Mixing
Shortest distance between two points , when you can't cross sphere in the middle
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You are given three coordinates and radius of sphere:
$$
begin{split}
x_1,y_1,z_1 &= text{ start point};\
x_2,y_2,z_2 &= text{ finish point};\
x_3,y_3,z_3 &= text{ sphere center};\
r &= text{ sphere radius};
end{split}
$$
Find shortest distance between start and finish without crossing sphere.
Start and end are not necessary on sphere.
On my observations we need to find two tangent lines which go through start(finish) and arc between tangent line-sphere touching point.
linear-algebra geometry
asked Jan 24 at 18:43
mantas kryzeviciusmantas kryzevicius
162
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add a comment |
$begingroup$
You are given three coordinates and radius of sphere:
$$
begin{split}
x_1,y_1,z_1 &= text{ start point};\
x_2,y_2,z_2 &= text{ finish point};\
x_3,y_3,z_3 &= text{ sphere center};\
r &= text{ sphere radius};
end{split}
$$
Find shortest distance between start and finish without crossing sphere.
Start and end are not necessary on sphere.
On my observations we need to find two tangent lines which go through start(finish) and arc between tangent line-sphere touching point.
linear-algebra geometry
asked Jan 24 at 18:43
mantas kryzeviciusmantas kryzevicius
162
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$begingroup$
@Blue I don't think this is a duplicate; in the linked question, it is assumed that the points are on the surface of the sphere, which is not assumed here.
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– Milo Brandt
Jan 24 at 18:59
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@MiloBrandt: Perhaps you're right; it's a little hard to tell, but I guess the question deserves the benefit of the doubt.
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– Blue
Jan 24 at 19:02
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@Blue: But those points of intersection will never be on the shortest path.
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– TonyK
Jan 24 at 20:23
add a comment |
$begingroup$
You are given three coordinates and radius of sphere:
$$
begin{split}
x_1,y_1,z_1 &= text{ start point};\
x_2,y_2,z_2 &= text{ finish point};\
x_3,y_3,z_3 &= text{ sphere center};\
r &= text{ sphere radius};
end{split}
$$
Find shortest distance between start and finish without crossing sphere.
Start and end are not necessary on sphere.
On my observations we need to find two tangent lines which go through start(finish) and arc between tangent line-sphere touching point.
linear-algebra geometry
asked Jan 24 at 18:43
mantas kryzeviciusmantas kryzevicius
162
$endgroup$
You are given three coordinates and radius of sphere:
$$
begin{split}
x_1,y_1,z_1 &= text{ start point};\
x_2,y_2,z_2 &= text{ finish point};\
x_3,y_3,z_3 &= text{ sphere center};\
r &= text{ sphere radius};
end{split}
$$
Find shortest distance between start and finish without crossing sphere.
Start and end are not necessary on sphere.
On my observations we need to find two tangent lines which go through start(finish) and arc between tangent line-sphere touching point.
linear-algebra geometry
linear-algebra geometry
asked Jan 24 at 18:43
mantas kryzeviciusmantas kryzevicius
162
asked Jan 24 at 18:43
mantas kryzeviciusmantas kryzevicius
162
edited Jan 24 at 19:23
mantas kryzevicius
asked Jan 24 at 18:43
mantas kryzeviciusmantas kryzevicius
162
asked Jan 24 at 18:43
mantas kryzeviciusmantas kryzevicius
162
asked Jan 24 at 18:43
mantas kryzeviciusmantas kryzevicius
162
162
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@Blue I don't think this is a duplicate; in the linked question, it is assumed that the points are on the surface of the sphere, which is not assumed here.
$endgroup$
– Milo Brandt
Jan 24 at 18:59
$begingroup$
@MiloBrandt: Perhaps you're right; it's a little hard to tell, but I guess the question deserves the benefit of the doubt.
$endgroup$
– Blue
Jan 24 at 19:02
$begingroup$
@Blue: But those points of intersection will never be on the shortest path.
$endgroup$
– TonyK
Jan 24 at 20:23
add a comment |
$begingroup$
@Blue I don't think this is a duplicate; in the linked question, it is assumed that the points are on the surface of the sphere, which is not assumed here.
$endgroup$
– Milo Brandt
Jan 24 at 18:59
$begingroup$
@MiloBrandt: Perhaps you're right; it's a little hard to tell, but I guess the question deserves the benefit of the doubt.
$endgroup$
– Blue
Jan 24 at 19:02
$begingroup$
@Blue: But those points of intersection will never be on the shortest path.
$endgroup$
– TonyK
Jan 24 at 20:23
$begingroup$
@Blue I don't think this is a duplicate; in the linked question, it is assumed that the points are on the surface of the sphere, which is not assumed here.
$endgroup$
– Milo Brandt
Jan 24 at 18:59
$begingroup$
@Blue I don't think this is a duplicate; in the linked question, it is assumed that the points are on the surface of the sphere, which is not assumed here.
$endgroup$
– Milo Brandt
Jan 24 at 18:59
$begingroup$
@MiloBrandt: Perhaps you're right; it's a little hard to tell, but I guess the question deserves the benefit of the doubt.
$endgroup$
– Blue
Jan 24 at 19:02
$begingroup$
@MiloBrandt: Perhaps you're right; it's a little hard to tell, but I guess the question deserves the benefit of the doubt.
$endgroup$
– Blue
Jan 24 at 19:02
$begingroup$
@Blue: But those points of intersection will never be on the shortest path.
$endgroup$
– TonyK
Jan 24 at 20:23
$begingroup$
@Blue: But those points of intersection will never be on the shortest path.
$endgroup$
– TonyK
Jan 24 at 20:23
add a comment |
1 Answer
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The shortest path is included in the plane defined by the two points and the sphere center (or any plane through these points in case they are aligned).
Now you have to find the shortes path between two points that avoids a circle.
answered Jan 24 at 20:05
Yves DaoustYves Daoust
130k676229
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add a comment |
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1 Answer
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$begingroup$
The shortest path is included in the plane defined by the two points and the sphere center (or any plane through these points in case they are aligned).
Now you have to find the shortes path between two points that avoids a circle.
answered Jan 24 at 20:05
Yves DaoustYves Daoust
130k676229
$endgroup$
add a comment |
$begingroup$
The shortest path is included in the plane defined by the two points and the sphere center (or any plane through these points in case they are aligned).
Now you have to find the shortes path between two points that avoids a circle.
answered Jan 24 at 20:05
Yves DaoustYves Daoust
130k676229
$endgroup$
add a comment |
$begingroup$
The shortest path is included in the plane defined by the two points and the sphere center (or any plane through these points in case they are aligned).
Now you have to find the shortes path between two points that avoids a circle.
answered Jan 24 at 20:05
Yves DaoustYves Daoust
130k676229
$endgroup$
The shortest path is included in the plane defined by the two points and the sphere center (or any plane through these points in case they are aligned).
Now you have to find the shortes path between two points that avoids a circle.
answered Jan 24 at 20:05
Yves DaoustYves Daoust
130k676229
answered Jan 24 at 20:05
Yves DaoustYves Daoust
130k676229
answered Jan 24 at 20:05
Yves DaoustYves Daoust
130k676229
answered Jan 24 at 20:05
Yves DaoustYves Daoust
130k676229
130k676229
add a comment |
add a comment |
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$begingroup$
@Blue I don't think this is a duplicate; in the linked question, it is assumed that the points are on the surface of the sphere, which is not assumed here.
$endgroup$
– Milo Brandt
Jan 24 at 18:59
$begingroup$
@MiloBrandt: Perhaps you're right; it's a little hard to tell, but I guess the question deserves the benefit of the doubt.
$endgroup$
– Blue
Jan 24 at 19:02
$begingroup$
@Blue: But those points of intersection will never be on the shortest path.
$endgroup$
– TonyK
Jan 24 at 20:23