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Diffstat (limited to 'docs/source')
| -rw-r--r-- | docs/source/operations/projections/merc.rst | 53 |
1 files changed, 31 insertions, 22 deletions
diff --git a/docs/source/operations/projections/merc.rst b/docs/source/operations/projections/merc.rst index 063d02dc..62aa5bc9 100644 --- a/docs/source/operations/projections/merc.rst +++ b/docs/source/operations/projections/merc.rst @@ -6,7 +6,9 @@ Mercator The Mercator projection is a cylindrical map projection that origins from the 15th century. It is widely recognized as the first regularly used map projection. -The projection is conformal which makes it suitable for navigational purposes. +The projection has the property that a rhumb line, a course of constant +heading, projects to a straight line. This makes it suitable for +navigational purposes. +---------------------+----------------------------------------------------------+ @@ -38,8 +40,10 @@ Usage Applications should be limited to equatorial regions, but is frequently used for navigational charts with latitude of true scale (:option:`+lat_ts`) specified within or near chart's boundaries. -Often inappropriately used for world maps since the regions near the poles -cannot be shown :cite:`Evenden1995`. +It is considered to be inappropriate for world maps because of the gross +distortions in area; for example the projected area of Greenland is +larger than that of South America, despite the fact that Greenland's +area is :math:`\frac18` that of South America :cite:`Snyder1987`. Example using latitude of true scale:: @@ -78,8 +82,6 @@ Parameters Mathematical definition ####################### -The formulas describing the Mercator projection are all taken from G. Evenden's libproj manuals :cite:`Evenden2005`. - Spherical form ************** For the spherical form of the projection we introduce the scaling factor: @@ -93,11 +95,14 @@ Forward projection .. math:: - x = k_0 \lambda + x = k_0R \lambda; \qquad y = k_0R \psi .. math:: - y = k_0 \ln \left[ \tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right] + \psi &= \log \tan \biggl(\frac{\pi}{4} + \frac{\phi}{2} \biggr)\\ + &= \sinh^{-1}\tan\phi + +The quantity :math:`\psi` is the isometric latitude. Inverse projection @@ -105,38 +110,43 @@ Inverse projection .. math:: - \lambda = \frac{x}{k_0} + \lambda = \frac{x}{k_0R}; \qquad \psi = \frac{y}{k_0R} .. math:: - \phi = \frac{\pi}{2} - 2 \arctan \left[ e^{-y/k_0} \right] + \phi &= \frac{\pi}{2} - 2 \tan^{-1} \exp(-\psi)\\ + &= \tan^{-1}\sinh\psi -Ellisoidal form -*************** +Ellipsoidal form +**************** For the ellipsoidal form of the projection we introduce the scaling factor: .. math:: - k_0 = m\left( \phi_{ts} \right) + k_0 = m( \phi_{ts} ) + +where -where :math:`m\left(\phi\right)` is the parallel radius at latitude :math:`\phi`. +.. math:: -We also use the Isometric Latitude kernel function :math:`t()`. + m(\phi) = \frac{\cos\phi}{\sqrt{1 - e^2\sin^2\phi}} -.. note:: - m() and t() should be described properly on a separate page about the theory of projections on the ellipsoid. +:math:`a\,m(\phi)` is the radius of the circle of latitude :math:`\phi`. Forward projection ================== .. math:: - x = k_0 \lambda + x = k_0 a \lambda; \qquad y = k_0 a \psi .. math:: - y = k_0 \ln t \left( \phi \right) + \psi &= \log\tan\biggl(\frac\pi4 + \frac{\phi}2\biggr) + -\frac12 e + \log \biggl(\frac{1 + e \sin\phi}{1 - e \sin\phi}\biggr)\\ + &= \sinh^{-1}\tan\phi - e \tanh^{-1}(e \sin\phi) Inverse projection @@ -144,11 +154,10 @@ Inverse projection .. math:: - \lambda = \frac{x}{k_0} - -.. math:: + \lambda = \frac{x}{k_0 a}; \quad \psi = \frac{y}{k_0 a} - \phi = t^{-1}\left[ e^{ -y/k_0 } \right] +The latitude :math:`\phi` is found by inverting the equation for +:math:`\psi` iteratively. Further reading ############### |