<html>
<head>
<meta http-equiv=Content-Type content="text/html; charset=iso-2022-jp">
<meta name=Generator content="Microsoft Word 11 (filtered)">
<style>
<!--
/* Font Definitions */
@font-face
{font-family:Tahoma;
panose-1:2 11 6 4 3 5 4 4 2 4;}
@font-face
{font-family:"MS PGothic";
panose-1:0 0 0 0 0 0 0 0 0 0;}
@font-face
{font-family:"MS UI Gothic";
panose-1:0 0 0 0 0 0 0 0 0 0;}
/* Style Definitions */
p.MsoNormal, li.MsoNormal, div.MsoNormal
{margin:0cm;
margin-bottom:.0001pt;
font-size:12.0pt;
font-family:"MS PGothic";}
a:link, span.MsoHyperlink
{color:blue;
text-decoration:underline;}
a:visited, span.MsoHyperlinkFollowed
{color:#606420;
text-decoration:underline;}
span.EmailStyle17
{font-family:Arial;
color:navy;}
@page Section1
{size:595.3pt 841.9pt;
margin:72.0pt 90.0pt 72.0pt 90.0pt;}
div.Section1
{page:Section1;}
/* List Definitions */
ol
{margin-bottom:0cm;}
ul
{margin-bottom:0cm;}
-->
</style>
</head>
<body lang=EN-GB link=blue vlink="#606420">
<div class=Section1>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>Daan, Clifford,</span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>Of course, deep down, I knew that the
scale varies along the central line of an oblique Mercator projection and I
suppose my suggestion was not theoretically sound. I should also say that I get
as much satisfaction as anybody when I get the chance to use Vincenty! </span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>Even Vincenty does not deal with antipodal
points, however!</span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>In my defence I would say that we were
talking about points scaled off a paper map! </span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>In my days as a surveyor in the early 70s –
do you remember Peter’s Tables and Facit mechanical calculators? – we used to use
the mean of the scale factors at the end points of lines to reduce measured
distances to the projection, and for the very longest lines we sometimes used
Simpson’s rule (but I never remember it making a sensible difference). That was
on at the edge of UTM zones. In those days it was really important to be able compute
on the projection. I once computed a traverse using the Mid Latitude formulae –
it took me two days, and I got the wrong answer! </span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>I bet the scale change on the central line
of a Hotine Oblique Mercator projection, even across California, could be
treated the same way! Do you have any figures?</span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'> </span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>Regards,</span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'> </span></font></p>
<div>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>Ron Russell<br>
Tel : 01823 270308 email : <a href="mailto:ron@russfam.freeserve.co.uk">ron@russfam.freeserve.co.uk</a>
</span></font></p>
</div>
<div>
<div class=MsoNormal align=center style='text-align:center'><font size=3
face="MS PGothic"><span lang=EN-US style='font-size:12.0pt'>
<hr size=2 width="100%" align=center tabindex=-1>
</span></font></div>
<p class=MsoNormal><b><font size=2 face=Tahoma><span lang=EN-US
style='font-size:10.0pt;font-family:Tahoma;font-weight:bold'>From:</span></font></b><font
size=2 face=Tahoma><span lang=EN-US style='font-size:10.0pt;font-family:Tahoma'>
proj-bounces@lists.maptools.org [mailto:proj-bounces@lists.maptools.org] <b><span
style='font-weight:bold'>On Behalf Of </span></b>Strebe@aol.com<br>
<b><span style='font-weight:bold'>Sent:</span></b> 17 October 2005 22:56<br>
<b><span style='font-weight:bold'>To:</span></b> proj@lists.maptools.org<br>
<b><span style='font-weight:bold'>Subject:</span></b> [Proj] Re: Mercator
Problem</span></font></p>
</div>
<p class=MsoNormal><font size=3 face="MS PGothic"><span style='font-size:12.0pt'> </span></font></p>
<p class=MsoNormal><font size=2 face="MS UI Gothic"><span style='font-size:
10.0pt;font-family:"MS UI Gothic"'><br>
Ron's proposal is perfectly sound on a spherical earth. Since the oblique
Mercator central line runs through the two points of interest, and since the
scale is constant and correct along that great circle, there's no need for
integration or scale factor corrections. Obviously it's more problematic on the
ellipsoid, given that none of the usual oblique formulations carry constant
scale along the transformed equator (and if I recall correctly, it's not
possible whilst retaining conformality), but even so, Hotine is close enough to
constant scale that it might suffice, depending on the accuracy needs and
assuming the two points are reasonably short of antipodal.<br>
<br>
If other calculations that Ron mentions aren't needed then clearly Cliff's
suggestion is the way to go. Why go out on a... erm... tangent?<br>
<br>
Regards,<br>
daan Strebe<br>
<br>
<br>
In a message dated 10/17/2005 2:42:58 PM </span></font><font size=2><span
lang=JA style='font-size:10.0pt'>太平洋夏時間</span></font><font size=2
face="MS UI Gothic"><span style='font-size:10.0pt;font-family:"MS UI Gothic"'>,
cjmce@lsu.edu writes:<br>
<br>
<br>
</span></font></p>
<p class=MsoNormal style='margin-bottom:12.0pt'><font size=2 color=black
face=Arial BACK="#ffffff" PTSIZE=10 FAMILY=SANSSERIF><span style='font-size:
10.0pt;font-family:Arial;color:black;background:white'>Ron,<br>
<br>
You're proposing a piecewise-integration along a grid line. With that,<br>
you'd also have to include the correction for scale factor at a point, as<br>
integrated by so many points of choice along the projected line. Scale<br>
factor along the Central Line of an Oblique Mercator is not going to<br>
obviate that need.<br>
<br>
It's far simpler to just call the subroutine in PROJ4 for an ellipsoidal<br>
geodesic between the two end points. (Once called the "Principal
Problem<br>
of Geodesy" in the 19th Century).<br>
<br>
Assuming you know how to program the subroutine calls, it's easier done<br>
than said.<br>
<br>
Clifford J. Mugnier<br>
Chief of Geodesy and<br>
Associate Director,<br>
CENTER FOR GEOINFORMATICS<br>
Department of Civil Engineering<br>
LOUISIANA STATE UNIVERSITY<br>
Baton Rouge, LA 70803<br>
Voice and Facsimile: (225) 578-8536 [Academic]<br>
Voice and Facsimile: (225) 578-4474 [Research]<br>
================================<br>
http://www.ASPRS.org/resources/GRIDS<br>
http://www.cee.lsu.edu/facultyStaff/mugnier/index.html<br>
================================<br>
<br>
<br>
<br>
What about using the Lat and Long of the endpoints to define the central<br>
line of an Oblique Mercator, with a scale factor of 1.0 on the central<br>
line?<br>
Then the distance can be calculated by Pythagoras, and other useful<br>
operations can easily be calculated - mid point, distance of a third point<br>
from the line and even the calculation of a buffer zone, all of which seem<br>
horrendous when working on the ellipsoid. (OK, the mid point is not too<br>
bad).<br>
<br>
Ron Russell<br>
Tel : 01823 270308 email : ron@russfam.freeserve.co.uk<br>
-----Original Message-----<br>
From: proj-bounces@lists.maptools.org<br>
[mailto:proj-bounces@lists.maptools.org] On Behalf Of Clifford J Mugnier<br>
Sent: 17 October 2005 20:52<br>
To: PROJ.4 and general Projections Discussions<br>
Subject: Re: [Proj] Mercator Problem</span></font></p>
<p class=MsoNormal><font size=2 color=black face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:black;background:white'><br>
<br>
</span></font></p>
</div>
</body>
</html>