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/
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