Documentation

# gcxgc

Intersection points for pairs of great circles

## Syntax

``[lat,lon] = gcxgc(lat1,lon1,az1,lat2,lon2,az2)``
``[lat,lon] = gcxgc(lat1,lon1,az1,lat2,lon2,az2,units)``
``latlon = gcxgc(___)``

## 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 `NaN`s 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 `NaN`s for the latitude and longitude coordinates of the intersection points. 