Angle AOB forms a central angle of circle O
A central angle is an angle which vertex is the center of a circle , and whose sides pass through a pair of points on the circle, thereby subtending an arc between those two points whose angle is (by definition) equal to the central angle itself. It is also known as the arc segment 's angular distance .
Coordinates [ ] On a sphere or ellipsoid , the central angle is delineated along a great circle .
The usually provided coordinates of a point on a sphere/ellipsoid is its conjugate latitude ("Lat"),
ϕ
{\displaystyle \phi\,\!}
longitude ("Long"),
λ
{\displaystyle \lambda \,\!}
σ
^
{\displaystyle \widehat{\sigma}\,\!}
transverse colatitude ("TvL"), and the central angle/angular distance is the difference between two TvLs,
Δ
σ
^
{\displaystyle \Delta\widehat{\sigma}\,\!}
Calculation of TvL [ ] The calculation of
σ
^
s
{\displaystyle \widehat{\sigma}_s\,\!}
σ
^
f
{\displaystyle \widehat{\sigma}_f\,\!}
V
s
,
V
f
,
V
w
,
V
c
:
S
t
a
n
d
p
o
i
n
t
,
f
o
r
e
p
o
i
n
t
,
w
o
r
k
i
n
g
,
c
o
w
o
r
k
i
n
g
v
a
l
u
e
s
;
{\displaystyle V_s,V_f,V_w,V_c:\mathrm{\;Standpoint,\ forepoint,\ working,\ coworking\ values};\,\!}
α
^
w
:
O
r
t
h
o
d
r
o
m
i
c
a
z
i
m
u
t
h
a
t
σ
^
w
;
{\displaystyle \widehat{\alpha}_w:\mathrm{\;Orthodromic\ azimuth\ at\ \widehat{\sigma}_w};\,\!}
.
(
sgn
(
V
)
=
|
V
|
⋅
V
−
1
;
sgn
→
(
V
)
=
sgn
(
sgn
(
V
)
+
1
2
)
(
sgn
(
0
)
=
0
;
sgn
→
(
0
)
=
+
1
)
)
.
{\displaystyle {}_{\color{white}.}\!\begin{pmatrix}\operatorname{sgn}(V)=|V|\!\cdot
V^{-1};\quad\overrightarrow{\operatorname{sgn}}(V)=\operatorname{sgn}\big(\operatorname{sgn}(V)+\frac{1}{2}\big)\\{}_{(\,\operatorname{sgn}(0)=0;\qquad\overrightarrow{\operatorname{sgn}}(0)=+1\,)}\end{pmatrix}{}_{\color{white}.}\!\!\,\!}
Δ
λ
=
λ
f
−
λ
s
;
{\displaystyle \Delta\lambda=\lambda_f-\lambda_s;\,\!}
.
(
If
ϕ
s
=
ϕ
f
=
0
, then
σ
^
s
=
π
−
|
Δ
λ
|
2
,
σ
^
f
=
π
+
|
Δ
λ
|
2
)
.
{\displaystyle {}_{\color{white}.}\!\left(\mbox{If } \phi_s=\phi_f=0\mbox{, then }\;\widehat{\sigma}_s=\frac{\pi-|\Delta\lambda|}{2},\;\widehat{\sigma}_f=\frac{\pi+|\Delta\lambda|}{2}\right){}_{\color{white}.}\!\!\,\!}
ϕ
w
=
ϕ
s
;
ϕ
c
=
ϕ
f
:
Get
σ
^
w
:
σ
^
s
=
σ
^
w
⋅
sgn
→
(
S
B
w
)
+
π
⋅
sgn
→
(
σ
^
w
)
sgn
(
1
−
sgn
→
(
S
B
w
)
)
;
{\displaystyle \begin{align}\phi_w=\phi_s;\;
&\phi_c=\phi_f\!\!:\mbox{Get}\;\widehat{\sigma}_w\!\!:\\
&\widehat{\sigma}_s=\widehat{\sigma}_w\!\cdot\overrightarrow{\mbox{sgn}}(S\!B_w)+\pi\!\cdot\overrightarrow{\mbox{sgn}}(\widehat{\sigma}_w)\mbox{sgn}(1-\overrightarrow{\mbox{sgn}}(S\!B_w));\end{align}\,\!}
ϕ
w
=
ϕ
f
;
ϕ
c
=
ϕ
s
:
Get
σ
^
w
:
σ
^
f
=
σ
^
w
⋅
sgn
→
(
−
S
B
w
)
+
π
⋅
sgn
→
(
σ
^
w
)
sgn
(
1
−
sgn
→
(
−
S
B
w
)
)
+
2
π
⋅
sgn
(
1
−
sgn
→
(
σ
^
w
−
σ
^
s
)
)
;
{\displaystyle \begin{align}\phi_w=\phi_f;\;
&\phi_c=\phi_s\!\!:\mbox{Get}\;\widehat{\sigma}_w\!\!:\\
&\widehat{\sigma}_f=\widehat{\sigma}_w\!\cdot\overrightarrow{\mbox{sgn}}(-S\!B_w)+\pi\!\cdot\overrightarrow{\mbox{sgn}}(\widehat{\sigma}_w)\mbox{sgn}(1-\overrightarrow{\mbox{sgn}}(-S\!B_w))\\
&\qquad\qquad\qquad\qquad\quad+2\pi\!\cdot\mbox{sgn}(1-\overrightarrow{\mbox{sgn}}(\widehat{\sigma}_w-\widehat{\sigma}_s));\end{align}\,\!}
_____________________________________________________________________
S
A
w
=
cos
(
ϕ
c
)
sin
(
Δ
λ
)
;
S
B
w
=
sin
(
ϕ
w
+
ϕ
c
)
sin
2
(
Δ
λ
2
)
+
sin
(
ϕ
c
−
ϕ
w
)
cos
2
(
Δ
λ
2
)
;
{\displaystyle \begin{matrix}S\!A_w=\cos(\phi_c)\sin(\Delta\lambda);\qquad\qquad\qquad\qquad\qquad\qquad\;\;\\S\!B_w=\sin(\phi_w+\phi_c)\sin^2(\frac{\Delta\lambda}{2})+\sin(\phi_c-\phi_w)\cos^2(\frac{\Delta\lambda}{2});\end{matrix}\,\!}
(
sin
2
(
Δ
σ
^
)
=
S
A
w
2
+
S
B
w
2
;
|
tan
(
a
^
w
)
|
=
|
S
A
w
S
B
w
|
)
{\displaystyle \left(\,\sin^2(\Delta\widehat{\sigma})={S\!A_w}^2+{S\!B_w}^2;\quad|\tan(\widehat{a}_w)|=\left|\frac{S\!A_w}{S\!B_w}\right|\,\right)\,\!}
σ
^
w
=
arctan
(
|
sec
(
a
^
w
)
|
tan
(
ϕ
w
)
)
=
arctan
(
|
sin
(
Δ
σ
^
)
S
B
w
|
tan
(
ϕ
w
)
)
,
=
arctan
(
S
A
w
2
+
S
B
w
2
|
S
B
w
|
tan
(
ϕ
w
)
)
.
{\displaystyle \begin{matrix}\widehat{\sigma}_w\!\!\!&=&\!\!\!\arctan\big(|\sec(\widehat{a}_w)|\tan(\phi_w)\big)=\arctan\!\left(\left|\frac{\sin(\Delta\widehat{\sigma})}{S\!B_w}\right|\tan(\phi_w)\right),\\&=&\!\!\!\!\!\!\arctan\!\left(\frac{\sqrt{{S\!A_w}^2+{S\!B_w}^2}}{|S\!B_w|}\tan(\phi_w)\right).\qquad\qquad\qquad\qquad\qquad\end{matrix}}
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Each point has at least two values, both a forward and reverse value.
Occupying great circle [ ] The arc path,
A
^
{\displaystyle \scriptstyle{\widehat{\Alpha}}\,\!}
Clairaut's constant
sin
(
A
^
)
=
|
cos
(
ϕ
w
)
sin
(
α
^
w
)
|
;
{\displaystyle \sin(\widehat{\Alpha})=\Big|\cos(\phi_w)\sin(\widehat{\alpha}_w)\Big|;\,\!}
From this and relationships to
σ
^
{\displaystyle \widehat{\sigma}\,\!}
A
^
=
|
arcsin
(
cos
(
ϕ
w
)
sin
(
α
^
w
)
)
|
=
|
arccos
(
sin
(
ϕ
w
)
sin
(
σ
^
w
)
)
|
,
=
|
arctan
(
cos
(
σ
^
w
)
tan
(
α
^
w
)
)
|
=
|
arctan
(
sin
(
α
^
w
)
sin
(
σ
^
w
)
cot
(
ϕ
w
)
)
|
.
{\displaystyle \begin{align}\widehat{\Alpha}
&=\Big|\arcsin\big(\cos(\phi_w)\sin(\widehat{\alpha}_w)\big)\Big|\!\!\!&&=\Big|\arccos\left(\frac{\sin(\phi_w)}{\sin(\widehat{\sigma}_w)}\right)\Big|,\\
&=\Big|\arctan\big(\cos(\widehat{\sigma}_w)\tan(\widehat{\alpha}_w)\big)\Big|\!\!\!&&=\Big|\arctan\big(\sin(\widehat{\alpha}_w)\sin(\widehat{\sigma}_w)\cot(\phi_w)\big)\Big|.\end{align}\,\!}
Angular distance formulary [ ] The angular distance can be calculated either directly as the TvL difference, or via the common coordinates (here, either SAw , SBw value set can be used):
.
Δ
σ
^
=
σ
^
f
−
σ
^
s
,
=
arcsin
(
S
A
2
+
S
B
2
)
,
(
c
a
n
o
n
l
y
f
i
n
d
t
h
e
f
i
r
s
t
q
u
a
d
r
a
n
t
,
i
.
e
.
,
u
p
t
o
90
∘
)
=
arccos
(
sin
(
ϕ
s
)
sin
(
ϕ
f
)
+
cos
(
ϕ
s
)
cos
(
ϕ
f
)
cos
(
Δ
λ
)
)
,
(
n
o
t
r
e
c
o
m
m
e
n
d
e
d
f
o
r
s
m
a
l
l
a
n
g
l
e
s
,
d
u
e
t
o
r
o
u
n
d
i
n
g
e
r
r
o
r
)
=
arctan
(
S
A
2
+
S
B
2
sin
(
ϕ
s
)
sin
(
ϕ
f
)
+
cos
(
ϕ
s
)
cos
(
ϕ
f
)
cos
(
Δ
λ
)
)
,
.
{\displaystyle \begin{align}{}_{\color{white}.}\\\Delta\widehat{\sigma}
&=\widehat{\sigma}_f\;-\;\widehat{\sigma}_s,\\
&=\arcsin\!\left(\sqrt{{S\!A}^2+{S\!B}^2}\,\right),\\
&\quad{}^{\mathit{(can\,only\,find\,the\,first\,quadrant,\,i.e.,\;up\,to\,90^\circ)}}\\
&=\arccos\!\Big(\sin(\phi_s)\sin(\phi_f)+\cos(\phi_s)\cos(\phi_f)\cos(\Delta\lambda)\,\Big),\\
&\quad{}^{\mathit{(not\,recommended\,for\,small\,angles,\;due\,to\,rounding\,error)}}\\
&=\arctan\!\left(\frac{\sqrt{{S\!A}^2+{S\!B}^2}}{\sin(\phi_s)\sin(\phi_f)+\cos(\phi_s)\cos(\phi_f)\cos(\Delta\lambda)}\right),\\{}^{\color{white}.}\end{align}\,\!}
and, using half-angles,
.
=
2
arcsin
(
sin
2
(
ϕ
f
−
ϕ
s
2
)
+
cos
(
ϕ
s
)
cos
(
ϕ
f
)
sin
2
(
Δ
λ
2
)
)
,
=
2
arccos
(
cos
2
(
ϕ
f
−
ϕ
s
2
)
−
cos
(
ϕ
s
)
cos
(
ϕ
f
)
sin
2
(
Δ
λ
2
)
)
,
=
2
arctan
(
sin
2
(
ϕ
f
−
ϕ
s
2
)
+
cos
(
ϕ
s
)
cos
(
ϕ
f
)
sin
2
(
Δ
λ
2
)
cos
2
(
ϕ
f
−
ϕ
s
2
)
−
cos
(
ϕ
s
)
cos
(
ϕ
f
)
sin
2
(
Δ
λ
2
)
)
.
.
{\displaystyle \begin{align}{}_{\color{white}.}\\
&=2\arcsin\!\left(\sqrt{\sin^2\!\left(\frac{\phi_f-\phi_s}{2}\right)+\cos(\phi_s)\cos(\phi_f)\sin^2\!\left(\frac{\Delta\lambda}{2}\right)}\,\right),\\
&=2\arccos\!\left(\sqrt{\cos^2\!\left(\frac{\phi_f-\phi_s}{2}\right)-\cos(\phi_s)\cos(\phi_f)\sin^2\!\left(\frac{\Delta\lambda}{2}\right)}\,\right),\\
&=2\arctan\!\left(\sqrt{\frac{\sin^2\left(\frac{\phi_f-\phi_s}{2}\right)+\cos(\phi_s)\cos(\phi_f)\sin^2\Big(\frac{\Delta\lambda}{2}\Big)}{\cos^2\left(\frac{\phi_f-\phi_s}{2}\right)-\cos(\phi_s)\cos(\phi_f)\sin^2\!\Big(\frac{\Delta\lambda}{2}\Big)}}\,\right).\\{}^{\color{white}.}\end{align}\,\!}
It can, as well, be found by means of finding the chord length via Cartesian subtraction :[ 1]
Δ
X
=
cos
(
ϕ
f
)
cos
(
λ
f
)
−
cos
(
ϕ
s
)
cos
(
λ
s
)
;
Δ
Y
=
cos
(
ϕ
f
)
sin
(
λ
f
)
−
cos
(
ϕ
s
)
sin
(
λ
s
)
;
Δ
Z
=
sin
(
ϕ
f
)
−
sin
(
ϕ
s
)
;
C
h
=
(
Δ
X
)
2
+
(
Δ
Y
)
2
+
(
Δ
Z
)
2
;
Δ
σ
^
=
2
arcsin
(
C
h
2
)
.
{\displaystyle \begin{align}
&\Delta{X}=\cos(\phi_f)\cos(\lambda_f) - \cos(\phi_s)\cos(\lambda_s);\\
&\Delta{Y}=\cos(\phi_f)\sin(\lambda_f) - \cos(\phi_s)\sin(\lambda_s);\\
&\Delta{Z}=\sin(\phi_f) - \sin(\phi_s);\\
&C_h=\sqrt{(\Delta{X})^2+(\Delta{Y})^2+(\Delta{Z})^2};\\
&\Delta\widehat{\sigma}=2\arcsin\left(\frac{C_h}{2}\right).\end{align}\,\!}
Also, by using Cartesian products rather than differences, the origin of the spherical cosine for sides
Π
X
=
cos
(
ϕ
s
)
cos
(
ϕ
f
)
cos
(
λ
s
)
cos
(
λ
f
)
;
Π
Y
=
cos
(
ϕ
s
)
cos
(
ϕ
f
)
sin
(
λ
s
)
sin
(
λ
f
)
;
Π
Z
=
sin
(
ϕ
s
)
sin
(
ϕ
f
)
;
Π
X
+
Π
Y
cos
(
ϕ
s
)
cos
(
ϕ
f
)
=
cos
(
λ
s
)
cos
(
λ
f
)
+
sin
(
λ
s
)
sin
(
λ
f
)
=
cos
(
Δ
λ
)
;
Δ
σ
^
=
arccos
(
Π
X
+
Π
Y
+
Π
Z
)
=
arccos
(
Π
Z
+
(
Π
X
+
Π
Y
)
)
,
=
arccos
(
sin
(
ϕ
s
)
sin
(
ϕ
f
)
+
cos
(
ϕ
s
)
cos
(
ϕ
f
)
cos
(
Δ
λ
)
)
.
{\displaystyle \begin{align}
{\scriptstyle{\Pi}}X&=\cos(\phi_s)\cos(\phi_f)\cos(\lambda_s)\cos(\lambda_f);\\
{\scriptstyle{\Pi}}Y&=\cos(\phi_s)\cos(\phi_f)\sin(\lambda_s)\sin(\lambda_f);\\
{\scriptstyle{\Pi}}Z&=\sin(\phi_s)\sin(\phi_f);\\
\frac{{\scriptstyle{\Pi}}X\!\!+\!{\scriptstyle{\Pi}}Y}{\cos(\phi_s)\cos(\phi_f)}&=\cos(\lambda_s)\cos(\lambda_f)+\sin(\lambda_s)\sin(\lambda_f)=\cos(\Delta\lambda);\\
\Delta\widehat{\sigma}&=\arccos\Big({\scriptstyle{\Pi}}X+{\scriptstyle{\Pi}}Y+{\scriptstyle{\Pi}}Z\Big)
=\arccos\Big({\scriptstyle{\Pi}}Z+\big({\scriptstyle{\Pi}}X+{\scriptstyle{\Pi}}Y\big)\Big),\\
&=\arccos\Big(\sin(\phi_s)\sin(\phi_f)+\cos(\phi_s)\cos(\phi_f)\cos(\Delta\lambda)\Big).\end{align}\,\!}
There is also a logarithmical form:
.
N
=
exp
(
ln
(
cos
(
ϕ
f
−
ϕ
s
2
)
sin
(
ϕ
s
+
ϕ
f
2
)
)
−
ln
(
tan
(
|
Δ
λ
|
2
)
)
)
;
{\displaystyle {}_{\color{white}.}\;\mathbb{N}=\exp\left(\ln\!\left(\frac{\cos\left(\frac{\phi_f-\phi_s}{2}\right)}{\sin\left(\frac{\phi_s+\phi_f}{2}\right)}\right)-\ln\left(\tan\Big(\frac{|\Delta\lambda|}{2}\Big)\right)\right);\,\!}
.
D
=
exp
(
ln
(
sin
(
|
ϕ
f
−
ϕ
s
|
2
)
cos
(
ϕ
s
+
ϕ
f
2
)
)
−
ln
(
tan
(
|
Δ
λ
|
2
)
)
)
;
{\displaystyle {}_{\color{white}.}\;\mathbb{D}=\exp\left(\ln\!\left(\frac{\sin\left(\frac{|\phi_f-\phi_s|}{2}\right)}{\cos\left(\frac{\phi_s+\phi_f}{2}\right)}\right)-\ln\left(\tan\Big(\frac{|\Delta\lambda|}{2}\Big)\right)\right);\,\!}
.
Δ
σ
^
=
2
arctan
(
|
exp
(
ln
(
sin
(
arctan
(
N
)
)
sin
(
arctan
(
D
)
)
)
+
ln
(
tan
(
|
ϕ
f
−
ϕ
s
|
2
)
)
)
|
)
.
{\displaystyle {}_{\color{white}.}\quad\!\Delta\widehat{\sigma}=2\arctan\!\left(\,\left|\exp\left(\ln\!\left(\frac{\sin(\arctan(\mathbb{N}))}{\sin(\arctan(\mathbb{D}))}\right)+\ln\left(\tan\Big(\frac{|\phi_f-\phi_s|}{2}\Big)\right)\right)\right|\,\right).\,\!}
See also [ ] References [ ] External links [ ] 
