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    Re: Mathematics Question
    From: George Huxtable
    Date: 2006 Mar 30, 10:15 +0100

    | Robert Eno wrote:
    |
    | > Just to make sure I got this right, please have a look at the diagram
    | > on the
    | > following website:
    | >
    | > http://whatis.techtarget.com/definition/0,,sid9_gci528813,00.html
    | >
    | > Is the lens-shaped figure in red called a steradian or does this term
    | > refer
    | > only to the cone shaped figure in the centre of the sphere?  I am looking
    | > for the name of the red lined shape.
    
    and Herbert Prinz answered
     It appears that I misunderstood your question. You said "arc" and I
    | thought "angle". What you really wanted was the name for the curved
    | surface of a spherical cap (i.e. the surface without the flat bottom),
    | right? I am not sure whether there is a name for that, at least I am not
    | aware of  any.
    
    ============================
    
    I think Herbert has just coined the appropriate term that refers to an area across the surface of a
    sphere, equivalent to an arc along the periphery of a circle, when he called it a "spherical cap".
    "Cap" is a word that describes precisely what Robert Eno requires; and indeed I have seen it so used
    in technical articles (can't recall where). It should be adopted. It has a rather different meaning
    to "solid angle", in that solid angle refers to what is subtended at the centre of the sphere by
    such a cap at its surface. Just as with a circle, in which an arc, on the periphery, has a different
    meaning to an angle subtended at the centre, though one refers to the other, and they are measured
    in the same units.
    
    Going back to solid angle, a physicist would normally measure it in "steradians". There are 4.pi
    steradians in a sphere, just as there are 2.pi radians in a circle. So the whole sky occupies about
    12.5 steradians to an observer floating in space, and about half that to someone at sea. Solid angle
    can also be measured in square degrees; more properly degrees-squared, a patch in the sky that
    subtends 1 degree by 1 degree, or a corresponding area if it doesn't have a square shape. The Sun,
    and the Moon, each occupies about 0.2 degrees-squared.
    
    There are other expressions for solid-angle in folk-speech. For describing the smallness of a solid
    angle, recall the familiar cliche of an ominous cloud which starts off "no bigger than a man's
    hand". That's quite a good unit of solid angle, in that it refers to an well-known area (that of the
    human hand) at a defined distance (the length of the human arm).
    
    Another expression, for expressing the largeness of a solid angle, often referring to a growing area
    of blue sky, appears in the cliche "big enough to patch a Dutchman's trousers". Not quite such a
    good unit, as it's unclear just how far away said Dutchman should be beheld from. Nevertheless, the
    "Dutchman's trouser" might make a picturesque, and large, unit of solid angle. Apologies to all
    Dutchmen, of course.
    
    George.
    
    ==================
    
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
    

       
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