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June 14

intersection of a plane and sphere

Given a,b,c,d and the 3D body {ax+by+cz<d; x^2+y^2+z^2<1}. compute its volume.--Exx8 (talk) 20:11, 14 June 2020 (UTC)[reply]

@Exx8: Please Do your own homework. --CiaPan (talk) 20:19, 14 June 2020 (UTC)[reply]

this is not part of a homework, but a question in a test that nobody has been able to solve.--Exx8 (talk) 21:09, 14 June 2020 (UTC)[reply]

IMVHO the '3D body' defined above is actually 2-D, just immersed in a 3-D space, hence its volume is zero. But I may be wrong.... --CiaPan (talk) 20:29, 14 June 2020 (UTC)[reply]

How do you reckon that? The two inequalities define an open half-space and an open ball respectively. Their intersection can be empty but I don't see how it can be flat. —Tamfang (talk) 01:03, 15 June 2020 (UTC)[reply]

@Tamfang: Well, the answer is surprisingly simple: there were NO two inequalities there when I was writing it -- one of them was an equation, making the solution a part of a sphere: Special:Diff/962568983#intersection of a plane and sphere.

OP silently fixed that later, which made my comment look stupid. --CiaPan (talk) 11:47, 15 June 2020 (UTC)[reply]

Evil! —Tamfang (talk) 13:37, 15 June 2020 (UTC)[reply]

:-) CiaPan (talk) 14:05, 15 June 2020 (UTC)[reply]

See Spherical cap for hints on calculating the volume, and Distance from a point to a plane for hints to calculate the cap's height. You'll also need to devise a way to identify the part of the sphere on the correct side of the plane. --CiaPan (talk) 21:28, 14 June 2020 (UTC)[reply]

Since that plane (actually boundary of half-space) goes through the origin, doesn't it bisect the sphere? Did you want to add a constant? 2602:24A:DE47:BB20:50DE:F402:42A6:A17D (talk) 08:51, 15 June 2020 (UTC)[reply]

The boundary plane defined by

ax+by+cz=d

does not go through the origin unless

d=0

; it is at a distance

d(a^{2}+b^{2}+c^{2})^{-{\frac {1}{2}}}

from the origin. --Lambiam 10:46, 15 June 2020 (UTC)[reply]

...which can be also written in a form of

d/{\sqrt {a^{2}+b^{2}+c^{2}}},

possibly more legible for some readers. :) --CiaPan (talk) 15:47, 15 June 2020 (UTC)[reply]

@Lambiam: We both forgot to use an absolute value: the distance is

{\color {red}\left|d\right|}(a^{2}+b^{2}+c^{2})^{-{\frac {1}{2}}}={\frac {\color {red}\left|d\right|}{\sqrt {a^{2}+b^{2}+c^{2}}}}

and the value with a sign indicates whether the origin point is 'before' the plane or a plane is 'before' the origin, with respect to the normal vector

[a\ b\ c]^{T}

(and the sign of the

d

term). --CiaPan (talk) 14:31, 16 June 2020 (UTC)[reply]

Oops. Mea culpa. --Lambiam 18:13, 16 June 2020 (UTC)[reply]

I'm pretty sure you want the signed distance anyway. If d<0 you get a cap and if d>0 the you get the complement of a cap, but the same volume formula applies. Actually what you really want is h, the distance from the plane the pole at the center of the cap; you get a normal cap when h<r and the complement of a cap when h>r. The curve y=πx²(3r-x)/3 is symmetric about the point (r, 2πr²/3), so the same formula works for a cap and a cap complement. --RDBury (talk) 15:44, 17 June 2020 (UTC)[reply]