-- Lune and lens: correct definitions?

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David W.Cantrell

Jan 20, 2008, 10:25:50 AM1/20/08
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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"


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!

David W. Cantrell

Alexander Bogomolny

Jan 24, 2008, 8:18:47 AM1/24/08
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David, hello:

Eric's definition is definitely not common. Yours is more close to my thinking:

1. Lune is a shape bounded by two circular arcs of "the same concavity".

2. Lens is a more general term for a shape bounded by two circular arcs, not necessarily concave.

I could not find a definition of "lens" in my library, except for common dictionaries. The word "lune" is described in

a. The Penguin Dictionary of Mathematics
b. The Harper/Collins Dictionary of Mathematics
c. Schwartzman's The Words of Mathematics
d. The American Heritage Dictionary
e. Webster's Collegiate Dictionary
f. The Merriam-Webster Dictionary

a-b-c 1. In addition, a-b give a second meaning to "lune" as a spherical region bounded by two great half-circles.

d-e defines "lune" as any region bounded by two circular arcs whether planar or spherival.

f does not define "lune".

The definition of "lens" in d-e-f is 3D and is more sophisticated permitting two opposite curved surfaces which need not be spherical.

Alex Bogomolny

David W.Cantrell

Jan 25, 2008, 11:18:57 AM1/25/08
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Robert Israel <isr...@math.MyUniversitysInitials.ca> wrote:

Thanks for your reply, Robert. It seems, by the way, that
1. your response never appeared in geometry.college
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!


Jan 25, 2008, 12:29:35 PM1/25/08
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Also take a look at http://en.wikipedia.org/wiki/Lune_%28mathematics%29

Johan E. Mebius


Jan 25, 2008, 1:02:27 PM1/25/08
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Another reference for your consideration:

David W.Cantrell

Jan 25, 2008, 4:24:27 PM1/25/08
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Cary <ca...@domain.invalid> wrote:
> Another reference for your consideration:
> <http://www.pballew.net/arithme5.html#lune>

Thank you. Curiously, earlier today, I sent an email to Pat Ballew,
alerting him to the existence of this thread, but without knowing then that
he had a web page about the lune!


David W.Cantrell

Jan 25, 2008, 4:24:23 PM1/25/08
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JEMebius <jeme...@xs4all.nl> wrote:

> >> "David W.Cantrell" <DWCan...@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"


Thanks for the suggestion.

In my article in alt.math.recreational, to which I had provided a link, I
had already given links to the Wikipedia entries for both lune and lens.

Their entry for lune says "In plane geometry, a lune is a concave area
bounded by two arcs, necessarily of unequal radii." But I must suspect that
that last condition originated with MathWorld. As such, I do not think it
supports any necessity that the radii be unequal, but rather merely shows
how "contagious" errors can be.


Pat Ballew

Mar 26, 2008, 9:13:41 AM3/26/08
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David and all,
I recently saw a problem about the solid formed by the intersection of two spheres, and wondered if that should be (could be/ is ever) called a three-dimensional lens?

Pat B

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