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    Re: Spherical Trig
    From: Alexandre Eremenko
    Date: 2005 Apr 5, 22:24 -0500

    Bill,
    
    > Focusing on the words "is
    > between"
    > does that literally mean the lower limit can approach
    > 180d and the upper
    > limit approach 540d, or can the sums actual equal
    > either 180d or 540d?
    
    It depends of what you exactly mean by a "triangle".
    Whether you admit triangles of zero area as genuine triangles
    or not, for example.
    (The excess over 180 is essentially the area. Up to a
    factor which depends on your choice of units for the area).
    
    Speaking of the maximum of 540, this statement
    (that 540 is the maximum) even stronger depends on what
    you mean by a "triangle". Usually, (in particular in CelNav)
    they only consider so-called CONVEX triangles, whose all
    angles are at most 180 each. For such a triangle, of course,
    the sum of the angles cannot be more than 3 times 180=540.
    Whether 540 can be reached, again depends on your definition:
    whether you agree to consider a hemisphere as a legitimate
    triangle or not (see my previous message).
    
    Mathematicians consider triangles which can be bigger than
    a hemisphere. And angles which are bigger than 180 and
    even bigger than 360:-)
    By the way, surprisingly, the complete answer to the question,
    what can the angles of
    a spherical triangle be, was found only very recently, and
    it is due to your
    Obedinet Servant:-)
    "Metrics of positive curvature with conic singularities
    on the sphere",
    Proc. Amer. Math. Soc., 132 (2004), 11, 3349--3355.
    But this is way out of the scope of this list:-)
    
    A
    
    
    
    > Bill
    >
    > > On Apr 5, 2005, at 4:08 PM, n s gurnell wrote:
    > >
    > >> Re Bill's question about the angles in a spherical triangle.
    > >> Sixty years ago the exams called "Principles for Second Mates" used to
    > >> have a
    > >> question:- "What is Spherical Excess?" I've forgotten the exact answer
    > >> but
    > >> someone might know. Old Timer
    > >
    > > The sum of the angles of a spherical triangle is between pi and 3 pi
    > > radians (180 deg and 540 deg). The amount by which it exceeds 180 deg
    > > is called the spherical excess and is denoted E. The difference between
    > > 2 pi radians (360 deg) and the sum of the side arc lengths a, b, and c
    > > is called the spherical defect and is denoted D.
    > >
    > > Girard's formula:
    > >
    > > Spherical Excess = E = A + B + C - pi
    > > where A,B, and C are angles measured in radians
    > > Surface Area = E * R^2
    > > where R is the radius of the sphere
    > >
    > > Dan
    >
    
    
    

       
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