Finding way-points Great-circle navigation

figure 2. great circle path between node (an equator crossing) , arbitrary point (φ,λ).

finally, calculate position , azimuth @ arbitrary point, p (see fig. 2), spherical version of direct geodesic problem. napier s rules give

.


)


sin
⁡
ϕ
=
cos
⁡

α

0


sin
⁡
σ
,


{\displaystyle {\color {white}.\,\qquad )}\sin \phi =\cos \alpha _{0}\sin \sigma ,}










tan
⁡
(
λ
−

λ

0


)



=



sin
⁡

α

0


sin
⁡
σ


cos
⁡
σ



,




tan
⁡
α



=



tan
⁡

α

0




cos
⁡
σ



.






{\displaystyle {\begin{aligned}\tan(\lambda -\lambda _{0})&={\frac {\sin \alpha _{0}\sin \sigma }{\cos \sigma }},\\\tan \alpha &={\frac {\tan \alpha _{0}}{\cos \sigma }}.\end{aligned}}}

the atan2 function should used determine σ01, λ, , α. example, find midpoint of path, substitute σ = ½(σ01 + σ02); alternatively find point distance d starting point, take σ = σ01 + d/r. likewise, vertex, point on great circle greatest latitude, found substituting σ = +½π. may convenient parameterize route in terms of longitude using

tan
⁡
ϕ
=
cot
⁡

α

0


sin
⁡
(
λ
−

λ

0


)
.


{\displaystyle \tan \phi =\cot \alpha _{0}\sin(\lambda -\lambda _{0}).}

latitudes @ regular intervals of longitude can found , resulting positions transferred mercator chart allowing great circle approximated series of rhumb lines. path determined in way gives great ellipse joining end points, provided coordinates



(
ϕ
,
λ
)


{\displaystyle (\phi ,\lambda )}

interpreted geographic coordinates on ellipsoid.

these formulas apply spherical model of earth. used in solving great circle on auxiliary sphere device finding shortest path, or geodesic, on ellipsoid of revolution; see article on geodesics on ellipsoid.

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