {d} \theta }}={\frac {\mathrm {d} r(\theta ,f)}{\mathrm {d} \theta }}}
S
m
=
M
m
f
=
d
R
(
θ
)
d
θ
{\displaystyle S_{m}={\frac {M_{m}}{f}}={\frac {\mathrm {d} R(\theta )}{\mathrm {d} \theta }}}
M
s
=
r
sin
θ
{\displaystyle M_{s}={\frac {r}{\sin \theta }}}
(
x
y
z
)
=
(
sin
ϕ
cos
ϕ
⋅
sin
λ
cos
ϕ
⋅
cos
λ
)
{\displaystyle {\begin{pmatrix}x&y&z\end{pmatrix}}={\begin{pmatrix}\sin \phi &{\cos \phi \cdot \sin \lambda }&{\cos \phi \cdot \cos \lambda }\end{pmatrix}}}
S
s
=
M
s
f
=
R
sin
θ
{\displaystyle S_{s}={\frac {M_{s}}{f}}={\frac {R}{\sin \theta }}}
D
=
S
m
S
s
{\displaystyle D={\frac {S_{m}}{S_{s}}}}
S
Ω
=
S
m
⋅
S
s
{\displaystyle S_{\Omega }=S_{m}\cdot S_{s}}
S
=
S
Ω
=
S
m
⋅
S
s
{\displaystyle S={{\sqrt {S}}_{\Omega }}={\sqrt {S_{m}\cdot S_{s}}}}
R
=
θ
+
r
3
θ
3
+
r
5
θ
5
+
.
.
.
{\displaystyle R=\theta +r_{3}\theta ^{3}\color {Periwinkle}+r_{5}\theta ^{5}+{}...}
S
m
=
1
+
s
m
2
θ
2
+
s
m
4
θ
4
+
.
.
.
=
1
+
3
r
3
θ
2
+
5
r
5
θ
4
+
.
.
.
{\displaystyle S_{m}=1+s_{m2}\theta ^{2}{\color {Periwinkle}{}+{}s_{m4}\theta ^{4}+{}...}=1+3r_{3}\theta ^{2}\color {Periwinkle}+5r_{5}\theta ^{4}+{}...}
sin
θ
=
θ
−
θ
3
6
+
θ
5
120
+
.
.
.
{\displaystyle \sin \theta =\theta -{\frac {\theta ^{3}}{6}}\color {Periwinkle}+{\frac {\theta ^{5}}{120}}+{}...}
S
s
=
1
+
s
s
2
θ
2
+
.
.
.
=
1
+
(
r
3
+
1
6
)
θ
2
+
.
.
.
{\displaystyle S_{s}=1+s_{s2}\theta ^{2}{\color {Periwinkle}{}+{}...}=1+(r_{3}+{\frac {1}{6}})\theta ^{2}\color {Periwinkle}+{}...}
D
=
1
+
d
2
θ
2
+
.
.
.
=
1
+
(
2
r
3
−
1
6
)
θ
2
+
.
.
.
{\displaystyle D=1+d_{2}\theta ^{2}{\color {Periwinkle}{}+{}...}=1+(2r_{3}-{\frac {1}{6}})\theta ^{2}\color {Periwinkle}+{}...}
S
=
1
+
s
2
θ
2
+
.
.
.
=
1
+
(
2
r
3
+
1
12
)
θ
2
+
.
.
.
{\displaystyle S=1+s_{2}\theta ^{2}{\color {Periwinkle}{}+{}...}=1+(2r_{3}+{\frac {1}{12}})\theta ^{2}\color {Periwinkle}+{}...}
S
Ω
=
1
+
s
Ω
2
θ
2
+
.
.
.
=
1
+
(
4
r
3
+
1
6
)
θ
2
+
.
.
.
{\displaystyle S_{\Omega }=1+s_{\Omega 2}\theta ^{2}{\color {Periwinkle}{}+{}...}=1+(4r_{3}+{\frac {1}{6}})\theta ^{2}{\color {Periwinkle}{}+{}...}}
C
=
S
m
⋅
0
,
5
sin
2
θ
−
R
R
sin
θ
{\displaystyle C={\frac {S_{m}\cdot 0{,}5\sin 2\theta -R}{R\sin \theta }}}
C
=
c
1
θ
+
.
.
.
=
(
r
3
−
2
3
)
θ
+
.
.
.
{\displaystyle C=c_{1}\theta {\color {Periwinkle}{}+{}...}=(r_{3}-{\frac {2}{3}})\theta \color {Periwinkle}{}+{}...}
P
1
=
(
p
1
x
p
1
y
p
1
z
)
=
(
tan
φ
sin
θ
cos
θ
)
{\displaystyle P1={\begin{pmatrix}p1_{x}\\p1_{y}\\p1_{z}\end{pmatrix}}={\begin{pmatrix}\tan \varphi \\\sin \theta \\\cos \theta \end{pmatrix}}}
P
2
=
(
p
2
x
p
2
y
p
2
z
)
=
(
sin
φ
cos
φ
⋅
sin
θ
cos
φ
⋅
cos
θ
)
{\displaystyle P2={\begin{pmatrix}p2_{x}\\p2_{y}\\p2_{z}\end{pmatrix}}={\begin{pmatrix}\sin \varphi \\\cos \varphi \cdot \sin \theta \\\cos \varphi \cdot \cos \theta \end{pmatrix}}}
cos
λ
=
(
sin
φ
cos
φ
⋅
sin
θ
cos
φ
⋅
cos
θ
)
∙
(
0
0
1
)
=
cos
φ
⋅
cos
θ
{\displaystyle \cos \lambda ={\begin{pmatrix}\sin \varphi \\\cos \varphi \cdot \sin \theta \\\cos \varphi \cdot \cos \theta \end{pmatrix}}\bullet {\begin{pmatrix}0\\0\\1\end{pmatrix}}=\cos \varphi \cdot \cos \theta }
λ
=
arccos
(
cos
φ
⋅
cos
θ
)
{\displaystyle \lambda =\arccos(\cos \varphi \cdot \cos \theta )}
P
3
=
(
p
3
x
p
3
y
)
=
R
(
λ
)
⋅
(
p
2
x
p
2
x
2
+
p
2
y
2
p
2
y
p
2
x
2
+
p
2
y
2
)
{\displaystyle P3={\begin{pmatrix}p3_{x}\\p3_{y}\end{pmatrix}}=R(\lambda )\cdot {\begin{pmatrix}{\frac {p2_{x}}{\sqrt {p2_{x}^{2}+p2_{y}^{2}}}}\\{\frac {p2_{y}}{\sqrt {p2_{x}^{2}+p2_{y}^{2}}}}\end{pmatrix}}}
p
3
y
=
R
(
λ
)
⋅
cos
φ
⋅
cos
θ
sin
2
φ
+
cos
2
φ
⋅
sin
2
θ
{\displaystyle p3_{y}={\frac {R(\lambda )\cdot \cos \varphi \cdot \cos \theta }{\sqrt {\sin ^{2}\varphi +\cos ^{2}\varphi \cdot \sin ^{2}\theta }}}}
P
3
=
(
p
3
x
p
3
y
)
=
R
(
λ
)
⋅
(
k
x
k
y
)
{\displaystyle P3={\begin{pmatrix}p3_{x}\\p3_{y}\end{pmatrix}}=R(\lambda )\cdot {\begin{pmatrix}k_{x}\\k_{y}\end{pmatrix}}}
k
x
=
p
1
x
p
1
x
2
+
p
1
y
2
=
tan
φ
tan
2
φ
+
sin
2
θ
{\displaystyle k_{x}={\frac {p1_{x}}{\sqrt {p1_{x}^{2}+p1_{y}^{2}}}}={\frac {\tan \varphi }{\sqrt {\tan ^{2}\varphi +\sin ^{2}\theta }}}}
k
y
=
p
1
y
p
1
x
2
+
p
1
y
2
=
sin
θ
tan
2
φ
+
sin
2
θ
{\displaystyle k_{y}={\frac {p1_{y}}{\sqrt {p1_{x}^{2}+p1_{y}^{2}}}}={\frac {\sin \theta }{\sqrt {\tan ^{2}\varphi +\sin ^{2}\theta }}}}
k
x
=
p
2
x
p
2
x
2
+
p
2
y
2
=
sin
φ
sin
2
φ
+
cos
2
φ
⋅
sin
2
θ
{\displaystyle k_{x}={\frac {p2_{x}}{\sqrt {p2_{x}^{2}+p2_{y}^{2}}}}={\frac {\sin \varphi }{\sqrt {\sin ^{2}\varphi +\cos ^{2}\varphi \cdot \sin ^{2}\theta }}}}
k
y
=
p
2
y
p
2
x
2
+
p
2
y
2
=
cos
φ
⋅
sin
θ
sin
2
φ
+
cos
2
φ
⋅
sin
2
θ
{\displaystyle k_{y}={\frac {p2_{y}}{\sqrt {p2_{x}^{2}+p2_{y}^{2}}}}={\frac {\cos \varphi \cdot \sin \theta }{\sqrt {\sin ^{2}\varphi +\cos ^{2}\varphi \cdot \sin ^{2}\theta }}}}
p
3
y
=
R
(
λ
)
⋅
k
y
{\displaystyle p3_{y}=R(\lambda )\cdot k_{y}}
Y
=
p
3
y
{\displaystyle Y=p3_{y}}
Y
=
R
[
arccos
(
cos
φ
⋅
cos
θ
)
]
⋅
sin
θ
tan
2
φ
+
sin
2
θ
{\displaystyle Y=R[\arccos(\cos \varphi \cdot \cos \theta )]\cdot {\frac {\sin \theta }{\sqrt {\tan ^{2}\varphi +\sin ^{2}\theta }}}}
Y
=
R
[
arccos
(
cos
φ
⋅
cos
θ
)
]
⋅
cos
φ
⋅
sin
θ
sin
2
φ
+
cos
2
φ
⋅
sin
2
θ
{\displaystyle Y=R[\arccos(\cos \varphi \cdot \cos \theta )]\cdot {\frac {\cos \varphi \cdot \sin \theta }{\sqrt {\sin ^{2}\varphi +\cos ^{2}\varphi \cdot \sin ^{2}\theta }}}}
Y
{
R
[
λ
(
θ
;
φ
)
]
;
k
y
(
θ
;
φ
)
}
{\displaystyle Y\lbrace R[\lambda (\theta ;\varphi )];k_{y}(\theta ;\varphi )\rbrace }
Y
=
R
⋅
k
y
{\displaystyle Y=R\cdot k_{y}}
d
Y
d
φ
=
d
R
d
φ
⋅
k
y
+
R
⋅
d
k
y
d
φ
{\displaystyle {\frac {\mathrm {d} Y}{\mathrm {d} \varphi }}={\frac {\mathrm {d} R}{\mathrm {d} \varphi }}\cdot k_{y}+R\cdot {\frac {\mathrm {d} k_{y}}{\mathrm {d} \varphi }}}
d
Y
d
φ
{\displaystyle {\tfrac {\mathrm {d} Y}{\mathrm {d} \varphi }}}
d
R
d
φ
{\displaystyle {\tfrac {\mathrm {d} R}{\mathrm {d} \varphi }}}
d
R
d
λ
{\displaystyle {\tfrac {\mathrm {d} R}{\mathrm {d} \lambda }}}
d
λ
d
φ
{\displaystyle {\tfrac {\mathrm {d} \lambda }{\mathrm {d} \varphi }}}
d
R
d
φ
=
d
R
d
λ
⋅
d
λ
d
φ
{\displaystyle {\frac {\mathrm {d} R}{\mathrm {d} \varphi }}={\frac {\mathrm {d} R}{\mathrm {d} \lambda }}\cdot {\frac {\mathrm {d} \lambda }{\mathrm {d} \varphi }}}
d
Y
d
φ
=
d
R
d
λ
⋅
d
λ
d
φ
⋅
k
y
+
R
⋅
d
k
y
d
φ
{\displaystyle {\frac {\mathrm {d} Y}{\mathrm {d} \varphi }}={\frac {\mathrm {d} R}{\mathrm {d} \lambda }}\cdot {\frac {\mathrm {d} \lambda }{\mathrm {d} \varphi }}\cdot k_{y}+R\cdot {\frac {\mathrm {d} k_{y}}{\mathrm {d} \varphi }}}
d
2
Y
d
φ
2
=
d
R
d
λ
⋅
(
d
2
λ
d
φ
2
⋅
k
y
+
d
λ
d
φ
⋅
d
k
y
d
φ
)
+
(
d
R
d
λ
⋅
d
λ
d
φ
⋅
d
k
y
d
φ
+
R
⋅
d
2
k
y
d
φ
2
)
{\displaystyle {\frac {\mathrm {d} ^{2}Y}{\mathrm {d} \varphi ^{2}}}={\frac {\mathrm {d} R}{\mathrm {d} \lambda }}\cdot \left({\frac {\mathrm {d} ^{2}\lambda }{\mathrm {d} \varphi ^{2}}}\cdot k_{y}+{\frac {\mathrm {d} \lambda }{\mathrm {d} \varphi }}\cdot {\frac {\mathrm {d} k_{y}}{\mathrm {d} \varphi }}\right)+\left({\frac {\mathrm {d} R}{\mathrm {d} \lambda }}\cdot {\frac {\mathrm {d} \lambda }{\mathrm {d} \varphi }}\cdot {\frac {\mathrm {d} k_{y}}{\mathrm {d} \varphi }}+R\cdot {\frac {\mathrm {d} ^{2}k_{y}}{\mathrm {d} \varphi ^{2}}}\right)}
d
2
Y
d
φ
2
=
d
R
d
λ
⋅
(
d
2
λ
d
φ
2
⋅
k
y
+
2
⋅
d
λ
d
φ
⋅
d
k
y
d
φ
)
+
R
⋅
d
2
k
y
d
φ
2
{\displaystyle {\frac {\mathrm {d} ^{2}Y}{\mathrm {d} \varphi ^{2}}}={\frac {\mathrm {d} R}{\mathrm {d} \lambda }}\cdot \left({\frac {\mathrm {d} ^{2}\lambda }{\mathrm {d} \varphi ^{2}}}\cdot k_{y}+2\cdot {\frac {\mathrm {d} \lambda }{\mathrm {d} \varphi }}\cdot {\frac {\mathrm {d} k_{y}}{\mathrm {d} \varphi }}\right)+R\cdot {\frac {\mathrm {d} ^{2}k_{y}}{\mathrm {d} \varphi ^{2}}}}
d
2
Y
d
φ
2
{\displaystyle {\tfrac {\mathrm {d} ^{2}Y}{\mathrm {d} \varphi ^{2}}}}
R
[
λ
(
θ
;
φ
=
0
)
]
=
R
(
θ
)
{\displaystyle R[\lambda (\theta ;\varphi \!=\!0)]=R(\theta )}
d
R
[
λ
(
θ
;
φ
=
0
)
]
d
λ
=
d
R
d
θ
=
S
m
{\displaystyle {\frac {\mathrm {d} R[\lambda (\theta ;\varphi \!=\!0)]}{\mathrm {d} \lambda }}={\frac {\mathrm {d} R}{\mathrm {d} \theta }}=S_{m}}
λ
(
θ
;
φ
=
0
)
=
θ
{\displaystyle \lambda (\theta ;\varphi \!=\!0)=\theta }
d
λ
d
φ
=
cos
θ
⋅
sin
φ
1
−
cos
2
θ
⋅
cos
2
φ
{\displaystyle {\frac {\mathrm {d} \lambda }{\mathrm {d} \varphi }}={\frac {\cos \theta \cdot \sin \varphi }{\sqrt {1-\cos ^{2}\theta \cdot \cos ^{2}\varphi }}}}
d
λ
d
φ
=
u
v
{\displaystyle {\frac {\mathrm {d} \lambda }{\mathrm {d} \varphi }}={\frac {u}{v}}}
d
2
λ
d
φ
2
=
d
u
d
φ
⋅
v
−
u
⋅
d
v
d
φ
v
2
{\displaystyle {\frac {\mathrm {d} ^{2}\lambda }{\mathrm {d} \varphi ^{2}}}={\frac {{\frac {\mathrm {d} u}{\mathrm {d} \varphi }}\cdot v-u\cdot {\frac {\mathrm {d} v}{\mathrm {d} \varphi }}}{v^{2}}}}
u
=
cos
θ
⋅
sin
φ
{\displaystyle u=\cos \theta \cdot \sin \varphi }
u
(
φ
=
0
)
=
0
{\displaystyle u(\varphi =0)=0}
v
=
1
−
cos
2
θ
⋅
cos
2
φ
{\displaystyle v={\sqrt {1-\cos ^{2}\theta \cdot \cos ^{2}\varphi }}}
v
(
φ
=
0
)
=
1
−
cos
2
θ
=
sin
θ
{\displaystyle v(\varphi =0)={\sqrt {1-\cos ^{2}\theta }}=\sin \theta }
v
2
=
1
−
cos
2
θ
⋅
cos
2
φ
{\displaystyle v^{2}=1-\cos ^{2}\theta \cdot \cos ^{2}\varphi }
v
2
(
φ
=
0
)
=
1
−
cos
2
θ
=
sin
2
θ
{\displaystyle v^{2}(\varphi =0)=1-\cos ^{2}\theta =\sin ^{2}\theta }
d
u
d
φ
=
cos
θ
⋅
cos
φ
{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} \varphi }}=\cos \theta \cdot \cos \varphi }
d
u
(
φ
=
0
)
d
φ
=
cos
θ
{\displaystyle {\frac {\mathrm {d} u(\varphi =0)}{\mathrm {d} \varphi }}=\cos \theta }
d
v
d
φ
=
1
2
⋅
1
−
cos
2
θ
⋅
cos
2
φ
⋅
−
2
cos
2
θ
⋅
cos
φ
⋅
−
sin
φ
=
cos
2
θ
cos
φ
⋅
sin
φ
1
−
cos
2
θ
⋅
cos
2
φ
{\displaystyle {\frac {\mathrm {d} v}{\mathrm {d} \varphi }}={\frac {1}{2\cdot {\sqrt {1-\cos ^{2}\theta \cdot \cos ^{2}\varphi }}}}\cdot -2\cos ^{2}\theta \cdot \cos \varphi \cdot -\sin \varphi ={\frac {\cos ^{2}\theta \cos \varphi \cdot \sin \varphi }{\sqrt {1-\cos ^{2}\theta \cdot \cos ^{2}\varphi }}}}
d
v
d
φ
=
0
{\displaystyle {\frac {\mathrm {d} v}{\mathrm {d} \varphi }}=0}
d
λ
d
φ
=
cos
θ
⋅
sin
φ
⋅
(
1
−
cos
2
θ
⋅
cos
2
φ
)
−
1
2
{\displaystyle {\frac {\mathrm {d} \lambda }{\mathrm {d} \varphi }}=\cos \theta \cdot \sin \varphi \cdot (1-\cos ^{2}\theta \cdot \cos ^{2}\varphi )^{-{\frac {1}{2}}}}
d
λ
d
φ
=
cos
θ
⋅
u
v
{\displaystyle {\frac {\mathrm {d} \lambda }{\mathrm {d} \varphi }}=\cos \theta \cdot uv}
d
λ
(
θ
;
φ
=
0
)
d
φ
=
0
{\displaystyle {\frac {\mathrm {d} \lambda (\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi }}=0}
d
2
λ
d
φ
2
{\displaystyle {\tfrac {\mathrm {d} ^{2}\lambda }{\mathrm {d} \varphi ^{2}}}}
d
2
λ
d
φ
2
=
cos
θ
⋅
(
d
u
d
φ
v
+
u
d
v
d
φ
)
{\displaystyle {\frac {\mathrm {d} ^{2}\lambda }{\mathrm {d} \varphi ^{2}}}=\cos \theta \cdot ({\frac {\mathrm {d} u}{\mathrm {d} \varphi }}v+u{\frac {\mathrm {d} v}{\mathrm {d} \varphi }})}
u
=
sin
φ
{\displaystyle u=\sin \varphi }
u
(
θ
;
φ
=
0
)
=
0
{\displaystyle u(\theta ;\varphi \!=\!0)=0}
d
u
d
φ
=
cos
φ
{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} \varphi }}=\cos \varphi }
d
u
(
θ
;
φ
=
0
)
d
φ
=
1
{\displaystyle {\frac {\mathrm {d} u(\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi }}=1}
v
=
(
1
−
cos
2
θ
⋅
cos
2
φ
)
−
1
2
{\displaystyle v=(1-\cos ^{2}\theta \cdot \cos ^{2}\varphi )^{-{\frac {1}{2}}}}
v
(
θ
;
φ
=
0
)
=
1
sin
θ
{\displaystyle v(\theta ;\varphi \!=\!0)={\frac {1}{\sin \theta }}}
d
v
d
φ
=
−
1
2
(
1
−
cos
2
θ
⋅
cos
2
φ
)
−
3
2
⋅
−
2
cos
2
θ
⋅
cos
φ
⋅
−
sin
φ
=
−
1
2
cos
2
θ
⋅
sin
2
φ
(
1
−
cos
2
θ
⋅
cos
2
φ
)
3
2
{\displaystyle {\frac {\mathrm {d} v}{\mathrm {d} \varphi }}=-{\frac {1}{2}}(1-\cos ^{2}\theta \cdot \cos ^{2}\varphi )^{-{\frac {3}{2}}}\cdot -2\cos ^{2}\theta \cdot \cos \varphi \cdot -\sin \varphi ={\frac {-{\frac {1}{2}}\cos ^{2}\theta \cdot \sin 2\varphi }{(1-\cos ^{2}\theta \cdot \cos ^{2}\varphi )^{\frac {3}{2}}}}}
d
v
(
θ
;
φ
=
0
)
d
φ
=
0
{\displaystyle {\frac {\mathrm {d} v(\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi }}=0}
d
2
λ
(
θ
;
φ
=
0
)
d
φ
2
=
cos
θ
⋅
(
1
⋅
1
sin
θ
+
0
⋅
0
)
=
cot
θ
{\displaystyle {\frac {\mathrm {d} ^{2}\lambda (\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi ^{2}}}=\cos \theta \cdot (1\cdot {\frac {1}{\sin \theta }}+0\cdot 0)=\cot \theta }
k
y
=
sin
θ
sin
2
θ
+
tan
2
φ
{\displaystyle k_{y}={\frac {\sin \theta }{\sqrt {\sin ^{2}\theta +\tan ^{2}\varphi }}}}
k
y
=
u
v
{\displaystyle k_{y}=uv}
d
k
y
d
φ
{\displaystyle {\tfrac {\mathrm {d} k_{y}}{\mathrm {d} \varphi }}}
d
k
y
d
φ
=
d
u
d
φ
v
+
u
d
v
d
φ
{\displaystyle {\frac {\mathrm {d} k_{y}}{\mathrm {d} \varphi }}={\frac {\mathrm {d} u}{\mathrm {d} \varphi }}v+u{\frac {\mathrm {d} v}{\mathrm {d} \varphi }}}
u
=
sin
θ
{\displaystyle u=\sin \theta }
d
u
d
φ
=
0
{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} \varphi }}=0}
v
=
(
sin
2
θ
+
tan
2
φ
)
−
1
2
{\displaystyle v=(\sin ^{2}\theta +\tan ^{2}\varphi )^{-{\frac {1}{2}}}}
v
(
θ
;
φ
=
0
)
=
1
sin
θ
{\displaystyle v(\theta ;\varphi \!=\!0)={\frac {1}{\sin \theta }}}
d
v
d
φ
=
−
1
2
(
sin
2
θ
+
tan
2
φ
)
−
3
2
⋅
2
tan
φ
⋅
1
cos
2
φ
=
−
sin
φ
cos
3
φ
⋅
(
sin
2
θ
+
tan
2
φ
)
−
3
2
{\displaystyle {\frac {\mathrm {d} v}{\mathrm {d} \varphi }}=-{\frac {1}{2}}(\sin ^{2}\theta +\tan ^{2}\varphi )^{-{\frac {3}{2}}}\cdot 2\tan \varphi \cdot {\frac {1}{\cos ^{2}\varphi }}={\frac {-\sin \varphi }{\cos ^{3}\varphi }}\cdot (\sin ^{2}\theta +\tan ^{2}\varphi )^{-{\frac {3}{2}}}}
d
v
(
θ
;
φ
=
0
)
d
φ
=
0
{\displaystyle {\frac {\mathrm {d} v(\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi }}=0}
d
k
y
d
φ
=
−
sin
θ
⋅
sin
φ
cos
3
φ
⋅
(
sin
2
θ
+
tan
2
φ
)
−
3
2
{\displaystyle {\frac {\mathrm {d} k_{y}}{\mathrm {d} \varphi }}={\frac {-\sin \theta \cdot \sin \varphi }{\cos ^{3}\varphi }}\cdot (\sin ^{2}\theta +\tan ^{2}\varphi )^{-{\frac {3}{2}}}}
d
k
y
(
θ
;
φ
=
0
)
d
φ
=
0
{\displaystyle {\frac {\mathrm {d} k_{y}(\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi }}=0}
d
2
k
y
d
φ
2
{\displaystyle {\tfrac {\mathrm {d} ^{2}k_{y}}{\mathrm {d} \varphi ^{2}}}}
d
u
d
φ
{\displaystyle {\tfrac {\mathrm {d} u}{\mathrm {d} \varphi }}}
d
v
d
φ
{\displaystyle {\tfrac {\mathrm {d} v}{\mathrm {d} \varphi }}}
d
w
d
φ
{\displaystyle {\tfrac {\mathrm {d} w}{\mathrm {d} \varphi }}}
d
k
y
d
φ
=
−
sin
θ
⋅
u
v
w
{\displaystyle {\frac {\mathrm {d} k_{y}}{\mathrm {d} \varphi }}=-\sin \theta \cdot uvw}
d
2
k
y
d
φ
2
=
−
sin
θ
⋅
(
d
u
d
φ
v
w
+
u
d
v
d
φ
w
+
u
v
d
w
d
φ
)
{\displaystyle {\frac {\mathrm {d} ^{2}k_{y}}{\mathrm {d} \varphi ^{2}}}=-\sin \theta \cdot ({\frac {\mathrm {d} u}{\mathrm {d} \varphi }}vw+u{\frac {\mathrm {d} v}{\mathrm {d} \varphi }}w+uv{\frac {\mathrm {d} w}{\mathrm {d} \varphi }})}
u
=
sin
φ
{\displaystyle u=\sin \varphi }
u
(
θ
;
φ
=
0
)
=
0
{\displaystyle u(\theta ;\varphi \!=\!0)=0}
d
u
d
φ
=
cos
φ
{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} \varphi }}=\cos \varphi }
d
u
(
θ
;
φ
=
0
)
d
φ
=
1
{\displaystyle {\frac {\mathrm {d} u(\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi }}=1}
v
=
cos
−
3
φ
{\displaystyle v=\cos ^{-3}\varphi }
v
(
θ
;
φ
=
0
)
=
1
{\displaystyle v(\theta ;\varphi \!=\!0)=1}
d
v
d
φ
=
sin
φ
3
cos
4
φ
{\displaystyle {\frac {\mathrm {d} v}{\mathrm {d} \varphi }}={\frac {\sin \varphi }{3\cos ^{4}\varphi }}}
d
v
(
θ
;
φ
=
0
)
d
φ
=
0
{\displaystyle {\frac {\mathrm {d} v(\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi }}=0}
w
=
(
sin
2
θ
+
tan
2
φ
)
−
3
2
{\displaystyle w=(\sin ^{2}\theta +\tan ^{2}\varphi )^{-{\frac {3}{2}}}}
w
(
θ
;
φ
=
0
)
=
sin
−
3
θ
{\displaystyle w(\theta ;\varphi \!=\!0)=\sin ^{-3}\theta }
d
w
d
φ
=
−
3
sin
φ
cos
3
φ
⋅
(
s
i
n
2
θ
+
tan
2
φ
)
5
2
{\displaystyle {\frac {\mathrm {d} w}{\mathrm {d} \varphi }}={\frac {-3\sin \varphi }{\cos ^{3}\varphi \cdot (sin^{2}\theta +\tan ^{2}\varphi )^{\frac {5}{2}}}}}
d
w
(
θ
;
φ
=
0
)
d
φ
=
0
{\displaystyle {\frac {\mathrm {d} w(\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi }}=0}
d
2
k
y
(
θ
;
φ
=
0
)
d
φ
2
=
−
sin
θ
⋅
(
1
⋅
1
⋅
1
sin
3
θ
+
0
⋅
0
⋅
w
+
0
⋅
v
⋅
0
)
=
−
1
sin
2
θ
{\displaystyle {\frac {\mathrm {d} ^{2}k_{y}(\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi ^{2}}}=-\sin \theta \cdot (1\cdot 1\cdot {\frac {1}{\sin ^{3}\theta }}+0\cdot 0\cdot w+0\cdot v\cdot 0)={\frac {-1}{\sin ^{2}\theta }}}
d
2
Y
(
θ
;
φ
=
0
)
d
φ
2
=
S
m
⋅
(
cot
θ
⋅
1
+
2
⋅
0
⋅
0
)
+
R
⋅
−
1
sin
2
θ
=
S
m
⋅
cot
θ
−
R
sin
2
θ
=
S
m
⋅
0
,
5
sin
2
θ
−
R
sin
2
θ
{\displaystyle {\frac {\mathrm {d} ^{2}Y(\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi ^{2}}}=S_{m}\cdot \left(\cot \theta \cdot 1+2\cdot 0\cdot 0\right)+R\cdot {\frac {-1}{\sin ^{2}\theta }}=S_{m}\cdot \cot \theta -{\frac {R}{\sin ^{2}\theta }}={\frac {S_{m}\cdot 0{,}5\sin 2\theta -R}{\sin ^{2}\theta }}}
r
=
f
⋅
R
{\displaystyle r=f\cdot R}
y
=
f
⋅
Y
{\displaystyle y=f\cdot Y}
d
2
y
d
φ
2
=
f
⋅
d
2
Y
d
φ
2
{\displaystyle {\frac {\mathrm {d} ^{2}y}{d\varphi ^{2}}}=f\cdot {\frac {\mathrm {d} ^{2}Y}{d\varphi ^{2}}}}
d
2
y
(
θ
;
φ
=
0
)
d
φ
2
=
M
m
⋅
0
,
5
sin
2
θ
−
r
sin
2
θ
{\displaystyle {\frac {\mathrm {d} ^{2}y(\theta ;\varphi \!=\!0)}{\mathrm {d} \varphi ^{2}}}={\frac {M_{m}\cdot 0{,}5\sin 2\theta -r}{\sin ^{2}\theta }}}
Tabelle 3: Hin- und Rückprojektion mit fundamentalen Funktionen
Projektionstyp
Projektionsrichtung
Rückprojektion
Abbildung
gnomonisch
r
x
y
=
g
{\displaystyle r_{xy}=g}
z
=
1
{\displaystyle z=1}
g
=
r
x
y
z
{\displaystyle g={\frac {r_{xy}}{z}}}
Fischauge
winkeltreu
r
x
y
=
s
{\displaystyle r_{xy}=s}
z
=
1
−
s
2
4
{\displaystyle z=1-{\frac {s^{2}}{4}}}
s
=
2
r
x
y
z
+
r
{\displaystyle s={\frac {2r_{xy}}{z+r}}}
linear
r
x
y
=
l
{\displaystyle r_{xy}=l}
z
=
l
tan
l
{\displaystyle z={\frac {l}{\tan l}}}
r
x
y
=
sin
l
{\displaystyle r_{xy}=\sin l}
z
=
cos
l
{\displaystyle z=\cos l}
l
=
2
arctan
r
x
y
z
+
r
{\displaystyle l=2\arctan {\frac {r_{xy}}{z+r}}}
flächentreu
r
x
y
=
a
{\displaystyle r_{xy}=a}
z
=
2
−
a
2
4
−
a
2
{\displaystyle z={\frac {2-a^{2}}{\sqrt {4-a^{2}}}}}
r
x
y
=
a
1
−
a
2
4
{\displaystyle r_{xy}=a{\sqrt {1-{\frac {a^{2}}{4}}}}}
z
=
1
−
a
2
2
{\displaystyle z=1-{\frac {a^{2}}{2}}}
a
=
r
x
y
2
r
(
z
+
r
)
{\displaystyle a=r_{xy}{\sqrt {\frac {2}{r(z+r)}}}}
orthografisch
r
x
y
=
o
{\displaystyle r_{xy}=o}
z
=
1
−
o
2
{\displaystyle z={\sqrt {1-o^{2}}}}
o
=
r
x
y
r
{\displaystyle o={\frac {r_{xy}}{r}}}
Die Formeln für die Rückprojektion erzeugen Bildschalen, die per Parallelprojektion wieder das Bild ergeben. Im Fall der linearen und der flächentreuen Projektion kann die Bildschale für den Polwinkel von 180° nicht berechnet werden, weil dann eine Division durch Null stattfindet. Für die beiden Projektionen sind die Tabellenzellen zweigeteilt. Auf der linken Hälfte entsteht eine Bildschale, bei der der z-Wert in der Nähe vom 180°-rxy -Wert ins Unendliche geht. Auf der rechten Hälfte sind die Formeln so skaliert, dass bei 180° keine Division durch Null stattfindet. Es ergibt sich eine andere Bildschale, die nicht mehr so einfach in das Bild überführt werden kann. Das ist aber auch gar nicht nötig, denn das erledigt die Formel in der Spalte Abbildung . Wichtig ist, dass die originale Blickrichtung im Raum wiederhergestellt wird. Der Abstand zur Eintrittspupille ist dabei beliebig.
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![{\displaystyle Y=R[\arccos(\cos \varphi \cdot \cos \theta )]\cdot {\frac {\sin \theta }{\sqrt {\tan ^{2}\varphi +\sin ^{2}\theta }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/869934773c781e39af0f12482e59ef68ebe03a8d)
![{\displaystyle Y=R[\arccos(\cos \varphi \cdot \cos \theta )]\cdot {\frac {\cos \varphi \cdot \sin \theta }{\sqrt {\sin ^{2}\varphi +\cos ^{2}\varphi \cdot \sin ^{2}\theta }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e96d4201294c06d14ec522380ccf7e997bf7b04)
![{\displaystyle Y\lbrace R[\lambda (\theta ;\varphi )];k_{y}(\theta ;\varphi )\rbrace }](https://wikimedia.org/api/rest_v1/media/math/render/svg/791df8302fcab5edfc49969acd426c5248dfb211)
![{\displaystyle R[\lambda (\theta ;\varphi \!=\!0)]=R(\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6070f431ea0de5645d0314424ce582cf364bc4f)
![{\displaystyle {\frac {\mathrm {d} R[\lambda (\theta ;\varphi \!=\!0)]}{\mathrm {d} \lambda }}={\frac {\mathrm {d} R}{\mathrm {d} \theta }}=S_{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19ccd50da5f7a7fefecf3a110cad1e989444e31f)
