Mixing
Group Action of $SO(3)$ on unit sphere
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We know that $SO(3)$ acts transitively on $S^2$. That is, given any two points $p_1,p_2in S^2$, one can find an element in $SO(3)$ mapping $p_1$ to $p_2$. My question is, given two sets of points $(a_1,a_2)$ and $(b_1,b_2)$ on $S^2$ such that the induced distance on sphere are the same, can one necessarily find an element in $SO(3)$ mapping one set to the other?
general-topology differential-topology geometric-group-theory
asked Jul 15 at 2:59
Chun Gan
30718
add a comment |Â
up vote
1
down vote
favorite
We know that $SO(3)$ acts transitively on $S^2$. That is, given any two points $p_1,p_2in S^2$, one can find an element in $SO(3)$ mapping $p_1$ to $p_2$. My question is, given two sets of points $(a_1,a_2)$ and $(b_1,b_2)$ on $S^2$ such that the induced distance on sphere are the same, can one necessarily find an element in $SO(3)$ mapping one set to the other?
general-topology differential-topology geometric-group-theory
asked Jul 15 at 2:59
Chun Gan
30718
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
We know that $SO(3)$ acts transitively on $S^2$. That is, given any two points $p_1,p_2in S^2$, one can find an element in $SO(3)$ mapping $p_1$ to $p_2$. My question is, given two sets of points $(a_1,a_2)$ and $(b_1,b_2)$ on $S^2$ such that the induced distance on sphere are the same, can one necessarily find an element in $SO(3)$ mapping one set to the other?
general-topology differential-topology geometric-group-theory
asked Jul 15 at 2:59
Chun Gan
30718
We know that $SO(3)$ acts transitively on $S^2$. That is, given any two points $p_1,p_2in S^2$, one can find an element in $SO(3)$ mapping $p_1$ to $p_2$. My question is, given two sets of points $(a_1,a_2)$ and $(b_1,b_2)$ on $S^2$ such that the induced distance on sphere are the same, can one necessarily find an element in $SO(3)$ mapping one set to the other?
general-topology differential-topology geometric-group-theory
asked Jul 15 at 2:59
Chun Gan
30718
asked Jul 15 at 2:59
Chun Gan
30718
asked Jul 15 at 2:59
Chun Gan
30718
asked Jul 15 at 2:59
Chun Gan
30718
30718
add a comment |Â
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
2
down vote
accepted
Yes; this is just elementary geometry. We may first choose a rotation sending $a_1$ to $b_1$, and so may assume $a_1=b_1$. Then we can choose a rotation about the diameter of the sphere through $a_1$ that maps $a_2$ to $b_2$. (Explicitly, rotate by the angle between the geodesic from $a_1$ to $a_2$ and the geodesic from $a_1$ to $b_2$.)
answered Jul 15 at 3:08
Eric Wofsey
163k12189300
I'm wondering what is the largest number of points that we can perform this?
– Chun Gan
Jul 15 at 3:29
I think my actual question is, when can we uniquely determine an element in $SO(3)$ by specifying the number of points that we match. More generally, how about $SO(n)$, $SU(n)$? Thanks a lot!
– Chun Gan
Jul 15 at 3:33
2
An element of $SO(n)$ (or $SU(n)$) is uniquely determined by where it sends any collection of $n-1$ linearly independent vectors, since if you take an $n$th vector orthogonal to those $n-1$ its image is uniquely determined by needing to remain orthogonal to them and produce a determinant of $1$.
– Eric Wofsey
Jul 15 at 3:59
add a comment |Â
up vote
1
down vote
Yes. Take $a_3 = a_1 times a_2$ and $b_3 = b_1 times b_2$. Define the map $T: mathbb R^3 to mathbb R^3$ by the formula $T(a_i) = b_i$. This map is a oriented isometry of $mathbb R^3$ and therefore belongs to $SO(3)$.
answered Jul 15 at 3:09
Hugocito
1,6451019
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Yes; this is just elementary geometry. We may first choose a rotation sending $a_1$ to $b_1$, and so may assume $a_1=b_1$. Then we can choose a rotation about the diameter of the sphere through $a_1$ that maps $a_2$ to $b_2$. (Explicitly, rotate by the angle between the geodesic from $a_1$ to $a_2$ and the geodesic from $a_1$ to $b_2$.)
answered Jul 15 at 3:08
Eric Wofsey
163k12189300
I'm wondering what is the largest number of points that we can perform this?
– Chun Gan
Jul 15 at 3:29
I think my actual question is, when can we uniquely determine an element in $SO(3)$ by specifying the number of points that we match. More generally, how about $SO(n)$, $SU(n)$? Thanks a lot!
– Chun Gan
Jul 15 at 3:33
2
An element of $SO(n)$ (or $SU(n)$) is uniquely determined by where it sends any collection of $n-1$ linearly independent vectors, since if you take an $n$th vector orthogonal to those $n-1$ its image is uniquely determined by needing to remain orthogonal to them and produce a determinant of $1$.
– Eric Wofsey
Jul 15 at 3:59
add a comment |Â
up vote
2
down vote
accepted
Yes; this is just elementary geometry. We may first choose a rotation sending $a_1$ to $b_1$, and so may assume $a_1=b_1$. Then we can choose a rotation about the diameter of the sphere through $a_1$ that maps $a_2$ to $b_2$. (Explicitly, rotate by the angle between the geodesic from $a_1$ to $a_2$ and the geodesic from $a_1$ to $b_2$.)
answered Jul 15 at 3:08
Eric Wofsey
163k12189300
I'm wondering what is the largest number of points that we can perform this?
– Chun Gan
Jul 15 at 3:29
I think my actual question is, when can we uniquely determine an element in $SO(3)$ by specifying the number of points that we match. More generally, how about $SO(n)$, $SU(n)$? Thanks a lot!
– Chun Gan
Jul 15 at 3:33
2
An element of $SO(n)$ (or $SU(n)$) is uniquely determined by where it sends any collection of $n-1$ linearly independent vectors, since if you take an $n$th vector orthogonal to those $n-1$ its image is uniquely determined by needing to remain orthogonal to them and produce a determinant of $1$.
– Eric Wofsey
Jul 15 at 3:59
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Yes; this is just elementary geometry. We may first choose a rotation sending $a_1$ to $b_1$, and so may assume $a_1=b_1$. Then we can choose a rotation about the diameter of the sphere through $a_1$ that maps $a_2$ to $b_2$. (Explicitly, rotate by the angle between the geodesic from $a_1$ to $a_2$ and the geodesic from $a_1$ to $b_2$.)
answered Jul 15 at 3:08
Eric Wofsey
163k12189300
Yes; this is just elementary geometry. We may first choose a rotation sending $a_1$ to $b_1$, and so may assume $a_1=b_1$. Then we can choose a rotation about the diameter of the sphere through $a_1$ that maps $a_2$ to $b_2$. (Explicitly, rotate by the angle between the geodesic from $a_1$ to $a_2$ and the geodesic from $a_1$ to $b_2$.)
answered Jul 15 at 3:08
Eric Wofsey
163k12189300
answered Jul 15 at 3:08
Eric Wofsey
163k12189300
answered Jul 15 at 3:08
Eric Wofsey
163k12189300
answered Jul 15 at 3:08
Eric Wofsey
163k12189300
163k12189300
I'm wondering what is the largest number of points that we can perform this?
– Chun Gan
Jul 15 at 3:29
I think my actual question is, when can we uniquely determine an element in $SO(3)$ by specifying the number of points that we match. More generally, how about $SO(n)$, $SU(n)$? Thanks a lot!
– Chun Gan
Jul 15 at 3:33
2
An element of $SO(n)$ (or $SU(n)$) is uniquely determined by where it sends any collection of $n-1$ linearly independent vectors, since if you take an $n$th vector orthogonal to those $n-1$ its image is uniquely determined by needing to remain orthogonal to them and produce a determinant of $1$.
– Eric Wofsey
Jul 15 at 3:59
add a comment |Â
I'm wondering what is the largest number of points that we can perform this?
– Chun Gan
Jul 15 at 3:29
I think my actual question is, when can we uniquely determine an element in $SO(3)$ by specifying the number of points that we match. More generally, how about $SO(n)$, $SU(n)$? Thanks a lot!
– Chun Gan
Jul 15 at 3:33
2
An element of $SO(n)$ (or $SU(n)$) is uniquely determined by where it sends any collection of $n-1$ linearly independent vectors, since if you take an $n$th vector orthogonal to those $n-1$ its image is uniquely determined by needing to remain orthogonal to them and produce a determinant of $1$.
– Eric Wofsey
Jul 15 at 3:59
I'm wondering what is the largest number of points that we can perform this?
– Chun Gan
Jul 15 at 3:29
I'm wondering what is the largest number of points that we can perform this?
– Chun Gan
Jul 15 at 3:29
I think my actual question is, when can we uniquely determine an element in $SO(3)$ by specifying the number of points that we match. More generally, how about $SO(n)$, $SU(n)$? Thanks a lot!
– Chun Gan
Jul 15 at 3:33
I think my actual question is, when can we uniquely determine an element in $SO(3)$ by specifying the number of points that we match. More generally, how about $SO(n)$, $SU(n)$? Thanks a lot!
– Chun Gan
Jul 15 at 3:33
2
2
An element of $SO(n)$ (or $SU(n)$) is uniquely determined by where it sends any collection of $n-1$ linearly independent vectors, since if you take an $n$th vector orthogonal to those $n-1$ its image is uniquely determined by needing to remain orthogonal to them and produce a determinant of $1$.
– Eric Wofsey
Jul 15 at 3:59
An element of $SO(n)$ (or $SU(n)$) is uniquely determined by where it sends any collection of $n-1$ linearly independent vectors, since if you take an $n$th vector orthogonal to those $n-1$ its image is uniquely determined by needing to remain orthogonal to them and produce a determinant of $1$.
– Eric Wofsey
Jul 15 at 3:59
add a comment |Â
up vote
1
down vote
Yes. Take $a_3 = a_1 times a_2$ and $b_3 = b_1 times b_2$. Define the map $T: mathbb R^3 to mathbb R^3$ by the formula $T(a_i) = b_i$. This map is a oriented isometry of $mathbb R^3$ and therefore belongs to $SO(3)$.
answered Jul 15 at 3:09
Hugocito
1,6451019
add a comment |Â
up vote
1
down vote
Yes. Take $a_3 = a_1 times a_2$ and $b_3 = b_1 times b_2$. Define the map $T: mathbb R^3 to mathbb R^3$ by the formula $T(a_i) = b_i$. This map is a oriented isometry of $mathbb R^3$ and therefore belongs to $SO(3)$.
answered Jul 15 at 3:09
Hugocito
1,6451019
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Yes. Take $a_3 = a_1 times a_2$ and $b_3 = b_1 times b_2$. Define the map $T: mathbb R^3 to mathbb R^3$ by the formula $T(a_i) = b_i$. This map is a oriented isometry of $mathbb R^3$ and therefore belongs to $SO(3)$.
answered Jul 15 at 3:09
Hugocito
1,6451019
Yes. Take $a_3 = a_1 times a_2$ and $b_3 = b_1 times b_2$. Define the map $T: mathbb R^3 to mathbb R^3$ by the formula $T(a_i) = b_i$. This map is a oriented isometry of $mathbb R^3$ and therefore belongs to $SO(3)$.
answered Jul 15 at 3:09
Hugocito
1,6451019
answered Jul 15 at 3:09
Hugocito
1,6451019
answered Jul 15 at 3:09
Hugocito
1,6451019
answered Jul 15 at 3:09
Hugocito
1,6451019
1,6451019
add a comment |Â
add a comment |Â
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