Find the line that is closest to 4 skew lines
$begingroup$
If I have 4 skew lines in $mathbb{R}^3$, how can I find the line $L_c$, that is closest to all of them?
I know that with 3 skew lines, there is always a line that intersects all of them, in fact infinite many:
- Construct a plane with $L_1$ and and arbitrary point on $L_2$,
- intersect the plane with $L_3$ to get a second point,
- the line through both points intersects $L_1$,$L_2$ and $L_3$
With 4 skew lines, there is at most two intersecting lines (I think, though I have not found a way to construct that yet). Assuming there is none, how do I find the line that is closest to them all?
This is related to a previous question by me: I am trying to find the axis for a surface of rotation, given 4 random points on the surface and some measurement error.
geometry euclidean-geometry 3d geometric-construction
asked Dec 12 '18 at 10:25
HugoRuneHugoRune
1338
$endgroup$
add a comment |
$begingroup$
If I have 4 skew lines in $mathbb{R}^3$, how can I find the line $L_c$, that is closest to all of them?
I know that with 3 skew lines, there is always a line that intersects all of them, in fact infinite many:
- Construct a plane with $L_1$ and and arbitrary point on $L_2$,
- intersect the plane with $L_3$ to get a second point,
- the line through both points intersects $L_1$,$L_2$ and $L_3$
With 4 skew lines, there is at most two intersecting lines (I think, though I have not found a way to construct that yet). Assuming there is none, how do I find the line that is closest to them all?
This is related to a previous question by me: I am trying to find the axis for a surface of rotation, given 4 random points on the surface and some measurement error.
geometry euclidean-geometry 3d geometric-construction
asked Dec 12 '18 at 10:25
HugoRuneHugoRune
1338
$endgroup$
add a comment |
$begingroup$
If I have 4 skew lines in $mathbb{R}^3$, how can I find the line $L_c$, that is closest to all of them?
I know that with 3 skew lines, there is always a line that intersects all of them, in fact infinite many:
- Construct a plane with $L_1$ and and arbitrary point on $L_2$,
- intersect the plane with $L_3$ to get a second point,
- the line through both points intersects $L_1$,$L_2$ and $L_3$
With 4 skew lines, there is at most two intersecting lines (I think, though I have not found a way to construct that yet). Assuming there is none, how do I find the line that is closest to them all?
This is related to a previous question by me: I am trying to find the axis for a surface of rotation, given 4 random points on the surface and some measurement error.
geometry euclidean-geometry 3d geometric-construction
asked Dec 12 '18 at 10:25
HugoRuneHugoRune
1338
$endgroup$
If I have 4 skew lines in $mathbb{R}^3$, how can I find the line $L_c$, that is closest to all of them?
I know that with 3 skew lines, there is always a line that intersects all of them, in fact infinite many:
- Construct a plane with $L_1$ and and arbitrary point on $L_2$,
- intersect the plane with $L_3$ to get a second point,
- the line through both points intersects $L_1$,$L_2$ and $L_3$
With 4 skew lines, there is at most two intersecting lines (I think, though I have not found a way to construct that yet). Assuming there is none, how do I find the line that is closest to them all?
This is related to a previous question by me: I am trying to find the axis for a surface of rotation, given 4 random points on the surface and some measurement error.
geometry euclidean-geometry 3d geometric-construction
geometry euclidean-geometry 3d geometric-construction
asked Dec 12 '18 at 10:25
HugoRuneHugoRune
1338
asked Dec 12 '18 at 10:25
HugoRuneHugoRune
1338
edited Dec 12 '18 at 10:31
HugoRune
asked Dec 12 '18 at 10:25
HugoRuneHugoRune
1338
asked Dec 12 '18 at 10:25
HugoRuneHugoRune
1338
asked Dec 12 '18 at 10:25
HugoRuneHugoRune
1338
1338
add a comment |
add a comment |
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