Solid angle subtended by an ellipse
$begingroup$
This question introduces a formula for the evaluation of the solid angle corresponding to an ellipsoid. However, it is an approximation. Instead, this paper represents a general surface formula for conical surfaces as follows
$$Omega = 2pi - oint dl sqrt{vec{u}^2 - (vec{s} cdot vec{u})^{2}},$$
where $s$ is a parametric curve, whose projected curve can be parameterized by the angle (being a linear distance on the surface of
the unit sphere) $l$, so that the curve on the sphere is defined as $vec{s}(l)$, and
$$vec{u}:= frac{d^{2}}{dl^{2}}vec{s}.$$
The paper computed the solid angle corresponding to an sphere, and I need to use the formula for the case of an ellipse as below. However, I'm not clear about the $vec{u}$ in this case. In particular, the parameterization of an ellipsoid is
$$begin{align}
x & = a rho sin(theta) sin(varphi), \
y & = b rho cos(theta) sin(varphi), \
z & = c rho cos(varphi).
end{align}$$
So, one reads
$$vec{u} = frac{d^{2}vec{s}(varphi)}{dl^{2}} = frac{d^{2}vec{s}(varphi)}{dvarphi^{2}}(frac{dvarphi}{dl})^{2}$$
In the sphere case, we have $dl = sin{theta}dvarphi$. But what is the evaluation of $frac{dvarphi}{dl}$ in the ellipsoid case?
differential-geometry solid-angle
asked Jan 23 at 19:53
PintonPinton
133
$endgroup$
add a comment |
$begingroup$
This question introduces a formula for the evaluation of the solid angle corresponding to an ellipsoid. However, it is an approximation. Instead, this paper represents a general surface formula for conical surfaces as follows
$$Omega = 2pi - oint dl sqrt{vec{u}^2 - (vec{s} cdot vec{u})^{2}},$$
where $s$ is a parametric curve, whose projected curve can be parameterized by the angle (being a linear distance on the surface of
the unit sphere) $l$, so that the curve on the sphere is defined as $vec{s}(l)$, and
$$vec{u}:= frac{d^{2}}{dl^{2}}vec{s}.$$
The paper computed the solid angle corresponding to an sphere, and I need to use the formula for the case of an ellipse as below. However, I'm not clear about the $vec{u}$ in this case. In particular, the parameterization of an ellipsoid is
$$begin{align}
x & = a rho sin(theta) sin(varphi), \
y & = b rho cos(theta) sin(varphi), \
z & = c rho cos(varphi).
end{align}$$
So, one reads
$$vec{u} = frac{d^{2}vec{s}(varphi)}{dl^{2}} = frac{d^{2}vec{s}(varphi)}{dvarphi^{2}}(frac{dvarphi}{dl})^{2}$$
In the sphere case, we have $dl = sin{theta}dvarphi$. But what is the evaluation of $frac{dvarphi}{dl}$ in the ellipsoid case?
differential-geometry solid-angle
asked Jan 23 at 19:53
PintonPinton
133
$endgroup$
add a comment |
$begingroup$
This question introduces a formula for the evaluation of the solid angle corresponding to an ellipsoid. However, it is an approximation. Instead, this paper represents a general surface formula for conical surfaces as follows
$$Omega = 2pi - oint dl sqrt{vec{u}^2 - (vec{s} cdot vec{u})^{2}},$$
where $s$ is a parametric curve, whose projected curve can be parameterized by the angle (being a linear distance on the surface of
the unit sphere) $l$, so that the curve on the sphere is defined as $vec{s}(l)$, and
$$vec{u}:= frac{d^{2}}{dl^{2}}vec{s}.$$
The paper computed the solid angle corresponding to an sphere, and I need to use the formula for the case of an ellipse as below. However, I'm not clear about the $vec{u}$ in this case. In particular, the parameterization of an ellipsoid is
$$begin{align}
x & = a rho sin(theta) sin(varphi), \
y & = b rho cos(theta) sin(varphi), \
z & = c rho cos(varphi).
end{align}$$
So, one reads
$$vec{u} = frac{d^{2}vec{s}(varphi)}{dl^{2}} = frac{d^{2}vec{s}(varphi)}{dvarphi^{2}}(frac{dvarphi}{dl})^{2}$$
In the sphere case, we have $dl = sin{theta}dvarphi$. But what is the evaluation of $frac{dvarphi}{dl}$ in the ellipsoid case?
differential-geometry solid-angle
asked Jan 23 at 19:53
PintonPinton
133
$endgroup$
This question introduces a formula for the evaluation of the solid angle corresponding to an ellipsoid. However, it is an approximation. Instead, this paper represents a general surface formula for conical surfaces as follows
$$Omega = 2pi - oint dl sqrt{vec{u}^2 - (vec{s} cdot vec{u})^{2}},$$
where $s$ is a parametric curve, whose projected curve can be parameterized by the angle (being a linear distance on the surface of
the unit sphere) $l$, so that the curve on the sphere is defined as $vec{s}(l)$, and
$$vec{u}:= frac{d^{2}}{dl^{2}}vec{s}.$$
The paper computed the solid angle corresponding to an sphere, and I need to use the formula for the case of an ellipse as below. However, I'm not clear about the $vec{u}$ in this case. In particular, the parameterization of an ellipsoid is
$$begin{align}
x & = a rho sin(theta) sin(varphi), \
y & = b rho cos(theta) sin(varphi), \
z & = c rho cos(varphi).
end{align}$$
So, one reads
$$vec{u} = frac{d^{2}vec{s}(varphi)}{dl^{2}} = frac{d^{2}vec{s}(varphi)}{dvarphi^{2}}(frac{dvarphi}{dl})^{2}$$
In the sphere case, we have $dl = sin{theta}dvarphi$. But what is the evaluation of $frac{dvarphi}{dl}$ in the ellipsoid case?
differential-geometry solid-angle
differential-geometry solid-angle
asked Jan 23 at 19:53
PintonPinton
133
asked Jan 23 at 19:53
PintonPinton
133
asked Jan 23 at 19:53
PintonPinton
133
asked Jan 23 at 19:53
PintonPinton
133
asked Jan 23 at 19:53
PintonPinton
133
133
add a comment |
add a comment |
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