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    Re: Mathematics Question
    From: Herbert Prinz
    Date: 2006 Mar 30, 11:26 -0500

    George Huxtable wrote:
    >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".
    I did not coin that. This is what I understand most people will call the
    solid body. It's the special case of a segment of a sphere that reaches
    all the way to the "end". In German, definitely "Kugelkappe". But I
    thought (in my second message) that Robert wanted the name of the
    surface only, since he had asked for the "two-dimensional equivalent" of
    an arc. For this, I know no name.
    >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.
    Right. On the otherhand, just how different are these meanings really?
    How do we get a grasp of that thing "at the centre" that we call an
    angle. It's actually a ratio of arcs. We *say* two lines intersect at an
    angle of 30 deg, and we *see* these two lines intersecting each other at
    30 deg, but the former is a way of speaking and the latter is a vague
    image. The only way to assign mathematical meaning is by saying that if
    we draw a circle around the point of intersection, the arc that is cut
    off by the lines is to the full circle as 30 to 360. (Or as whatever it
    is to 2*R*Pi, if you prefer radians.) Therefore, in a way the angle *is*
    the arc that's cut off on the unit circle. You may ask, but what if I
    define the angle analytically? Take the scalar product of two unit
    vectors on these lines and you get cos(x), where x may be defined as the
    "angle", with an appropriate analytical definition of cos. True, but
    when you are done, it turns out that this cos function will behave
    exactly in the way so that it fits in with those diagrams that the
    ancient Greeks drew about arcs and their chords. (Sin being the half
    chord of the double angle, versin being the maximum distance of the
    chord from the arc, and finally cos being what remains after you
    subtract the versin from the radius.)
    So, trigonometric function are functions that relate the length of an
    arc to the length of some straight line. You can see this in statements
    like "For small x, sin(x) = x", or in the fact that we call the inverse
    functions of sin, cos, etc. the arc functions.
    All this long winded explanation by way of excusing why I brought up
    stereo angles, when in fact, Robert asked about a surface. He asked
    about the two dimensional equivalent of an arc. Although all angles are
    arcs, not all arcs need to be interpreted as angles. My mistake.
    Herbert Prinz

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