Re: [Boost-bugs] [Boost C++ Libraries] #11928: surveyor area method is inaccurate

Subject: Re: [Boost-bugs] [Boost C++ Libraries] #11928: surveyor area method is inaccurate
From: Boost C++ Libraries (noreply_at_[hidden])
Date: 2016-01-26 15:51:27

#11928: surveyor area method is inaccurate
-----------------------------------------+---------------------------
  Reporter:  Charles Karney <charles@…>  |      Owner:  barendgehrels
      Type:  Bugs                        |     Status:  new
 Milestone:  To Be Determined            |  Component:  geometry
   Version:  Boost 1.58.0                |   Severity:  Problem
Resolution:                              |   Keywords:  area
-----------------------------------------+---------------------------

Comment (by Charles Karney <charles@…>):

 Thanks for the reference to Abel Flint (a great name!).  You will see on
 pp. 59-63 that he describes the trapezium rule and ''not'' the so-called
 surveyor's method!  Most of his contemporaries who were mathematicians
 would have recognized this as such and not, therefore, have seen it as
 some special technique invented by a surveyor.

 The analysis of why

   sum x,,i,, y,,i+1,, - x,,i+1,, y,,i,,

 is typically subject to worse round off errors than

   sum (x,,i+1,, - x,,i,,) (y,,i+1,, + y,,i,,)

 goes as follows.  Each formula contains

    1. products
    2. differences in forming each term in the sum
    3. differences in forming the sum itself

 In the surveyors formula 2 follows 1 and so the mild round off errors
 from the products are sometimes catastrophically magnified.  In the
 trapezium rule 2 precedes 1 and there is ''no'' round off error if
 x,,i+1,,
 and x,,i,, are within a factor of 2 of each other (which is a common
 situation).

 Both formulas are subject to potentially dangerous round off with 3.
 Centering the numbers might ameliorate this problem.  Alternatively,
 merely switching to the other trapezium rule involving (y,,i+1,, - y,,i,,)
 would help, e.g., in the case of UTM coordinates where often the
 northings are much larger than the eastings.

-- 
Ticket URL: <https://svn.boost.org/trac/boost/ticket/11928#comment:2>
Boost C++ Libraries <http://www.boost.org/>
Boost provides free peer-reviewed portable C++ source libraries.