What is the spherical parametrization of an ellipsoid NOT centered in the origin?
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I would like to know how to parametrize an ellipsoid not centered in the origin, but with its axes parallel to the main axes of the reference system.
The result I am looking for would be an expression of the distance of a point on the surface from the origin, given the azimuth and elevation (or any possible two angles of a spherical coordinate system).
I have found on wikipedia a similar formula for an ellipse. The given formula accounts also for rotation, which I don't necessarily need.
geometry 3d quadratics
edited Sep 14 '14 at 17:51
MattAllegro
2,58751432
asked Sep 14 '14 at 17:39
MatteoMatteo
161
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add a comment |
$begingroup$
I would like to know how to parametrize an ellipsoid not centered in the origin, but with its axes parallel to the main axes of the reference system.
The result I am looking for would be an expression of the distance of a point on the surface from the origin, given the azimuth and elevation (or any possible two angles of a spherical coordinate system).
I have found on wikipedia a similar formula for an ellipse. The given formula accounts also for rotation, which I don't necessarily need.
geometry 3d quadratics
edited Sep 14 '14 at 17:51
MattAllegro
2,58751432
asked Sep 14 '14 at 17:39
MatteoMatteo
161
$endgroup$
add a comment |
$begingroup$
I would like to know how to parametrize an ellipsoid not centered in the origin, but with its axes parallel to the main axes of the reference system.
The result I am looking for would be an expression of the distance of a point on the surface from the origin, given the azimuth and elevation (or any possible two angles of a spherical coordinate system).
I have found on wikipedia a similar formula for an ellipse. The given formula accounts also for rotation, which I don't necessarily need.
geometry 3d quadratics
edited Sep 14 '14 at 17:51
MattAllegro
2,58751432
asked Sep 14 '14 at 17:39
MatteoMatteo
161
$endgroup$
I would like to know how to parametrize an ellipsoid not centered in the origin, but with its axes parallel to the main axes of the reference system.
The result I am looking for would be an expression of the distance of a point on the surface from the origin, given the azimuth and elevation (or any possible two angles of a spherical coordinate system).
I have found on wikipedia a similar formula for an ellipse. The given formula accounts also for rotation, which I don't necessarily need.
geometry 3d quadratics
geometry 3d quadratics
edited Sep 14 '14 at 17:51
MattAllegro
2,58751432
asked Sep 14 '14 at 17:39
MatteoMatteo
161
edited Sep 14 '14 at 17:51
MattAllegro
2,58751432
asked Sep 14 '14 at 17:39
MatteoMatteo
161
edited Sep 14 '14 at 17:51
MattAllegro
2,58751432
edited Sep 14 '14 at 17:51
MattAllegro
2,58751432
edited Sep 14 '14 at 17:51
MattAllegro
2,58751432
2,58751432
asked Sep 14 '14 at 17:39
MatteoMatteo
161
asked Sep 14 '14 at 17:39
MatteoMatteo
161
asked Sep 14 '14 at 17:39
MatteoMatteo
161
161
add a comment |
add a comment |
1 Answer
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$begingroup$
The parametrization for an ellipsoid is
$$begin{align}
x & = a rho sin(theta) sin(varphi), \
y & = b rho cos(theta) sin(varphi), \
z & = c rho cos(varphi).
end{align}$$
To translate it parallel to the axes simply add $(x_0,y_0,z_0)$:
$$begin{align}
x & = x_0 + a rho sin(theta) sin(varphi), \
y & = y_0 + b rho cos(theta) sin(varphi), \
z & = z_0 + c rho cos(varphi).
end{align}$$
We have
$$frac{(x-x_0)^2}{a^2} + frac{(y-y_0)^2}{b^2} + frac{(z-z_0)^2}{c^2} = 1.$$
answered Sep 14 '14 at 18:01
Mark FantiniMark Fantini
4,86041936
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add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The parametrization for an ellipsoid is
$$begin{align}
x & = a rho sin(theta) sin(varphi), \
y & = b rho cos(theta) sin(varphi), \
z & = c rho cos(varphi).
end{align}$$
To translate it parallel to the axes simply add $(x_0,y_0,z_0)$:
$$begin{align}
x & = x_0 + a rho sin(theta) sin(varphi), \
y & = y_0 + b rho cos(theta) sin(varphi), \
z & = z_0 + c rho cos(varphi).
end{align}$$
We have
$$frac{(x-x_0)^2}{a^2} + frac{(y-y_0)^2}{b^2} + frac{(z-z_0)^2}{c^2} = 1.$$
answered Sep 14 '14 at 18:01
Mark FantiniMark Fantini
4,86041936
$endgroup$
add a comment |
$begingroup$
The parametrization for an ellipsoid is
$$begin{align}
x & = a rho sin(theta) sin(varphi), \
y & = b rho cos(theta) sin(varphi), \
z & = c rho cos(varphi).
end{align}$$
To translate it parallel to the axes simply add $(x_0,y_0,z_0)$:
$$begin{align}
x & = x_0 + a rho sin(theta) sin(varphi), \
y & = y_0 + b rho cos(theta) sin(varphi), \
z & = z_0 + c rho cos(varphi).
end{align}$$
We have
$$frac{(x-x_0)^2}{a^2} + frac{(y-y_0)^2}{b^2} + frac{(z-z_0)^2}{c^2} = 1.$$
answered Sep 14 '14 at 18:01
Mark FantiniMark Fantini
4,86041936
$endgroup$
add a comment |
$begingroup$
The parametrization for an ellipsoid is
$$begin{align}
x & = a rho sin(theta) sin(varphi), \
y & = b rho cos(theta) sin(varphi), \
z & = c rho cos(varphi).
end{align}$$
To translate it parallel to the axes simply add $(x_0,y_0,z_0)$:
$$begin{align}
x & = x_0 + a rho sin(theta) sin(varphi), \
y & = y_0 + b rho cos(theta) sin(varphi), \
z & = z_0 + c rho cos(varphi).
end{align}$$
We have
$$frac{(x-x_0)^2}{a^2} + frac{(y-y_0)^2}{b^2} + frac{(z-z_0)^2}{c^2} = 1.$$
answered Sep 14 '14 at 18:01
Mark FantiniMark Fantini
4,86041936
$endgroup$
The parametrization for an ellipsoid is
$$begin{align}
x & = a rho sin(theta) sin(varphi), \
y & = b rho cos(theta) sin(varphi), \
z & = c rho cos(varphi).
end{align}$$
To translate it parallel to the axes simply add $(x_0,y_0,z_0)$:
$$begin{align}
x & = x_0 + a rho sin(theta) sin(varphi), \
y & = y_0 + b rho cos(theta) sin(varphi), \
z & = z_0 + c rho cos(varphi).
end{align}$$
We have
$$frac{(x-x_0)^2}{a^2} + frac{(y-y_0)^2}{b^2} + frac{(z-z_0)^2}{c^2} = 1.$$
answered Sep 14 '14 at 18:01
Mark FantiniMark Fantini
4,86041936
answered Sep 14 '14 at 18:01
Mark FantiniMark Fantini
4,86041936
answered Sep 14 '14 at 18:01
Mark FantiniMark Fantini
4,86041936
answered Sep 14 '14 at 18:01
Mark FantiniMark Fantini
4,86041936
4,86041936
add a comment |
add a comment |
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