Wikipedia:Reference desk/Archives/Mathematics/2008 March 2

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March 2

Geometry of an eclipse

Thinking about the fact that the sun and moon subtend nearly equal angles at the earth, so that a total eclipse of the sun is just about possible, I wondered if the following sort of problem has an exact solution. In 3D space, take a light source at the origin, a sphere of radius r at (2,0,0) and one of radius 3 at (8,0,0). Elementary geometry makes it obvious that the larger sphere just fails to be illuminated when r=9/8, but what proportion of its surface is lit when r=1, say, or for general r ≤ 9/8?…86.132.235.43 (talk) 13:08, 2 March 2008 (UTC)

How did you calculate the r=9/8 figure? I get r=3/4... It boils down to similar triangles - the smaller sphere is 4 times closer, so needs to be 4 times smaller. As for what proportion of the surface of the larger sphere will be in shadow, you need to work out the surface area of the part of the sphere with radius less than 4r. I think its:

$\int _{\sqrt {9-16r^{2}}}^{3}2\pi xdy$ where $y={\sqrt {9-x^{2}}}$

Then divide that by the surface area of a sphere of radius 3, which is $4\pi 3^{2}=36\pi$ --Tango (talk) 13:57, 2 March 2008 (UTC)

Sorry, I meant the smaller sphere centred at (3,0,0), but the principle is the same.86.132.235.43 (talk) —Preceding unsigned comment added by 84.67.149.226 (talk) 17:30, 2 March 2008 (UTC)

That would require r=2, surely, not r=9/8? --Tango (talk) 17:41, 2 March 2008 (UTC)

No, anonymous is right. 3/8 of 3 is 9/8, so a 9/8-radius sphere centered at (3,0,0) would be just large enough to cover the 3-radius sphere with shadow. Anyway, all this discussion is pretty useless for real eclipses, because the Sun is not a single point of light. —Keenan Pepper 03:51, 3 March 2008 (UTC)

Ah, I misread the correction. I moved the larger sphere to (3,0,0), not the smaller sphere - sorry! --Tango (talk) 11:39, 3 March 2008 (UTC)

The portion of the farther sphere that is not illuminated is a spherical cap. Noting that tangents to a circle are perpendicular to radii, the triangle formed by the origin, the limb of the closer sphere, and the center of that sphere is a right triangle, and so the half angle of the cone of blocked light is $\arcsin {\frac {r}{x}}$, where x and r describe the position and radius of the closer sphere. Then construct the analogous triangle for the farther sphere: the origin, the edge of the first sphere's shadow, and the center of the farther sphere. This is not a right triangle in general, since the shadow is only tangent to the farther sphere in the "precise coverage" case. Applying the law of sines and the fact that triangle angles sum to π radians, we get that the angle that the half shadow subtends at the center of the farther sphere is $\theta =\pi -\arcsin {\frac {r}{x}}-\arcsin {\frac {rX}{Rx}}$, where capital letters apply to the farther sphere. The height of the spherical cap is $R(1-\cos \theta )=R\left(1+\cos \left(\arcsin {\frac {r}{x}}+\arcsin {\frac {rX}{Rx}}\right)\right)$, its surface area is proportional. You may want to compare to the case of no shadow, where the cap height is $R-{\frac {R^{2}}{X}}$; divide the two heights to get the portion of the potentially-illuminated surface that actually is illuminated. --Tardis (talk) 19:27, 3 March 2008 (UTC)

Rational solutions to Y² = aX² + bX +c

OK, I've found the solutions for a a perfect square and c a perfect square, but the final part of the problem I'm working on makes the restriction $a+b+c=\Box$. Can someone give me a pointer on the algebra for this one... —Preceding unsigned comment added by Donald Hosek (talk • contribs) 16:30, 2 March 2008 (UTC)

It might help to change variables to eliminate the linear term: $X\mapsto X+{\frac {b}{2a}}$. You'll then have an equation of the form Y² = aX² + c which should be easier to solve. --Tango (talk) 16:37, 2 March 2008 (UTC)

I don't understand the notation for the restriction; aren't a, b and c given? What is the meaning of the box? After the reduction suggested by Tango the eqaution can be seen as a generalization of Pell's equation. To make the problem Diophantine, rewrite it as Y² = aX² + cZ² with a new unknown Z. I have not looked at it, but I suspect that the algebra that is relevant for solving Pell's equation, interesting as it is, is not going to help much here.  --Lambiam 19:31, 2 March 2008 (UTC)

I think the box means "a square". I think a, b and c are parameters - Donald is looking for a general solution. --Tango (talk) 19:50, 2 March 2008 (UTC)

If a+b+c is a square, then you have a rational solution for X=1. And it's easy to see in general that, once you have a solution, you can find all of them (consider the lines with rational slopes through this one point, and consider the second intersection point with the given parabola). Hum, we may want to write something on the rational parametrization of conics, I can't seem to find an article about this. A paragraph at conic? Bikasuishin (talk) 23:02, 2 March 2008 (UTC)

Aha, that's the magic point there. I should be able to get to the solution from that. I may be able to tackle the rational parametrization thing down the line. I would think it belongs in Diophantine equation. Donald Hosek (talk) 19:26, 3 March 2008 (UTC)

I'm not sure where we should put this. Actually, the statement itself is already in the encyclopedia at Severi-Brauer variety, but no proof is given, and it's probably not needed there considering the readership of an article like that. But it's a bit too specific for Diophantine equation, I believe. I'll do some article reading and ponder this (not that adding those two lines is a big deal, but anyway). Bikasuishin (talk) 00:25, 4 March 2008 (UTC)

Why 10?

Why is our numerical system based off of 10? Historical info? Thanks much--UhOhFeeling (talk) 20:43, 2 March 2008 (UTC)

It is said it is because we have ten digits on our hands. See decimal for more info. — Kieff | Talk 21:16, 2 March 2008 (UTC)

Great, Thanks much!--UhOhFeeling (talk) 21:30, 2 March 2008 (UTC)