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diff --git a/docs/source/operations/projections/index.rst b/docs/source/operations/projections/index.rst index 48420971..4f76dab2 100644 --- a/docs/source/operations/projections/index.rst +++ b/docs/source/operations/projections/index.rst @@ -126,6 +126,7 @@ Projections map the spherical 3D space to a flat 2D space. robin rouss rpoly + s2 sch sinu somerc diff --git a/docs/source/operations/projections/s2.rst b/docs/source/operations/projections/s2.rst new file mode 100644 index 00000000..46ae715e --- /dev/null +++ b/docs/source/operations/projections/s2.rst @@ -0,0 +1,120 @@ +.. _s2: + +******************************************************************************** +S2 +******************************************************************************** + ++---------------------+----------------------------------------------------------+ +| **Classification** | Miscellaneous | ++---------------------+----------------------------------------------------------+ +| **Available forms** | Forward and inverse, ellipsoidal | ++---------------------+----------------------------------------------------------+ +| **Defined area** | Global | ++---------------------+----------------------------------------------------------+ +| **Alias** | s2 | ++---------------------+----------------------------------------------------------+ +| **Domain** | 2D | ++---------------------+----------------------------------------------------------+ +| **Input type** | Geodetic coordinates | ++---------------------+----------------------------------------------------------+ +| **Output type** | Projected coordinates | ++---------------------+----------------------------------------------------------+ + +.. versionadded:: 8.2 + +The S2 projection, like the Quadrilateralized Spherical Cube (QSC) projection, projects +a sphere surface onto the six sides of a cube: + +.. image:: ../../../images/qsc_concept.jpg + :width: 500 px + :align: center + :alt: Quadrilateralized Spherical Cube + +S2 was created by Google to represent geographic data on the whole earth. The documentation can be found +at `S2 Geometry <https://s2geometry.io/>`_ It works by first +projecting a point on the sphere to a face of the cube. These are called u,v-coordinates, and they are in [-1,1] x [-1,1]. +This step is followed by a non-linear transformation to normalize the area of rectangles on the sphere. There are three +different choices available for this transformation, meaning that S2 is a family of projections. The final output is in +s,t-coordinates, which are in [0,1] x [0,1]. +See the comments in `S2 Code <https://github.com/google/s2geometry/blob/0c4c460bdfe696da303641771f9def900b3e440f/src/s2/s2coords.h#L226>`_ +for an explanation of the tradeoff between speed and area-preservation. Note that the projection is azimuthal when "none" or +"linear" is selected for the area-normalization, but it is not azimuthal when "quadratic" or "tangent" is chosen. See +S2's `Earthcube page <https://s2geometry.io/resources/earthcube>`_ +to visualize the unfolded cube and the orientation of each face. + +In this implementation, the cube side is selected by choosing one of the following six projection centers: + ++-------------------------+--------------------+ +| ``+lat_0=0 +lon_0=0`` | front cube side | ++-------------------------+--------------------+ +| ``+lat_0=0 +lon_0=90`` | right cube side | ++-------------------------+--------------------+ +| ``+lat_0=0 +lon_0=180`` | back cube side | ++-------------------------+--------------------+ +| ``+lat_0=0 +lon_0=-90`` | left cube side | ++-------------------------+--------------------+ +| ``+lat_0=90`` | top cube side | ++-------------------------+--------------------+ +| ``+lat_0=-90`` | bottom cube side | ++-------------------------+--------------------+ + +The specific transformation can be chosen with the UVtoST parameter: + ++-------------------------+-----------------------------+ +| ``+UVtoST=linear`` | fastest, no normalization | ++-------------------------+-----------------------------+ +| ``+UVtoST=quadratic`` | fast, good normalization | ++-------------------------+-----------------------------+ +| ``+UVtoST=tangent`` | slowest, best normalization | ++-------------------------+-----------------------------+ +| ``+UVtoST=none`` | returns u,v-coordinates | ++-------------------------+-----------------------------+ + +Furthermore, this implementation allows the projection to be applied to ellipsoids. +A preceding shift to a sphere is performed automatically; see :cite:`LambersKolb2012` for details. +The output of the projection is in s,t-coordinates ([0,1] x [0,1]), so only the +eccentricity of the ellipse is taken into account: the absolute value of the axes does +not affect the output. + + +Usage +############################################################################### + +The following example uses S2 on the right face:: + + echo 90 0 | ../bin/proj +proj=s2 +lat_0=0 +lon_0=90 +ellps=WGS84 +UVtoST=linear + + 0.5 0.5 + +Explanation: + +* S2 projection is selected with ``+proj=s2``. +* The WGS84 ellipsoid is specified with ``+ellps=WGS84``. +* The cube side is selected with ``+lat_0=... +lon_0=...``. +* The normalization transformation is selected with ``+UVtoST=...``. + +Parameters +################################################################################ + +.. note:: All parameters for the projection are optional. + +.. include:: ../options/lon_0.rst + +.. include:: ../options/lat_0.rst + +.. include:: ../options/ellps.rst + +.. option:: +UVtoST=<value> + + The area-normalization transformation. Choose from {linear, quadratic, tangent, none} + + *Defaults to "quadratic".* + +.. include:: ../options/x_0.rst + +.. include:: ../options/y_0.rst + +Further reading +################################################################################ + +#. `S2's Website <https://s2geometry.io/>`_ |