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Parameterizations of the unit simplex in $mathbbR^3$
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The unit simplex in $mathbbR^3$ is
$$Delta^3 = left(t_1,t_2,t_3)inmathbbR^3mid t_1+t_2+t_3 = 1 mbox and t_i ge 0 mbox for all iright$$
When trying to describe it parametrically, an obvious choice is
$$ (x,y,1-x-y)$$
over an approriate two-dimensional domain. This particular parameterization is not symmetric, as the third coordinate plays a different role than the first two.
Are there any other parameterizations of the unit simplex $Delta^3$ which are more natural? For example, ones in which the distance to the edge can be easily read. If so, I would appreciate some examples. Thank you!
geometry coordinate-systems simplex
asked Jul 24 at 15:01
user1337
16.5k42989
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up vote
3
down vote
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The unit simplex in $mathbbR^3$ is
$$Delta^3 = left(t_1,t_2,t_3)inmathbbR^3mid t_1+t_2+t_3 = 1 mbox and t_i ge 0 mbox for all iright$$
When trying to describe it parametrically, an obvious choice is
$$ (x,y,1-x-y)$$
over an approriate two-dimensional domain. This particular parameterization is not symmetric, as the third coordinate plays a different role than the first two.
Are there any other parameterizations of the unit simplex $Delta^3$ which are more natural? For example, ones in which the distance to the edge can be easily read. If so, I would appreciate some examples. Thank you!
geometry coordinate-systems simplex
asked Jul 24 at 15:01
user1337
16.5k42989
1
You could first transform to barycentric coordinates and then use the parameterization you mention
– dbx
Jul 24 at 15:05
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
The unit simplex in $mathbbR^3$ is
$$Delta^3 = left(t_1,t_2,t_3)inmathbbR^3mid t_1+t_2+t_3 = 1 mbox and t_i ge 0 mbox for all iright$$
When trying to describe it parametrically, an obvious choice is
$$ (x,y,1-x-y)$$
over an approriate two-dimensional domain. This particular parameterization is not symmetric, as the third coordinate plays a different role than the first two.
Are there any other parameterizations of the unit simplex $Delta^3$ which are more natural? For example, ones in which the distance to the edge can be easily read. If so, I would appreciate some examples. Thank you!
geometry coordinate-systems simplex
asked Jul 24 at 15:01
user1337
16.5k42989
The unit simplex in $mathbbR^3$ is
$$Delta^3 = left(t_1,t_2,t_3)inmathbbR^3mid t_1+t_2+t_3 = 1 mbox and t_i ge 0 mbox for all iright$$
When trying to describe it parametrically, an obvious choice is
$$ (x,y,1-x-y)$$
over an approriate two-dimensional domain. This particular parameterization is not symmetric, as the third coordinate plays a different role than the first two.
Are there any other parameterizations of the unit simplex $Delta^3$ which are more natural? For example, ones in which the distance to the edge can be easily read. If so, I would appreciate some examples. Thank you!
geometry coordinate-systems simplex
asked Jul 24 at 15:01
user1337
16.5k42989
asked Jul 24 at 15:01
user1337
16.5k42989
asked Jul 24 at 15:01
user1337
16.5k42989
asked Jul 24 at 15:01
user1337
16.5k42989
16.5k42989
1
You could first transform to barycentric coordinates and then use the parameterization you mention
– dbx
Jul 24 at 15:05
add a comment |Â
1
You could first transform to barycentric coordinates and then use the parameterization you mention
– dbx
Jul 24 at 15:05
1
1
You could first transform to barycentric coordinates and then use the parameterization you mention
– dbx
Jul 24 at 15:05
You could first transform to barycentric coordinates and then use the parameterization you mention
– dbx
Jul 24 at 15:05
add a comment |Â
1 Answer
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Take $$t_1=sin^2thetacos^2phi\t_2=sin^2thetasin^2phi\t_3=cos^2theta$$
answered Jul 24 at 15:54
Mostafa Ayaz
8,5373630
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Take $$t_1=sin^2thetacos^2phi\t_2=sin^2thetasin^2phi\t_3=cos^2theta$$
answered Jul 24 at 15:54
Mostafa Ayaz
8,5373630
add a comment |Â
up vote
1
down vote
Take $$t_1=sin^2thetacos^2phi\t_2=sin^2thetasin^2phi\t_3=cos^2theta$$
answered Jul 24 at 15:54
Mostafa Ayaz
8,5373630
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Take $$t_1=sin^2thetacos^2phi\t_2=sin^2thetasin^2phi\t_3=cos^2theta$$
answered Jul 24 at 15:54
Mostafa Ayaz
8,5373630
Take $$t_1=sin^2thetacos^2phi\t_2=sin^2thetasin^2phi\t_3=cos^2theta$$
answered Jul 24 at 15:54
Mostafa Ayaz
8,5373630
answered Jul 24 at 15:54
Mostafa Ayaz
8,5373630
answered Jul 24 at 15:54
Mostafa Ayaz
8,5373630
answered Jul 24 at 15:54
Mostafa Ayaz
8,5373630
8,5373630
add a comment |Â
add a comment |Â
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1
You could first transform to barycentric coordinates and then use the parameterization you mention
– dbx
Jul 24 at 15:05