Science > Physics > Independent/Dependent Phases 19.4: Which Quadric Intersections Yield (Phase) 1/2
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Science > Physics |
| User: |
"OsherD" |
| Date: |
14 Oct 2005 12:09:11 AM |
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Independent/Dependent Phases 19.4: Which Quadric Intersections Yield (Phase) 1/2 |
From Osher Doctorow
If it turns out that the only quadrics whose intersections reduce to
expressions with factor 1/2 are the ellipsoid (including sphere) and
cone, then since the cone and ellipsoid (especially the sphere) are
intuitively "opposites", the association of 1/2 with phase differences
in the previous sections of this thread becomes even more interesting.
We have to exclude some "obviously" poorly compared cases, such as for
example quadrics with different a, b, and/or c coefficients. The
terminology a, b, c refers to the following equations of quadrics.
1) x^2/a^2 + y^2/b^2 + z^2/c^2 = 1 (ellipsoid)
2) x^2/a^2 + y^2/b^2 - z^2/c^2 = 1 (hyperboloid of one sheet)
3) x^2/a^2 - y^2/b^2 - z^2/c^2 = 1 (hyperboloid of two sheets)
4) x^2/a^2 + y^2/b^2 - z^2/c^2 = 0 (cone)
5) x^2/a^2 + y^2/b^2 - cz = 0 (paraboloid)
6) y^2/a^2 - x^2/b^2 = cz (hyperbolic paraboloid)
Readers can see that if (1) is intersected with either (2) or (3), then
all terms vanish so that 2z^2/c^2 = 0 or else 2y^2/b^2 + 2z^2/c^2 = 0
which again yields each term as 0. Not so with the cone and ellipsoid,
which yields 2z^2/c^2 = 1 so z^2/c^2 = 1/2. If the ellipsoid is
intersected with the paraboloid, we get z^2/c^2 - cz = 1 which doesn't
involve 1/2. For the ellipsoid intersection with the hyperbolic
paraboloid, we get:
7) x^2/a^2 + x^2/b^2 + y^2/b^2 - y^2/a^2 + z^2/c^2 - cz = 1
which again has no 1/2 factor in general.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 19.4: Which Quadric Intersections Yield (Phase) 1/2 |
14 Oct 2005 12:30:14 AM |
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From Osher Doctorow
We could replace the particular hyperbolic paraboloid of (6) by:
1) x^2/a^2 - y^2/b^2 = cz
and then intersecting with the ellipsoid:
2) x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
yields:
3) 2y^2/b^2 -cz + z^2/c^2 = 1
If we divide through by 2, there are too many terms of make the role of
1/2 clear.
Next, intersecting the hyperboloid of one sheet with the hyperboloid of
two sheets yields 2y^2/b^2 =0 and a similar intersection of the
hyperboloid of one sheet with the cone yields 1 = 0 which is
impossible. The intersection of the hyperboloid of one sheet with the
paraboloid yields -z^2/c^2 + cz = 1 which has no 1/2 factor, while the
intersection of the former with the hyperbolic paraboloid of (1) above
has the same difficulty of too many terms of make the role of 1/2
clear.
The intersection of the hyperboloid of two sheets with the cone yields:
4) -2y^2/b^2 = 1
which is impossible since the left hand side is nonpositive and the
right hand side is 1.
The other two intersections of the hyperboloid of two sheets remaining
are similar to earlier non-1/2 or non-clear 1/2 situations.
The cone and the paraboloid intersected yields -z^2.c^2 + cz = 0 which
has nothing to do with 1/2.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 19.4: Which Quadric Intersections Yield (Phase) 1/2 |
14 Oct 2005 01:02:00 AM |
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From Osher Doctorow
In the sentence after equation (3) of last time, the word "of" should
be "to".
It might be asked whether it is legitimate to exclude a case where too
many terms have 1/2 on the grounds of "not having a clear
interpretation". Perhaps the words "not having a simple
interpretation" would be better. If 1/2 occurs in 2 or 3 terms instead
of the simple one term occurrence in the ellipsoid-cone intersection
z^2/c^2 = 1/2, I'd say that the clarity issue is rather
straightforward.
We still have one case left: the paraboloid intersected with the
hyperbolic paraboloid, which yields:
1) 2y^2/b^2 = 0
So we're left with one quadric intersection with a simple 1/2 factor or
term: the ellipsoid (including sphere) intersection with the cone.
This is the one arguably intuitive case of opposites (ellipsoid and
cone), which in turn suggests that their opposite nature is that of a
phase difference (phase in the sense of liquid, solid, gas, plasma,
Bose-Einstein condensate, superfluid, superconductor, liquid crystal,
and arguably black hole among other possibilities).
Osher Doctorow
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