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| Guest |
 Posted: Thu Mar 03, 2005 11:58 am |
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Hi All,
Is there a way to calculate the volume encased by gaussian function of
a certain value? For example, an asymmetric dumbbell can be described by
Exp[-0.5((x-1.5)^2+y^2+z^2)] + 0.75*Exp[-0.5((x+1.5)^2+y^2+z^2)] = 0.5.
I think that something like the marching cubes algorithm may work, but
don't know if there is any other more elegant way to calculate the
the area of this isosurface and the volume it surrounds.
Any help would be greatly appreciated.
Minhhuy |
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| Robert Israel |
| Posted: Thu Mar 03, 2005 3:57 pm |
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Guest
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In article <d07fne$rl4$1@knot.queensu.ca>,
<homh@momentum.chem.queensu.ca> wrote:
[quote:ec85da7e87]Is there a way to calculate the volume encased by gaussian function of
a certain value? For example, an asymmetric dumbbell can be described by
Exp[-0.5((x-1.5)^2+y^2+z^2)] + 0.75*Exp[-0.5((x+1.5)^2+y^2+z^2)] = 0.5.
[/quote:ec85da7e87]
This would be a solid of revolution obtained by revolving the curve
Exp[-0.5((x-1.5)^2+y^2)] + 0.75*Exp[-0.5((x+1.5)^2+y^2)] = 0.5
for y > 0, about the x axis. Surprisingly, this can be solved for y:
y = 1/2 sqrt(8 ln(2 exp(3x)+3/2) - (2x+3)^2)
So the volume is pi int_a^b y^2 dx = pi (F(b) - F(a)) where a and b are the
solutions of your equation with y = 0, and
3 2
9 x x 3 x 3
F(x) = - --- - ---- - ---- - 2/3 ln(2/3) ln( 1 - --------------)
4 3 2 4 exp(3 x) + 3
3 2 2
+ 2/3 dilog(--------------) + 1/3 ln(--------------)
4 exp(3 x) + 3 4 exp(3 x) + 3
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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| Guest |
| Posted: Fri Mar 04, 2005 6:08 pm |
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Robert Israel <israel@math.ubc.ca> wrote:
[quote:240e51bd3e]In article <d07fne$rl4$1@knot.queensu.ca>,
homh@momentum.chem.queensu.ca> wrote:
[/quote:240e51bd3e]
[quote:240e51bd3e]Is there a way to calculate the volume encased by gaussian function of
a certain value? For example, an asymmetric dumbbell can be described by
Exp[-0.5((x-1.5)^2+y^2+z^2)] + 0.75*Exp[-0.5((x+1.5)^2+y^2+z^2)] = 0.5.
This would be a solid of revolution obtained by revolving the curve
Exp[-0.5((x-1.5)^2+y^2)] + 0.75*Exp[-0.5((x+1.5)^2+y^2)] = 0.5
for y > 0, about the x axis. Surprisingly, this can be solved for y:
y = 1/2 sqrt(8 ln(2 exp(3x)+3/2) - (2x+3)^2)
So the volume is pi int_a^b y^2 dx = pi (F(b) - F(a)) where a and b are the
solutions of your equation with y = 0, and
3 2
9 x x 3 x 3
F(x) = - --- - ---- - ---- - 2/3 ln(2/3) ln( 1 - --------------)
4 3 2 4 exp(3 x) + 3
3 2 2
+ 2/3 dilog(--------------) + 1/3 ln(--------------)
4 exp(3 x) + 3 4 exp(3 x) + 3
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
[/quote:240e51bd3e]
Hi Robert,
Many thanks for the prompt reply, actually I am looking for the solution
of a more general case
Sum_i^n A_i Exp[-a_i(x_i^2+y_i^2+z_i^2) = C
where the dumbbell is but the simplest case. That is, the isosurface formed by
a series of gaussians and the volume it surrounds.
Anyhow, I would be interested in how you arrived at the above, as a matter of
learning.
Regards,
Minhhuy |
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