From 02ef4ed32e15ea386ad3285e72e1bb7e4c188f05 Mon Sep 17 00:00:00 2001 From: Charles Karney Date: Thu, 22 Oct 2020 11:01:29 -0400 Subject: Flesh out the documentation for Mercator This addresses item 1 in issue #2387 Things to note: * I made "editorial" changes to the text. The virtues and vices of Mercator are a hot topic. So check these out. (I judged that the text I replaced to be pretty misleading.) * I include the radius of the sphere/ellipsoid in the formulas (and I did this also for my mods for tmerc documentation). Surely this is better than leaving the reader to figure out how this is introduced. * I include the "old-style" (ca 18th century) formulas and the newer ones in terms of hyperbolic functions. The former may be the familiar ones, but the latter are better for computation (more succinct, more accurate, faster, preserve parity). * For the inverse ellipsoidal transformation, I just say that the formula for psi is inverted iteratively. This is probably sufficient, but it could be expanded later. --- docs/source/operations/projections/merc.rst | 53 +++++++++++++++++------------ 1 file changed, 31 insertions(+), 22 deletions(-) (limited to 'docs/source') 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 ############### -- cgit v1.2.3 From 5b72015b0aaca836ee0e8bf1f2da193574fe8ab1 Mon Sep 17 00:00:00 2001 From: Charles Karney Date: Thu, 22 Oct 2020 13:53:45 -0400 Subject: merc.rst: revert to ln instead of log. Also define merc (resp. tmerc) as the conformal projection in which the equator (resp. a chosen meridan} projects to a straight line at constant scale. --- docs/source/operations/projections/merc.rst | 13 +++++++------ docs/source/operations/projections/tmerc.rst | 2 ++ 2 files changed, 9 insertions(+), 6 deletions(-) (limited to 'docs/source') diff --git a/docs/source/operations/projections/merc.rst b/docs/source/operations/projections/merc.rst index 62aa5bc9..c474009f 100644 --- a/docs/source/operations/projections/merc.rst +++ b/docs/source/operations/projections/merc.rst @@ -6,9 +6,10 @@ 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 has the property that a rhumb line, a course of constant -heading, projects to a straight line. This makes it suitable for -navigational purposes. +It is a conformal projection in which the equator projects to a straight +line at constant scale. 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. +---------------------+----------------------------------------------------------+ @@ -99,7 +100,7 @@ Forward projection .. math:: - \psi &= \log \tan \biggl(\frac{\pi}{4} + \frac{\phi}{2} \biggr)\\ + \psi &= \ln \tan \biggl(\frac{\pi}{4} + \frac{\phi}{2} \biggr)\\ &= \sinh^{-1}\tan\phi The quantity :math:`\psi` is the isometric latitude. @@ -143,9 +144,9 @@ Forward projection .. math:: - \psi &= \log\tan\biggl(\frac\pi4 + \frac{\phi}2\biggr) + \psi &= \ln\tan\biggl(\frac\pi4 + \frac{\phi}2\biggr) -\frac12 e - \log \biggl(\frac{1 + e \sin\phi}{1 - e \sin\phi}\biggr)\\ + \ln \biggl(\frac{1 + e \sin\phi}{1 - e \sin\phi}\biggr)\\ &= \sinh^{-1}\tan\phi - e \tanh^{-1}(e \sin\phi) diff --git a/docs/source/operations/projections/tmerc.rst b/docs/source/operations/projections/tmerc.rst index 3b2a0190..7fed3846 100644 --- a/docs/source/operations/projections/tmerc.rst +++ b/docs/source/operations/projections/tmerc.rst @@ -5,6 +5,8 @@ Transverse Mercator ******************************************************************************** The transverse Mercator projection in its various forms is the most widely used projected coordinate system for world topographical and offshore mapping. +It is a conformal projection in which a chosen meridian projects to a +straight line at constant scale. +---------------------+----------------------------------------------------------+ | **Classification** | Transverse and oblique cylindrical | -- cgit v1.2.3 From f472a974904024a0baf09a4a6d6e4f2ceef2a290 Mon Sep 17 00:00:00 2001 From: Charles Karney Date: Fri, 23 Oct 2020 08:20:17 -0400 Subject: merc.rst: change "origins from 15th cent" to 16th cent. --- docs/source/operations/projections/merc.rst | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) (limited to 'docs/source') diff --git a/docs/source/operations/projections/merc.rst b/docs/source/operations/projections/merc.rst index c474009f..7b6e13da 100644 --- a/docs/source/operations/projections/merc.rst +++ b/docs/source/operations/projections/merc.rst @@ -4,12 +4,12 @@ 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. -It is a conformal projection in which the equator projects to a straight -line at constant scale. 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. +The Mercator projection is a cylindrical map projection that origins +from the 16th century. It is widely recognized as the first regularly +used map projection. It is a conformal projection in which the equator +projects to a straight line at constant scale. 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. +---------------------+----------------------------------------------------------+ -- cgit v1.2.3