Comments on: Inverse Mercator projection http://www.johndcook.com/blog/2009/09/21/gudermannian/ The blog of John D. Cook Sat, 11 Feb 2012 01:10:06 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: Poor Mercator — The Endeavour http://www.johndcook.com/blog/2009/09/21/gudermannian/comment-page-1/#comment-114782 Poor Mercator — The Endeavour Mon, 14 Nov 2011 17:26:49 +0000 http://www.johndcook.com/blog/?p=3226#comment-114782 [...] The inverse of the Mercator projection, going from maps onto the globe, is more interesting. Aside from its geographical importance, it gives a way of relating circular and hyperbolic functions without using complex numbers. More details here. [...] [...] The inverse of the Mercator projection, going from maps onto the globe, is more interesting. Aside from its geographical importance, it gives a way of relating circular and hyperbolic functions without using complex numbers. More details here. [...]

]]> By: Journey away from the center of the Earth — The Endeavour http://www.johndcook.com/blog/2009/09/21/gudermannian/comment-page-1/#comment-103686 Journey away from the center of the Earth — The Endeavour Fri, 16 Sep 2011 13:33:00 +0000 http://www.johndcook.com/blog/?p=3226#comment-103686 [...] is the shape of the Earth? Finding distances from coordinates Mercator projection Inverse Mercator projection ? [...] [...] is the shape of the Earth? Finding distances from coordinates Mercator projection Inverse Mercator projection ? [...]

]]> By: John http://www.johndcook.com/blog/2009/09/21/gudermannian/comment-page-1/#comment-41778 John Thu, 15 Jul 2010 19:59:19 +0000 http://www.johndcook.com/blog/?p=3226#comment-41778 Spivak's book is one of my favorites. Spivak’s book is one of my favorites.

]]> By: John http://www.johndcook.com/blog/2009/09/21/gudermannian/comment-page-1/#comment-41777 John Thu, 15 Jul 2010 19:26:40 +0000 http://www.johndcook.com/blog/?p=3226#comment-41777 The gudermanian has another property: its derivative is the secant. That's no surprise if you know the inverse function theorem, but there's a nice footnote to this (from Spivak's "Calculus", second edition, page 383, problem 19.12), which kind of amplifies what your earlier posting said: "There is another expression for [the integral of secant], which is ... log(tan(x/2 + pi/4)). This last expression was actually the first one discovered, and was due, not to any mathematician's cleverness, but to a curious historical accident: In 1599 Wright computed nautical tables that amounted to definite integrals of sec. When the first tables for the logarithms of tangents were produced, the correspondence between the two tables was immediately noticed ( but remained unexplained until the invention of calculus). " [If your (I'm speaking to everyone here) calculus book didn't have cool things in it like this, you should pick up a copy of Spivak's book!] The gudermanian has another property: its derivative is the secant. That’s no surprise if you know the inverse function theorem, but there’s a nice footnote to this (from Spivak’s “Calculus”, second edition, page 383, problem 19.12), which kind of amplifies what your earlier posting said:

“There is another expression for [the integral of secant], which is … log(tan(x/2 + pi/4)). This last expression was actually the first one discovered, and was due, not to any mathematician’s cleverness, but to a curious historical accident: In 1599 Wright computed nautical tables that amounted to definite integrals of sec. When the first tables for the logarithms of tangents were produced, the correspondence between the two tables was immediately noticed ( but remained unexplained until the invention of calculus). ”

[If your (I'm speaking to everyone here) calculus book didn't have cool things in it like this, you should pick up a copy of Spivak's book!]

]]> By: Walking Randomly » Carnival of Maths #58 http://www.johndcook.com/blog/2009/09/21/gudermannian/comment-page-1/#comment-27114 Walking Randomly » Carnival of Maths #58 Mon, 09 Nov 2009 11:29:36 +0000 http://www.johndcook.com/blog/?p=3226#comment-27114 [...] Moving over to The Endeavour, we have two related submissions from John D Cook all about The Mercator Projection and its inverse. [...] [...] Moving over to The Endeavour, we have two related submissions from John D Cook all about The Mercator Projection and its inverse. [...]

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