Measuring 3D Curvatures and Angles

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.

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?

(4) If I haven't already answered this question, I'll ask you to write 
back and explain some more to me about what you're asking.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/