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- Given a latitude/longitude point, an angle from due North, and a distance, calculate the latitude/longitude of the new point. This is calculated with any function starting with `DirectCalc...`
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- Given two latitude/longitude points, calculate the distance between them, and the angles formed from due North to the line connecting the two points. This is calculated with any function starting with `InverseCalc...`
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- Given a latitude/longitude point, an angle from due North, and a distance, calculate the latitude/longitude of the new point. This is calculated with any function starting with `DirectCalc...()`
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- Given two latitude/longitude points, calculate the distance between them, and the angles formed from due North to the line connecting the two points. This is calculated with any function starting with `InverseCalc...()`
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- Given a set of points or edges that form a polygon, calculate the area of said polygon. This is done by calling `NewPolygonArea()`, adding the points, and finally calling the `Compute()` method to get both the area and the perimeter of the polygon.
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- Given a set of points or edges that form a polyline (a set of connected lines), calculate the perimeter of the line. This is done by calling `NewPolygonArea()` with `is_polyline` set to true, adding the points, and finally calling the `Compute()` method to get the length of the lines.
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@@ -83,6 +84,66 @@ The shortest distance found by solving the inverse problem is (obviously) unique
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The area of a geodesic polygon can be determined by summing $S_{12}$ for successive edges of the polygon ($S_{12}$ is negated so that clockwise traversal of a polygon gives a positive area). However, if the polygon encircles a pole, the sum must be adjusted by $\pm A/2$, where $A$ is the area of the full ellipsoid, with the sign chosen to place the result in $(-A/2, A/2]$.
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## The library Interface
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This section is mostly copied from the [python documentation](https://geographiclib.sourceforge.io/Python/doc/interface.html), and altered where necessary.
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### The Units
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All angles (latitude, longitude, azimuth, arc length) are measured in degrees with latitudes increasing northwards, longitudes increasing eastwards, and azimuths measured clockwise from north. For a point at a pole, the azimuth is defined by keeping the longitude fixed, writing $\phi = \pm (90 \degree − \varepsilon)$, and taking the limit $\varepsilon → 0+$
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### Geodesic Dictionary
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The results returned by `DirectCalc...()` and `InverseCalc...()` are structs that may contain some or all of the following:
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-**LatDeg** = $φ_1$, latitude of a point (degrees)
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-**LonDeg** = $λ_1$, longitude of a point (degrees)
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-**AziDeg** = $α_1$, azimuth of line at point (degrees)
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-**DistanceM** = $s_{12}$, distance from 1 to 2 (meters)
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-**ArcLengthDeg** = $σ_{12}$, arc length on auxiliary sphere from 1 to 2 (degrees)
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-**ReducedLengthM** = $m_{12}$, reduced length of geodesic (meters)
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-**M12** = $M_{12}$, geodesic scale at 2 relative to 1 (dimensionless)
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-**M21** = $M_{21}$, geodesic scale at 1 relative to 2 (dimensionless)
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-**S12M2** = $S_{12}$, area between geodesic and equator (meters2)
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### Outmaks and caps
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There are a number of functions provided by the API. Each returns a struct with the fields calculated. A method that returns a struct with fewer fields than another means fewer calculations were made. *For faster computation, call methods that return only the fields you need.*
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You can also specify your own subset of fields by using the `DirectCalcWithCapabilities()` or `InverseCalcWithCapabilities()` and passing in a `capabilities` (sometimes called `caps`) field. `capabilities` are obtained by `or`’ing together the following values
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-`EMPTY`, no capabilities, no output
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-`LATITUDE`, compute latitude, lat2
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-`LONGITUDE`, compute longitude, lon2
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-`AZIMUTH`, compute azimuths, azi1 and azi2
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-`DISTANCE`, compute distance, s12
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-`STANDARD`, all of the above
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-`DISTANCE_IN`, allow s12 to be used as input in the direct problem
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-`REDUCEDLENGTH`, compute reduced length, m12
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-`GEODESICSCALE`, compute geodesic scales, M12 and M21
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-`AREA`, compute area, S12
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-`ALL`, all of the above;
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-`LONG_UNROLL`, unroll longitudes
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`DISTANCE_IN` is a capability provided to the GeodesicLine constructor. It allows the position on the line to specified in terms of distance. (Without this, the position can only be specified in terms of the arc length.) This only makes sense in the caps parameter.
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`LONG_UNROLL` controls the treatment of longitude. If it is not set then the lon1 and lon2 fields are both reduced to the range $[−180\degree, 180\degree]$. If it is set, then lon1 is as given in the function call and (lon2 − lon1) determines how many times and in what sense the geodesic has encircled the ellipsoid. This only makes sense in the outmask parameter.
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### Restrictions on Parameters
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- Latitudes must lie in $[−90\degree, 90\degree]$. Latitudes outside this range are replaced by NaNs.
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- The distance $s_{12}$ is unrestricted. This allows geodesics to wrap around the ellipsoid. Such geodesics are no longer shortest paths. However they retain the property that they are the straightest curves on the surface.
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- Similarly, the spherical arc length $a_{12}$ is unrestricted.
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- Longitudes and azimuths are unrestricted; internally these are exactly reduced to the range $[−180\degree, 180\degree]$; but see also the LONG_UNROLL bit.
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- The equatorial radius a and the polar semi-axis b must both be positive and finite (this implies that $−\infty < f < 1)$.
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- The flattening $f$ should satisfy $f \in [−1/50,1/50]$ in order to retain full accuracy. This condition holds for most applications in geodesy.
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Reasonably accurate results can be obtained for $−0.2 \le f \le 0.2$. Here is a table of the approximate maximum error (expressed as a distance) for an ellipsoid with the same equatorial radius as the WGS84 ellipsoid and different values of the flattening.
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| $\text{abs}(f)$ | $\text{error}$ |
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|--------|--------|
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| 0.003 | 15 nm |
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| 0.01 | 25 nm |
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| 0.02 | 30 nm |
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| 0.05 | 10 μm |
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| 0.1 | 1.5 mm |
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| 0.2 | 300 mm |
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Here 1 nm = 1 nanometer = $10^{−9}$ m (not 1 nautical mile!)
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