Earth section paths

Earth section paths are plane curves defined by the intersection of an earth ellipsoid and a plane (ellipsoid plane sections). Common examples include the great ellipse (containing the center of the ellipsoid) and normal sections (containing an ellipsoid normal direction). Earth section paths are useful as approximate solutions for geodetic problems, the direct and inverse calculation of geographic distances. The rigorous solution of geodetic problems involves skew curves known as geodesics.

Inverse problem

The inverse problem for earth sections is: given two points,

P

1

{\displaystyle P_{1}}

and

P

2

{\displaystyle P_{2}}

on the surface of the reference ellipsoid, find the length,

s

12

{\displaystyle s_{12}}

, of the short arc of a spheroid section from

P

1

{\displaystyle P_{1}}

to

P

2

{\displaystyle P_{2}}

and also find the departure and arrival azimuths (angle from true north) of that curve,

α

1

{\displaystyle \alpha _{1}}

and

α

2

{\displaystyle \alpha _{2}}

. The figure to the right illustrates the notation used here. Let

P

k

{\displaystyle P_{k}}

have geodetic latitude

ϕ

k

{\displaystyle \phi _{k}}

and longitude

λ

k

{\displaystyle \lambda _{k}}

(k=1,2). This problem is best solved using analytic geometry in earth-centered, earth-fixed (ECEF) Cartesian coordinates.

Let

R

1

=

E

C

E

F

(

P

1

)

{\displaystyle R_{1}=\mathrm {ECEF} (P_{1})}

and

R

2

=

E

C

E

F

(

P

2

)

{\displaystyle R_{2}=\mathrm {ECEF} (P_{2})}

be the ECEF coordinates of the two points, computed using the geodetic to ECEF transformation discussed here.

Section plane

To define the section plane select any third point

R

0

{\displaystyle R_{0}}

not on the line from

R

1

{\displaystyle R_{1}}

to

R

2

{\displaystyle R_{2}}

.