crescent shapes
Mark Spahn
of Richard Carrington and the Tale of How Modern Astronomy Began".
Photos of solar eclipses in that book show that the Sun and Moon
are of nearly equal size when seen from Earth, so that when the
Moon occludes the Sun, the crescent that appears is nearly the
same as the crescent that is seen when a sphere is illuminated from
a light source that is off at an angle from the viewer.
But are they exactly the same?
Let's consider the two kinds of crescent:
(1) a circle-on-circle crescent, in which one circle is occluded
by another circle of the same size
(2) an illuminated-sphere crescent, that is, the crescent that
one sees when light shines onto a sphere from an angle
Question: Are these two kinds of crescent of exactly the same shape?
Or rather can one distinguish a circle-on-circle crescent from
an illuminated-sphere crescent?
Curiously, neither of these natural crescents is quite like the
crescent one sees in graphic designs, which is formed by a circle
being partly occluded by a *smaller* circle. For examples, see
http://www.crescentofbetrayal.com/proof.htm ,
which shows pictures of the flags of Algeria, Azerbaijan,
Pakistan, and Tunisia. For another star-in-crescent logo, see
http://www.nisshin-steel.co.jp/nisshin-steel/english/index.htm ,
which shows the logo of the Japanese steel company Nisshin Steel.
Why this particular logo? Well, an old name for this company was
Tsukiboshi, a Japanese proper name meaning Moon (tsuki = moon)
and Star (hoshi (-boshi) = star).
For some reason, this unnatural crescent is considered more
esthetically pleasing -- or maybe it's because such a crescent
has tips that are not so skinny as to easily be bent when a
crescent is made out of metal.
-- Mark Spahn (West Seneca, NY)
David W. Cantrell
>
> Let's consider the two kinds of crescent:
> (1) a circle-on-circle crescent, in which one circle is occluded
> by another circle of the same size
> (2) an illuminated-sphere crescent, that is, the crescent that
> one sees when light shines onto a sphere from an angle
>
> Question: Are these two kinds of crescent of exactly the same shape?
No.
> Or rather can one distinguish a circle-on-circle crescent from
> an illuminated-sphere crescent?
Yes. For the latter crescent, as projected onto a plane, one of the
boundaries is a semicircle (of course), while the other boundary is a
semiellipse. The major axis of the semiellipse is the same as the diameter
of the semicircle. Eccentricity of the semiellipse depends on the angle
from which the sphere is illuminated.
David
mensa...@aol.com
Cool. So, are the flags that feature a cresent
moon with a star correct? Aside from the star
inorrectly shown inside the cresent?
>
> David
David W. Cantrell
The flags do not accurately depict the crescent Moon. But I cannot say for
certain that depiction of the crescent Moon was their objective. And
there's always the matter of "artistic license".
Rather, the flags appear to show a lune.
See <http://mathworld.wolfram.com/Lune.html>. But I must express doubt
about the accuracy of the first two sentences there: "A lune is a plane
figure bounded by two circular arcs of unequal radii, i.e., a crescent. (By
contrast, a plane figure bounded by two circular arcs of equal radius is
known as a lens.)" My guess is that MathWorld's requirements of unequal
radii for a lune and of equal radii for a lens is incorrect. I would
suppose that either a lune or a lens could have equal or unequal radii, and
that the actual distinction is merely whether the figure is convex or not.
If convex, we have a "lens"; if not, we have a "lune".
The Wikipedia entries
<http://en.wikipedia.org/wiki/Lune_%28mathematics%29> and
<http://en.wikipedia.org/wiki/Lens_%28geometry%29> might also be
of interest.
David
Mark Spahn
"David W. Cantrell" <DWCan...@sigmaxi.net> wrote in message
news:20071111103546.529$q...@newsreader.com...
David,
Thank you for your very clear explanation.
It ought to be embarrassing, but I never before realized that
the tips of the crescent displayed by the Moon never extend beyond
the illuminated half of the Moon. Of course, this must be so,
because a light source shined on a sphere cannot illuminate
more than half of it (a semisphere).
I am waiting for an unovercast night to verify this insight.
I will also be on the lookout for crescent shapes on flags
and company logos. My impression is that these decorative
crescents *always* extend beyond a semicircle,
a shape which the real Moon never displays.
If somebody finds an exception, let us know.
-- Mark Spahn, would-be moon-gazer
The Qurqirish Dragon
> "David W. Cantrell" <DWCantr...@sigmaxi.net> wrote in messagenews:20071111103546.529$q...@newsreader.com...
Although the naked eye won't be able to see it, I wouldn't be
surprised if the crescent of the moon DOES go beyond the 1/2 way
point. Although slight at the distance of 1 AU, the fact that the sun
is larger than the moon means that some of the "back" side will be
illuminated. Similarly, if you have a light source that is smaller in
diameter than the illuminated body, then less than 1/2 will be lit
(and again, at extreme distances it is probably tought to measure.)
Of course, since the moon is not a perfect sphere, the irregularity of
the shape probably causes more distortion in the lit region than the
relative sizes of the sun and moon.