Math Help - Music of the Spheres

Quick question for the math folks - how do I find the dimensions of a given section of a sphere? If I have a sphere of radius s, and it intersects a plane at distance d from the center of the sphere, what is the radius of the circle formed at the intersection?Sort of like this:A highly practical question for me this weekend.

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If I have a sphere of radius s, and it intersects a plane at distance d from the center of the sphere, what is the radius of the circle formed at the intersection?

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r = \sqrt{s^2 - d^2}.I think the area of the "cap" formed isA = 2 \pi (s^2 - sd)but someone better check my work here.Note: In case you can't read TeX, \sqrt is the square root sign, with the argument in the curly brackets. \pi is 3.14159....

You have a line of length d from the center of the sphere to the center of the intersecting circle.A line of length r, where r is the radius of the intersecting circle, joins the center of that circle to a point on its circumference.Those two lines are perpendicular, so are the legs of a right angled triangle. The hypotenuse is a line connecting the center of the sphere to the point on the circumference. That gives a length s for the hypotenuse.The Pythagorian theorem gives us d^2 + r^2 = s^2.I hope that helps.edit: corrected typo.This message has been edited by nwr, 10-01-2005 03:16 PM

crash, ...Did this off the cuff. I'm no mathematician. Here's what I did:1. Since spheres are symmetric in 3D space, I can validly rotate my sphere / plane system to my coordinate system however I want, and not change the problem.2. I chose to orient the sphere / plane system so that the center of my sphere is at the CENTER of my XYZ coordinate axes, and so that the plane is perpendicular to XY plane AND perpendicular to the X axis (parallel to the Y axis). Like I said, I can do this because of symmetry.This makes the problem turn into a completely 2D problem:3. This made things SUP easy; easy geometry / trig:

circle function: x2 + y2 = r2
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so for point p,
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px2 + py2 = r2
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or
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d2 + r_12 = r2
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so

At least, that's what I came up with.BenAbE: crap I did a lot of work that the "smart" people didn't have to do oh well.This message has been edited by Ben, Saturday, 2005/10/01 01:21 PMThis message has been edited by Ben, Saturday, 2005/10/01 01:23 PM

Don't feel bad - that's the way we used to do it (on the cave walls) back in high school.

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People who think they have all the answers usually don't understand the questions.

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crap I did a lot of work that the "smart" people didn't have to do

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I don't know whether I'm one of the "smart people", but your method is exactly the one I used. I just didn't post (or describe) my sketch.Edited to make my meaning more clear.This message has been edited by Chiroptera, 01-Oct-2005 09:12 PM

can you tell us why?

Damn, yeah, that's the way I was trying to do it in my head, but I just couldn't fill in the algrebra for some reason. (Well, not "some reason" - the reason is that I suck at math, hardcore.)As to what I'm doing - anybody ever heard of "casemodding"? I'm cutting a hole in my computer so that a plasma globe I bought will project out of the front of the case, and I want it to be nice and snug. The result should be something akin to this:Which is where I got the idea. Anyway, thanks to everybody that replied. Now that you've laid it out for me, I'm embarassed that its so simple.

Cool. Are you going to have to shield it? My understanding is that plasma globes are powered by small Tesla coils.

I doubt it. I don't think that guy did. I'm no eletrical engineer, but I suspect several things are working in my favor:1) My case is a big grounded cage, basically.
2) It's big enough that the inverter circuit for the globe will be fairly far from any crucial components.
3) The inverter/transformer assembly is completely encased in this black plastic stuff.I think I'll be ok. I doubt this thing kicks out any worse RF than neon tubes, or florescent lights, or those cold-cathode kits that folks throw right into their cases.Should look pretty sweet. I'm keeping a build log; interested folks can catch it on my (rarely-updated) blog You Magnificent Bastard.