gcxgc

Intersection points for pairs of great circles

Description

example

[lat,lon] = gcxgc(lat1,lon1,az1,lat2,lon2,az2) returns in lat and lon the locations where pairs of great circles intersect. The great circles are defined using great circle notation, which consists of a point on the great circle and the azimuth at that point along which the great circle proceeds. For example, the first great circle in a pair would pass through the point (lat1,lon1) with an azimuth of az1 (in angular units).

For any pair of great circles, there are two possible intersection conditions: the circles are identical or they intersect exactly twice on the sphere.

[lat,lon] = gcxgc(lat1,lon1,az1,lat2,lon2,az2,units) specifies the angular units used for all inputs, where units is any valid angular unit.

latlon = gcxgc(___) returns a single output consisting of the concatenated latitude and longitude coordinates of the great circle intersection points.

Examples

Find Intersection Points of Two Great Circles

Given a great circle passing through (10ºN,13ºE) and proceeding on an azimuth of 10º, where does it intersect with a great circle passing through (0º, 20ºE), on an azimuth of -23º (that is, 337º)?

[newlat,newlon] = gcxgc(10,13,10,0,20,-23)
newlat =
14.3105  -14.3105

newlon =
13.7838  -166.2162

Note that the two intersection points are always antipodes of each other. As a simple example, consider the intersection points of two meridians, which are just great circles with azimuths of 0º or 180º:

[newlat,newlon] = gcxgc(10,13,0,0,20,180)
newlat =

-90    90

newlon =

0   180

The two meridians intersect at the North and South Poles, which is exactly correct.

Input Arguments

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Latitude or longitude coordinate of a point on the first great circle in each pair, specified as one of these values.

• A numeric scalar to find the intersection of a single pair of great circles.

• A n-element numeric vector to find the intersection of n pairs of great circles.

lat1 and lon1 must have the same length.

Example: 10

Example: [-10 20 90 -45]

Azimuth of the first great circle of each pair, in angular units, specified as one of these values.

• A positive numeric scalar to find the intersection of a single pair of great circles.

• A n-element vector of positive numbers to find the intersection of n pairs of great circles. The length of az1 matches the length of lat1 and lon1.

Example: 20

Example: [20 10 45 45]

Latitude or longitude coordinate of a point on the second great circle in each pair, specified as one of these values.

• A numeric scalar to find the intersection of a single pair of great circles.

• A n-element numeric vector to find the intersection of n pairs of great circles.

lat2 and lon2 must have the same length as lat1 and lon1.

Example: 3

Example: [3 30 85 -45]

Azimuth of the second great circle of each pair, in angular units, specified as one of these values.

• A positive numeric scalar to find the intersection of a single pair of great circles.

• A n-element vector of positive numbers to find the intersection of n pairs of great circles. The length of az2 matches the length of lat2 and lon2.

Example: 15

Example: [15 15 45 50]

Angular units, specified as 'degrees' or 'radians'.

Output Arguments

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Coordinates of great circle intersections, returned as one of the following.

• 2-element vector when you find the intersection of a single pair of great circles.

• n-by-2 matrix when you find the intersection of n pairs of great circles.

If a pair of great circles are identical, then gcxgc displays a warning and returns NaNs for the latitude and longitude coordinates of the intersection points.

Concatenated coordinates of great circle intersections, returned as one of the following. This output is identical to [lat lon].

• 4-element vector when you find the intersection of a single pair of great circles.

• n-by-4 matrix when you find the intersection of n pairs of great circles.

If a pair of great circles are identical, then gcxgc displays a warning and returns NaNs for the latitude and longitude coordinates of the intersection points. 