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### Measuring 3D Curvatures and Angles

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Date: 10/21/2001 at 03:12:59
From: Evan Maukonen
Subject: Measurement of 3-dimensional curvatures and angles

I have a few questions that I have been wondering about for the past
few years, and particularly so in the past few days, to the detriment
of my sanity.

(1) I would like to know what the exact term for 'solid angles' is.
From what little I have been able to dig up, I am led to believe that
they might be called steradians; however, information I have acquired
here leads me to believe that they may simply be a way of measuring
these 'solid angles'. If the former is true, then I would also like to
know how and in what unit they are measured. Also along those lines,
if steradians do happen to be the unit of measurement, is there a 3-
dimensional version of the degree that is also used? If so, I would
also like to know how conversions between the two (steradians and the
3-d degrees, that is) can be made.

(2) How does one measure the curvature of a sphere? Also, what is the
term for such a 'sperical curve'? What I mean is, in the way that you
measure 2-dimensional curves, is there a way to determine the
'curvature' of a sphere? If so, how, and in what units? Perhaps I am
wrong in thinking that we measure 2D curves at all, rather we measure
the angles that make the curves that are segments of a circle x. If
that is the case, how exactly are they quantified?

(3) What is the relation between the 'solid angles' and the 'spherical
curves' that such 'solid angles' create when they intersect a sphere
whose exact center is the same as the location of the vertex?

(4) What is the relation between whatever one might call the sector on
the sphere as described above and the sphere as a whole?

I apologize for the length and number of the questions I have posed,
and I wish you to know that I very much appreciate your time. Thank
you very much.
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Date: 10/21/2001 at 06:33:32
From: Doctor Mitteldorf
Subject: Re: Measurement of 3-dimensional curvatures and angles

Dear Evan,

No apology is called for. I love to answer questions from people like
you who have already put a lot of thought into a subject.

(1) A solid angle is called a solid angle, and steradians are the unit
of measurement. The measurement of solid angle is the area of a
section of the sphere divided by the square of the radius of that
sphere. Since a sphere has area 4pi*r^2, the number of steradians in a
full sphere is 4pi.

I've never heard of a "square degree" or "3-d degree". I can
guess the reason why. Suppose you try to define a "square degree"
as the solid angle of a one-degree by one-degree square on the
surface of a sphere. The problem is that that area depends on where
the patch is located. Think of latitude and longitude: If the patch
is where I am, at 40 degrees latitude, the area of the patch is about
(r/360)*(r/360)*cos(40). I say "about" because the whole patch can't
be at 40 degrees latitude. Maybe it extends north from 40 to 41
degrees, or south from 39 to 40 degrees.

So even if the patch is centered right on the equator, the area is not
EXACTLY (r/360)*(r/360). (It might be a good exercise in geometry to
figure out exactly what that area is.)

(2) Einstein was thinking just the way you're thinking when he was a
young man. He had already concocted the Theory of Relativity in 1905.
In order to devise the General Theory, he had to understand deeply
what curvature means in 3 dimensions. (Actually, he considered time
and extended the idea to 4 dimensions as well.) The whole process took
him 10 years, and it was 1915 before the General Theory was complete.

See the MacTutor History of Mathematics archive's article on General
Relativity:

http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/General_relativity.html

If you have a curved line that you can draw on a piece of paper (1-d
line curved in 2 dimensions), you can imagine bringing a circle up to
any point on that line. If the circle is just the right radius it will
"kiss" the line, and the curves will look, for a stretch, just as
close as they can possibly be. The mathematical term for two such
curves is "osculating", which means to kiss. You can define the radius
of curvature in terms of an osculating circle.

But in three dimensions things get more complicated. You would like to
define the radius of curvature of a surface in terms of an osculating
sphere, but there may not be any such thing. This is because the
curvature can be different when you look in different directions. A
sphere has two radii of curvature that are the same. A cylinder has a
single radius of curvature - there is no curvature in the other
direction, so the other radius is infinite. You can imagine an
ellipsoid with two different, finite radii of curvature at the same
point. A saddle actually has two radii of curvature of opposite signs
- one positive, the other negative.

If there are two radii of curvature for a 2-d surface in 3 space, how
many radii of curvature did Einstein have to consider in 4-space? It
makes our heads spin just to imagine.

(3) I'm not sure what you mean here. Does it answer your question to
say that a solid angle is defined as the area of any odd-shaped part
of a spherical surface, divided by the radius squared?

- Doctor Mitteldorf, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
College Higher-Dimensional Geometry
High School Higher-Dimensional Geometry

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