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diff --git a/docs/source/projections/qsc.rst b/docs/source/projections/qsc.rst index fb94bb25..1e406056 100644 --- a/docs/source/projections/qsc.rst +++ b/docs/source/projections/qsc.rst @@ -4,7 +4,155 @@ Quadrilateralized Spherical Cube ******************************************************************************** -.. image:: ./images/qsc.png ++---------------------+----------------------------------------------------------+ +| **Classification** | Azimuthal | ++---------------------+----------------------------------------------------------+ +| **Available forms** | Forward and inverse, elliptical projection | ++---------------------+----------------------------------------------------------+ +| **Defined area** | Global | ++---------------------+----------------------------------------------------------+ +| **Implemented by** | Martin Lambers | ++---------------------+----------------------------------------------------------+ +| **Options** | ++---------------------+----------------------------------------------------------+ +| `+lat_0` | Latitude (in degrees) of the view position. | ++---------------------+----------------------------------------------------------+ +| `+lon_0` | Longitude (in degrees) of the view position. | ++---------------------+----------------------------------------------------------+ + +The purpose of the Quadrilateralized Spherical Cube (QSC) projection is to project +a sphere surface onto the six sides of a cube: + +.. image:: ../../images/qsc_concept.jpg :scale: 50% - :alt: Quadrilateralized Spherical Cube + :align: center + :alt: Quadrilateralized Spherical Cube + +For this purpose, other alternatives can be used, notably :ref:`gnom` or +:ref:`healpix`. However, QSC projection has the following favorable properties: + +It is an equal-area projection, and at the same time introduces only limited angular +distortions. It treats all cube sides equally, i.e. it does not use different +projections for polar areas and equatorial areas. These properties make QSC +projection a good choice for planetary-scale terrain rendering. Map data can be +organized in quadtree structures for each cube side. See [LambersKolb2012]_ for an example. + +The QSC projection was introduced by [ONeilLaubscher1976]_, +building on previous work by [ChanONeil1975]_. For clarity: The +earlier QSC variant described in [ChanONeil1975]_ became known as the COBE QSC since it +was used by the NASA Cosmic Background Explorer (COBE) project; it is an approximately +equal-area projection and is not the same as the QSC projection. + +See also [CalabrettaGreisen2002]_ Sec. 5.6.2 and 5.6.3 for a description of both and +some analysis. + +In this implementation, the QSC projection projects onto one side of a circumscribed +cube. 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 | ++-------------------------+--------------------+ + +Furthermore, this implementation allows the projection to be applied to ellipsoids. +A preceding shift to a sphere is performed automatically; see [LambersKolb2012]_ for details. + + +Usage +############################################################################### + +The following example uses QSC projection via GDAL to create the six cube side +maps from a world map for the WGS84 ellipsoid:: + + gdalwarp -t_srs "+wktext +proj=qsc +units=m +ellps=WGS84 +lat_0=0 +lon_0=0" \ + -wo SOURCE_EXTRA=100 -wo SAMPLE_GRID=YES -te -6378137 -6378137 6378137 6378137 \ + worldmap.tiff frontside.tiff + + gdalwarp -t_srs "+wktext +proj=qsc +units=m +ellps=WGS84 +lat_0=0 +lon_0=90" \ + -wo SOURCE_EXTRA=100 -wo SAMPLE_GRID=YES -te -6378137 -6378137 6378137 6378137 \ + worldmap.tiff rightside.tiff + + gdalwarp -t_srs "+wktext +proj=qsc +units=m +ellps=WGS84 +lat_0=0 +lon_0=180" \ + -wo SOURCE_EXTRA=100 -wo SAMPLE_GRID=YES -te -6378137 -6378137 6378137 6378137 \ + worldmap.tiff backside.tiff + + gdalwarp -t_srs "+wktext +proj=qsc +units=m +ellps=WGS84 +lat_0=0 +lon_0=-90" \ + -wo SOURCE_EXTRA=100 -wo SAMPLE_GRID=YES -te -6378137 -6378137 6378137 6378137 \ + worldmap.tiff leftside.tiff + + gdalwarp -t_srs "+wktext +proj=qsc +units=m +ellps=WGS84 +lat_0=90 +lon_0=0" \ + -wo SOURCE_EXTRA=100 -wo SAMPLE_GRID=YES -te -6378137 -6378137 6378137 6378137 \ + worldmap.tiff topside.tiff + + gdalwarp -t_srs "+wktext +proj=qsc +units=m +ellps=WGS84 +lat_0=-90 +lon_0=0" \ + -wo SOURCE_EXTRA=100 -wo SAMPLE_GRID=YES -te -6378137 -6378137 6378137 6378137 \ + worldmap.tiff bottomside.tiff + + +Explanation: + +* QSC projection is selected with ``+wktext +proj=qsc``. +* The WGS84 ellipsoid is specified with ``+ellps=WGS84``. +* The cube side is selected with ``+lat_0=... +lon_0=...``. +* The ``-wo`` options are necessary for GDAL to avoid holes in the output maps. +* The ``-te`` option limits the extends of the output map to the major axis diameter + (from -radius to +radius in both x and y direction). These are the dimensions of one side + of the circumscribing cube. + + +The resulting images can be layed out in a grid like below. + + +.. |topside| image:: ../../images/qsc_topside.jpg + :scale: 50% + :align: middle + :alt: Top side + +.. |leftside| image:: ../../images/qsc_leftside.jpg + :scale: 50% + :align: middle + :alt: Left side + +.. |frontside| image:: ../../images/qsc_frontside.jpg + :scale: 50% + :align: middle + :alt: Front side + +.. |rightside| image:: ../../images/qsc_rightside.jpg + :scale: 50% + :align: middle + :alt: Right side + +.. |backside| image:: ../../images/qsc_backside.jpg + :scale: 50% + :align: middle + :alt: Back side + +.. |bottomside| image:: ../../images/qsc_bottomside.jpg + :scale: 50% + :align: middle + :alt: Bottom side + + ++------------+--------------+-------------+------------+ +| | |topside| | | | ++------------+--------------+-------------+------------+ +| |leftside| | |frontside| | |rightside| | |backside| | ++------------+--------------+-------------+------------+ +| | |bottomside| | | | ++------------+--------------+-------------+------------+ + +Further reading +################################################################################ +#. `Wikipedia <https://en.wikipedia.org/wiki/Quadrilateralized_spherical_cube>`_ +#. `NASA <https://lambda.gsfc.nasa.gov/product/cobe/skymap_info_new.cfm>`_ |