How to calculate rectangle tangent to sphere
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Given a rectangle $ABCD,$ how do I calculate points $A, B, C, ; text{and}; D;$ if I place the rectangle tangent to a sphere, centered at a given Latitude and Longitude, and given a "Rotation" which would be degrees clockwise from "North" on the sphere?
You can assume that the sphere is a Unit Sphere centered at the Origin.
I'm trying include this in a software application, so I would appreciate solutions suitable for programming, i.e. algebra or trigonometry rather than calculus.
Thank you!
geometry computational-geometry
edited Dec 11 '18 at 13:23
user376343
3,7883828
asked Dec 11 '18 at 12:57
vocalionechovocalionecho
91
$endgroup$
add a comment |
$begingroup$
Given a rectangle $ABCD,$ how do I calculate points $A, B, C, ; text{and}; D;$ if I place the rectangle tangent to a sphere, centered at a given Latitude and Longitude, and given a "Rotation" which would be degrees clockwise from "North" on the sphere?
You can assume that the sphere is a Unit Sphere centered at the Origin.
I'm trying include this in a software application, so I would appreciate solutions suitable for programming, i.e. algebra or trigonometry rather than calculus.
Thank you!
geometry computational-geometry
edited Dec 11 '18 at 13:23
user376343
3,7883828
asked Dec 11 '18 at 12:57
vocalionechovocalionecho
91
$endgroup$
add a comment |
$begingroup$
Given a rectangle $ABCD,$ how do I calculate points $A, B, C, ; text{and}; D;$ if I place the rectangle tangent to a sphere, centered at a given Latitude and Longitude, and given a "Rotation" which would be degrees clockwise from "North" on the sphere?
You can assume that the sphere is a Unit Sphere centered at the Origin.
I'm trying include this in a software application, so I would appreciate solutions suitable for programming, i.e. algebra or trigonometry rather than calculus.
Thank you!
geometry computational-geometry
edited Dec 11 '18 at 13:23
user376343
3,7883828
asked Dec 11 '18 at 12:57
vocalionechovocalionecho
91
$endgroup$
Given a rectangle $ABCD,$ how do I calculate points $A, B, C, ; text{and}; D;$ if I place the rectangle tangent to a sphere, centered at a given Latitude and Longitude, and given a "Rotation" which would be degrees clockwise from "North" on the sphere?
You can assume that the sphere is a Unit Sphere centered at the Origin.
I'm trying include this in a software application, so I would appreciate solutions suitable for programming, i.e. algebra or trigonometry rather than calculus.
Thank you!
geometry computational-geometry
geometry computational-geometry
edited Dec 11 '18 at 13:23
user376343
3,7883828
asked Dec 11 '18 at 12:57
vocalionechovocalionecho
91
edited Dec 11 '18 at 13:23
user376343
3,7883828
asked Dec 11 '18 at 12:57
vocalionechovocalionecho
91
edited Dec 11 '18 at 13:23
user376343
3,7883828
edited Dec 11 '18 at 13:23
user376343
3,7883828
edited Dec 11 '18 at 13:23
user376343
3,7883828
3,7883828
asked Dec 11 '18 at 12:57
vocalionechovocalionecho
91
asked Dec 11 '18 at 12:57
vocalionechovocalionecho
91
asked Dec 11 '18 at 12:57
vocalionechovocalionecho
91
91
add a comment |
add a comment |
1 Answer
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oldest
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$begingroup$
If you are aware of matrices, you can construct your rotation matrix from simple rotation matrices. Let's have rectangle parallel to plane $yz$ (with $z$ being the axis of spherical coordinates), sides parallel to $y$ and $z$ and center on the sphere (you can derive the coordinates of the vertices yourself).
We first rotate rectangle around its central point (axis $x$) by angle $varphi$:
$$
Phi=begin{pmatrix}
0 & 0 & 0 \
0 & cosvarphi& -sinvarphi\
0 & sinvarphi& phantom{-}cosvarphi
end{pmatrix}.
$$
Then we rotate the rectangle by given latitude angle (around axis $y$):
$$
Theta=begin{pmatrix}
phantom{-}costheta& 0& sintheta\
0 & 0 & 0 \
-sintheta& 0& cosvarphi
end{pmatrix}.
$$
Finally, we will rotate around axis $z$ by longitude angle $lambda$:
$$
Lambda=begin{pmatrix}
coslambda& -sinlambda&0\
sinlambda& phantom{-}coslambda & 0\
0 & 0 & 0 \
end{pmatrix}.
$$
The total rotation is the product of the matrices. Since we apply matrices to vector from left to right, the order is like this:
$$
R = Lambda Theta Phi
$$
From programming point of view, you can either derive the final equation for the corners on paper and then write it in code. Or preferably, add multiplication by matrix as a function your code and apply matrix multiplication on demand. The latter case will give you more flexibility and is probably less prone to bugs.
answered Dec 11 '18 at 14:10
Vasily MitchVasily Mitch
2,3141311
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add a comment |
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1 Answer
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1 Answer
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votes
$begingroup$
If you are aware of matrices, you can construct your rotation matrix from simple rotation matrices. Let's have rectangle parallel to plane $yz$ (with $z$ being the axis of spherical coordinates), sides parallel to $y$ and $z$ and center on the sphere (you can derive the coordinates of the vertices yourself).
We first rotate rectangle around its central point (axis $x$) by angle $varphi$:
$$
Phi=begin{pmatrix}
0 & 0 & 0 \
0 & cosvarphi& -sinvarphi\
0 & sinvarphi& phantom{-}cosvarphi
end{pmatrix}.
$$
Then we rotate the rectangle by given latitude angle (around axis $y$):
$$
Theta=begin{pmatrix}
phantom{-}costheta& 0& sintheta\
0 & 0 & 0 \
-sintheta& 0& cosvarphi
end{pmatrix}.
$$
Finally, we will rotate around axis $z$ by longitude angle $lambda$:
$$
Lambda=begin{pmatrix}
coslambda& -sinlambda&0\
sinlambda& phantom{-}coslambda & 0\
0 & 0 & 0 \
end{pmatrix}.
$$
The total rotation is the product of the matrices. Since we apply matrices to vector from left to right, the order is like this:
$$
R = Lambda Theta Phi
$$
From programming point of view, you can either derive the final equation for the corners on paper and then write it in code. Or preferably, add multiplication by matrix as a function your code and apply matrix multiplication on demand. The latter case will give you more flexibility and is probably less prone to bugs.
answered Dec 11 '18 at 14:10
Vasily MitchVasily Mitch
2,3141311
$endgroup$
add a comment |
$begingroup$
If you are aware of matrices, you can construct your rotation matrix from simple rotation matrices. Let's have rectangle parallel to plane $yz$ (with $z$ being the axis of spherical coordinates), sides parallel to $y$ and $z$ and center on the sphere (you can derive the coordinates of the vertices yourself).
We first rotate rectangle around its central point (axis $x$) by angle $varphi$:
$$
Phi=begin{pmatrix}
0 & 0 & 0 \
0 & cosvarphi& -sinvarphi\
0 & sinvarphi& phantom{-}cosvarphi
end{pmatrix}.
$$
Then we rotate the rectangle by given latitude angle (around axis $y$):
$$
Theta=begin{pmatrix}
phantom{-}costheta& 0& sintheta\
0 & 0 & 0 \
-sintheta& 0& cosvarphi
end{pmatrix}.
$$
Finally, we will rotate around axis $z$ by longitude angle $lambda$:
$$
Lambda=begin{pmatrix}
coslambda& -sinlambda&0\
sinlambda& phantom{-}coslambda & 0\
0 & 0 & 0 \
end{pmatrix}.
$$
The total rotation is the product of the matrices. Since we apply matrices to vector from left to right, the order is like this:
$$
R = Lambda Theta Phi
$$
From programming point of view, you can either derive the final equation for the corners on paper and then write it in code. Or preferably, add multiplication by matrix as a function your code and apply matrix multiplication on demand. The latter case will give you more flexibility and is probably less prone to bugs.
answered Dec 11 '18 at 14:10
Vasily MitchVasily Mitch
2,3141311
$endgroup$
add a comment |
$begingroup$
If you are aware of matrices, you can construct your rotation matrix from simple rotation matrices. Let's have rectangle parallel to plane $yz$ (with $z$ being the axis of spherical coordinates), sides parallel to $y$ and $z$ and center on the sphere (you can derive the coordinates of the vertices yourself).
We first rotate rectangle around its central point (axis $x$) by angle $varphi$:
$$
Phi=begin{pmatrix}
0 & 0 & 0 \
0 & cosvarphi& -sinvarphi\
0 & sinvarphi& phantom{-}cosvarphi
end{pmatrix}.
$$
Then we rotate the rectangle by given latitude angle (around axis $y$):
$$
Theta=begin{pmatrix}
phantom{-}costheta& 0& sintheta\
0 & 0 & 0 \
-sintheta& 0& cosvarphi
end{pmatrix}.
$$
Finally, we will rotate around axis $z$ by longitude angle $lambda$:
$$
Lambda=begin{pmatrix}
coslambda& -sinlambda&0\
sinlambda& phantom{-}coslambda & 0\
0 & 0 & 0 \
end{pmatrix}.
$$
The total rotation is the product of the matrices. Since we apply matrices to vector from left to right, the order is like this:
$$
R = Lambda Theta Phi
$$
From programming point of view, you can either derive the final equation for the corners on paper and then write it in code. Or preferably, add multiplication by matrix as a function your code and apply matrix multiplication on demand. The latter case will give you more flexibility and is probably less prone to bugs.
answered Dec 11 '18 at 14:10
Vasily MitchVasily Mitch
2,3141311
$endgroup$
If you are aware of matrices, you can construct your rotation matrix from simple rotation matrices. Let's have rectangle parallel to plane $yz$ (with $z$ being the axis of spherical coordinates), sides parallel to $y$ and $z$ and center on the sphere (you can derive the coordinates of the vertices yourself).
We first rotate rectangle around its central point (axis $x$) by angle $varphi$:
$$
Phi=begin{pmatrix}
0 & 0 & 0 \
0 & cosvarphi& -sinvarphi\
0 & sinvarphi& phantom{-}cosvarphi
end{pmatrix}.
$$
Then we rotate the rectangle by given latitude angle (around axis $y$):
$$
Theta=begin{pmatrix}
phantom{-}costheta& 0& sintheta\
0 & 0 & 0 \
-sintheta& 0& cosvarphi
end{pmatrix}.
$$
Finally, we will rotate around axis $z$ by longitude angle $lambda$:
$$
Lambda=begin{pmatrix}
coslambda& -sinlambda&0\
sinlambda& phantom{-}coslambda & 0\
0 & 0 & 0 \
end{pmatrix}.
$$
The total rotation is the product of the matrices. Since we apply matrices to vector from left to right, the order is like this:
$$
R = Lambda Theta Phi
$$
From programming point of view, you can either derive the final equation for the corners on paper and then write it in code. Or preferably, add multiplication by matrix as a function your code and apply matrix multiplication on demand. The latter case will give you more flexibility and is probably less prone to bugs.
answered Dec 11 '18 at 14:10
Vasily MitchVasily Mitch
2,3141311
answered Dec 11 '18 at 14:10
Vasily MitchVasily Mitch
2,3141311
answered Dec 11 '18 at 14:10
Vasily MitchVasily Mitch
2,3141311
answered Dec 11 '18 at 14:10
Vasily MitchVasily Mitch
2,3141311
2,3141311
add a comment |
add a comment |
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