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A golang port of [geographiclib](https://geographiclib.sourceforge.io/).
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###Aims
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## Aims
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- Mimic the [rust port](https://github.com/georust/geographiclib-rs), the [python port](https://pypi.org/project/geographiclib/), and the [java port](https://github.com/geographiclib/geographiclib-java) as closely as possible
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- Test as extensively as possible. Match all the rust, python, and java tests
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### Progress
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## Outline:
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-[Background: Geodesics on an Ellipsoid](#background-geodesics-on-an-ellipsoid)
- 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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## Long Explanation of Library
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This section is copied from the [python documentation](https://geographiclib.sourceforge.io/Python/doc/geodesics.html)
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### Introduction
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Consider an ellipsoid of revolution with equatorial radius $a$, polar semi-axis $b$, and flattening $f = (a − b)/a$ . Points on the surface of the ellipsoid are characterized by their latitude $\phi$ and longitude $\lambda$. (Note that latitude here means the geographical latitude, the angle between the normal to the ellipsoid and the equatorial plane).
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The shortest path between two points on the ellipsoid at $(\phi_1, \lambda_1)$ and $(\phi_2, \lambda_2)$ is called the geodesic. Its length is $s_{12}$ and the geodesic from point 1 to point 2 has forward azimuths $\alpha_1$ and $\alpha_2$ at the two end points. In this figure, we have $\lambda_{12} = \lambda_2 − \lambda_1$.
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A geodesic can be extended indefinitely by requiring that any sufficiently small segment is a shortest path; geodesics are also the straightest curves on the surface.
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### Solution of Geodesic Problems
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Traditionally two geodesic problems are considered:
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- the direct problem — given $\phi_1$, $\lambda_1$, $\alpha_1$, $s_{12}$, determine $\phi_2$, $\lambda_2$, and $\alpha_2$; this is solved by the various functions starting with `DirectCalc...`
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- the inverse problem — given $\phi_1$, $\lambda_1$, $\phi_2$, $\lambda_2$, determine $s_{12}$, $\alpha_1$, and $\alpha_2$; this is solved by the various functions starting with `InverseCalc...`
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### Aditional Properties
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The routines also calculate several other quantities of interest
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- $S_{12}$ is the area between the geodesic from point 1 to point 2 and the equator; i.e., it is the area, measured counter-clockwise, of the quadrilateral with corners $(\phi_1,\lambda_1), (0,\lambda_1), (0,\lambda_2)$, and $(\phi_2,\lambda_2)$. It is given in $\text{meters}^2$.
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- $m_{12}$, the reduced length of the geodesic is defined such that if the initial azimuth is perturbed by $\delta\alpha_1$ (radians) then the second point is displaced by $m_{12}\ \delta\alpha_1$ in the direction perpendicular to the geodesic. $m_{12}$ is given in meters. On a curved surface the reduced length obeys a symmetry relation, $m_{12} + m_{21} = 0$. On a flat surface, we have $m_{12} = s_{12}$.
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- $M_{12}$ and $M_{21}$ are geodesic scales. If two geodesics are parallel at point 1 and separated by a small distance $\delta t$, then they are separated by a distance $M_{12} \delta t$ at point 2. $M_{21}$ is defined similarly (with the geodesics being parallel to one another at point 2). $M_{12}$ and $M_{21}$ are dimensionless quantities. On a flat surface, we have $M_{12} = M_{21} = 1$.
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- $\sigma_{12}$ is the arc length on the auxiliary sphere. This is a construct for converting the problem to one in spherical trigonometry. The spherical arc length from one equator crossing to the next is always $180 \degree$.
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If points 1, 2, and 3 lie on a single geodesic, then the following addition rules hold:
The shortest distance found by solving the inverse problem is (obviously) uniquely defined. However, in a few special cases there are multiple azimuths which yield the same shortest distance. Here is a catalog of those cases:
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- $\phi_1 = −\phi_2$ (with neither point at a pole). If $\alpha_1 = \alpha_2$, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by setting
(This occurs when the longitude difference is near $\pm 180 \degree$ for oblate ellipsoids.)
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- $\lambda_2 = \lambda_1 \pm 180 \degree$ (with neither point at a pole). If $\alpha_1 = 0 \degree$ or $\pm 180 \degree$, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by setting
(This occurs when $\phi_2$ is near $−\phi_1$ for prolate ellipsoids.)
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- Points 1 and 2 at opposite poles. There are infinitely many geodesics which can be generated by setting $[\alpha1,\alpha2] \leftarrow [\alpha1,\alpha2] + [\delta,−\delta]$, for arbitrary $\delta$. (For spheres, this prescription applies when points 1 and 2 are antipodal.)
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- $s_{12} = 0$ (coincident points). There are infinitely many geodesics which can be generated by setting $[\alpha1,\alpha2] \leftarrow [\alpha1,\alpha2] + [\delta,\delta]$, for arbitrary $\delta$.
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### Area of a Polygon
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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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## GeographicLib API
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## Examples
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-[ ] Add Examples to README
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-[ ] Examples of direct case on Earth
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-[ ] Examples of direct case on Mars
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-[ ] Examples of inverse case
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-[ ] Examples of calculating an area
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## Progress
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-[X] Go translation
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-[X] Geomath
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-[X] Geomath tests
@@ -21,14 +107,4 @@ A golang port of [geographiclib](https://geographiclib.sourceforge.io/).
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-[X] Geodisic Inverse tests
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-[X] Polygon Area
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-[X] Polygon Area tests
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-[X] Add DirectAndInverse interface
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-[ ] Explain it in README
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-[ ] Add description of the direct and inverse problem, with pictures
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-[ ] Add Examples to README
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-[ ] Examples of direct case on Earth
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-[ ] Examples of direct case on Mars
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-[ ] Examples of inverse case
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-[ ] Examples of calculating an area
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-[ ] Consider switching from having units in names to using github.com/golang/geo/
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which uses types for units. It would perhaps involve more allocations; need to do some
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testing.
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-[ ] Consider switching from having units in names to using github.com/golang/geo/ which uses types for units. It would perhaps involve more allocations; need to do some testing.
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