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JEMebius
Posts:
913
Registered:
12/13/04


Re:  Lune and lens: correct definitions?  Also in Wikipedia
Posted:
Jan 25, 2008 12:29 PM


David W.Cantrell wrote: > Robert Israel <israel@math.MyUniversitysInitials.ca> wrote: >> "David W.Cantrell" <DWCantrell@sigmaxi.net> writes: >> >>> The notion of a crescent or lune was discussed recently in >>> alt.math.recreational. Interested readers might look at my second >>> posting (Nov. 12) in the thread "crescent shapes" >>> > <http://groups.google.com/group/alt.math.recreational/msg/377799ffe126a44c> >>> In particular, I expressed doubt about the definitions of lune and >>> lens given at MathWorld: <http://mathworld.wolfram.com/Lune.html> >>> and <http://mathworld.wolfram.com/Lens.html>. My guess is that the >>> distinction based on whether the radii are equal or not is solely >>> due to Eric Weisstein. OTOH, I am also slightly uneasy with my >>> suggestion there that the distinction between lune and lens should be >>> one of convexity. After all, an optical lens need not be convex in >>> section and, when the Moon is gibbous, the lighted part that we see >>> from Earth is convex. Partially due to my uncertainty, I have not yet >>> written to Eric to try to get his entries for lune and lens corrected. >>> If anyone can shed light on what is "correct", historically or >>> otherwise, I would appreciate it! >>> >>> My idea: >>> >>> Given two circular disks, A and B, having a nonempty intersection and >>> such that neither is entirely contained in the other, three regions are >>> formed. (Think of a Venn diagram.) One of the regions, A /\ B, is >>> convex; I suggest that "lens" be used to name that kind of shape. >>> Neither of the other two regions, AB and BA, is convex; I suggest >>> that "lune" be used to name that kind of shape. >>> >>> I also suggest that a circular disk itself should be considered a >>> degenerate case of both the lune and the lens. >>> >>> Comments please! > > Thanks for your reply, Robert. It seems, by the way, that > 1. your response never appeared in geometry.college > and > 2. although it did appear in sci.math, it is archived neither at Google > Groups nor the MathForum. > Does anyone have an explanation for 1. or 2.? > > Let me also take this opportunity to thank Alexander Bogomolny for his > reply in geometry.college and another person for his reply by private > email (which also mentioned the OED's definition of lune). > >> The Oxford English Dictionary has for lune: >> >> 1. Geom. The figure formed on a sphere or on a plane by two arcs of >> circles that enclose a space. > > I was concerned with the planar figure called lune. Nonetheless it's > interesting to see that the OED's use of spherical lune is surprisingly > loose, allowing arcs of circles which aren't great circles. > >> On the other hand, Maple 11's mathematical distionary has >> >> 1. a section of the surface of a sphere enclosed between two semicircles >> that intersect at diametrically opposite points on the sphere. > > That's what I expect for a spherical lune. > >> 2. a crescentshaped figure formed on a plane surface by the >> intersection of the arcs of two circles, such as the shaded section >> of the figure. >> >> (and the figure shows the region inside one circle and outside the >> other) > > I'm guessing that Maple 11's mathematical dictionary is the same as the > HarperCollins Dictionary of Mathematics by Borowski and Borwein (esp. if > the figure to which you referred happens to be called "Fig. 233"). > >> Neither of these has lens (in its geometrical meaning). But OED >> does have the geological meaning of "lens": >> >> A body of ore or rock similar in shape to a biconvex lens. >> >> The distinction on whether the radii are equal seems pretty clearly >> bogus. > > Glad you agree. That was my primary contention. > >> Although of course an optical lens does not need to be convex, I think >> the first lenses were. The word "lens" in Latin means "lentil", and >> those are convex. Also the OED has a quote from Newton's "Opticks": >> >> A Glass spherically Convex on both sides (usually called a Lens). >> >> Although it's true the moon sometimes appears convex, the shape that is >> popularly connected with the moon is the crescent. So I'd agree that >> the distinction should be one of convexity. > > Thanks again to all who replied! > David
Also take a look at http://en.wikipedia.org/wiki/Lune_%28mathematics%29
Johan E. Mebius



