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Calculating differential geometry of a sphere using integral method
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I have a sphere(sphere at interface) that have two different patch(P, S). The sphere can rotate and translate(up-down) at the air-water interface. The angle alpha defines the position of the sphere (alpha increase means the sphere goes towards the air). The angle beta defines the rotation of the sphere. So the area of any patch into any phase changes with the position and orientation(example: area of P inside air phase will change with alpha and beta angle). I know that the surface area of a sphere in a spherical coordinate system $f(r,theta, phi)$ is $int_{theta=0}^pi int_{phi=0}^{2pi} (rsintheta dphi)(r dtheta) $
Now if I want to calculate the the portion of the patch P that is in the air what will be the limits of my integrals? I am looking for the limits because with that I will be able to calculate the area of patch p into air for any position and orientation.
integration geometry differential-geometry
asked Jan 25 at 1:00
TanvirTanvir
136
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add a comment |
$begingroup$
I have a sphere(sphere at interface) that have two different patch(P, S). The sphere can rotate and translate(up-down) at the air-water interface. The angle alpha defines the position of the sphere (alpha increase means the sphere goes towards the air). The angle beta defines the rotation of the sphere. So the area of any patch into any phase changes with the position and orientation(example: area of P inside air phase will change with alpha and beta angle). I know that the surface area of a sphere in a spherical coordinate system $f(r,theta, phi)$ is $int_{theta=0}^pi int_{phi=0}^{2pi} (rsintheta dphi)(r dtheta) $
Now if I want to calculate the the portion of the patch P that is in the air what will be the limits of my integrals? I am looking for the limits because with that I will be able to calculate the area of patch p into air for any position and orientation.
integration geometry differential-geometry
asked Jan 25 at 1:00
TanvirTanvir
136
$endgroup$
add a comment |
$begingroup$
I have a sphere(sphere at interface) that have two different patch(P, S). The sphere can rotate and translate(up-down) at the air-water interface. The angle alpha defines the position of the sphere (alpha increase means the sphere goes towards the air). The angle beta defines the rotation of the sphere. So the area of any patch into any phase changes with the position and orientation(example: area of P inside air phase will change with alpha and beta angle). I know that the surface area of a sphere in a spherical coordinate system $f(r,theta, phi)$ is $int_{theta=0}^pi int_{phi=0}^{2pi} (rsintheta dphi)(r dtheta) $
Now if I want to calculate the the portion of the patch P that is in the air what will be the limits of my integrals? I am looking for the limits because with that I will be able to calculate the area of patch p into air for any position and orientation.
integration geometry differential-geometry
asked Jan 25 at 1:00
TanvirTanvir
136
$endgroup$
I have a sphere(sphere at interface) that have two different patch(P, S). The sphere can rotate and translate(up-down) at the air-water interface. The angle alpha defines the position of the sphere (alpha increase means the sphere goes towards the air). The angle beta defines the rotation of the sphere. So the area of any patch into any phase changes with the position and orientation(example: area of P inside air phase will change with alpha and beta angle). I know that the surface area of a sphere in a spherical coordinate system $f(r,theta, phi)$ is $int_{theta=0}^pi int_{phi=0}^{2pi} (rsintheta dphi)(r dtheta) $
Now if I want to calculate the the portion of the patch P that is in the air what will be the limits of my integrals? I am looking for the limits because with that I will be able to calculate the area of patch p into air for any position and orientation.
integration geometry differential-geometry
integration geometry differential-geometry
asked Jan 25 at 1:00
TanvirTanvir
136
asked Jan 25 at 1:00
TanvirTanvir
136
edited Jan 25 at 1:36
Tanvir
asked Jan 25 at 1:00
TanvirTanvir
136
asked Jan 25 at 1:00
TanvirTanvir
136
asked Jan 25 at 1:00
TanvirTanvir
136
136
add a comment |
add a comment |
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