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Topic: -- Lune and lens: correct definitions?
Replies: 9   Last Post: Mar 26, 2008 4:10 AM

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Registered: 12/13/04
Re: -- Lune and lens: correct definitions? - Also in Wikipedia
Posted: Jan 25, 2008 12:29 PM
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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, A-B and B-A, 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 crescent-shaped 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

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