Mixing
Delaunay triangulation in $mathbb R^d$: Empty sphere property works for all $k$-faces?
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This is (it seems to me) a well-known fact, but I am struggling to find a reference.
Let $X={x_1,dots, x_n}subset mathbb R^d$ be a set of points. Then the following is true:
Subset $Fsubset X$, with $mathrm{card}F=k+1$, forms a $k$-face of the Delaunay triangulation of $X$ iff there exists an empty $d$-dimensional sphere on which all points of $F$ lie.
I am interested specifically in the case $d=3,k=1$, i.e. an edge of Delaunay tetrahedrization. If you only provide a reference for this case, I will accept it as an answer.
I only found some proofs of the case $d=2,k=1$ and then a number of people using this fact without a source.
Is there a reference for this fact? Or is the proof in fact so elementary that there simply is no reference?
reference-request computational-geometry triangulation voronoi-diagram
asked Jan 3 at 14:07
Dahn JahnDahn Jahn
2,35811837
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add a comment |
$begingroup$
This is (it seems to me) a well-known fact, but I am struggling to find a reference.
Let $X={x_1,dots, x_n}subset mathbb R^d$ be a set of points. Then the following is true:
Subset $Fsubset X$, with $mathrm{card}F=k+1$, forms a $k$-face of the Delaunay triangulation of $X$ iff there exists an empty $d$-dimensional sphere on which all points of $F$ lie.
I am interested specifically in the case $d=3,k=1$, i.e. an edge of Delaunay tetrahedrization. If you only provide a reference for this case, I will accept it as an answer.
I only found some proofs of the case $d=2,k=1$ and then a number of people using this fact without a source.
Is there a reference for this fact? Or is the proof in fact so elementary that there simply is no reference?
reference-request computational-geometry triangulation voronoi-diagram
asked Jan 3 at 14:07
Dahn JahnDahn Jahn
2,35811837
$endgroup$
$begingroup$
Can you check the book Triangulations - Structures for Algorithms and Applications by Jesús A. De Loera, Jörg Rambau, and Francisco Santos? I suppose they discuss this.
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– toric_actions
Jan 3 at 19:19
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@toric_actions I can, but the book is not specifically about Delaunay triangulations and only mentions the empty spheres on one page. I'll try looking (since Delaunay is a special case of regular triangulations, which are also in the book). Do you perhaps have a more concrete pointer?
$endgroup$
– Dahn Jahn
Jan 3 at 19:51
add a comment |
$begingroup$
This is (it seems to me) a well-known fact, but I am struggling to find a reference.
Let $X={x_1,dots, x_n}subset mathbb R^d$ be a set of points. Then the following is true:
Subset $Fsubset X$, with $mathrm{card}F=k+1$, forms a $k$-face of the Delaunay triangulation of $X$ iff there exists an empty $d$-dimensional sphere on which all points of $F$ lie.
I am interested specifically in the case $d=3,k=1$, i.e. an edge of Delaunay tetrahedrization. If you only provide a reference for this case, I will accept it as an answer.
I only found some proofs of the case $d=2,k=1$ and then a number of people using this fact without a source.
Is there a reference for this fact? Or is the proof in fact so elementary that there simply is no reference?
reference-request computational-geometry triangulation voronoi-diagram
asked Jan 3 at 14:07
Dahn JahnDahn Jahn
2,35811837
$endgroup$
This is (it seems to me) a well-known fact, but I am struggling to find a reference.
Let $X={x_1,dots, x_n}subset mathbb R^d$ be a set of points. Then the following is true:
Subset $Fsubset X$, with $mathrm{card}F=k+1$, forms a $k$-face of the Delaunay triangulation of $X$ iff there exists an empty $d$-dimensional sphere on which all points of $F$ lie.
I am interested specifically in the case $d=3,k=1$, i.e. an edge of Delaunay tetrahedrization. If you only provide a reference for this case, I will accept it as an answer.
I only found some proofs of the case $d=2,k=1$ and then a number of people using this fact without a source.
Is there a reference for this fact? Or is the proof in fact so elementary that there simply is no reference?
reference-request computational-geometry triangulation voronoi-diagram
reference-request computational-geometry triangulation voronoi-diagram
asked Jan 3 at 14:07
Dahn JahnDahn Jahn
2,35811837
asked Jan 3 at 14:07
Dahn JahnDahn Jahn
2,35811837
edited Jan 3 at 19:59
Dahn Jahn
asked Jan 3 at 14:07
Dahn JahnDahn Jahn
2,35811837
asked Jan 3 at 14:07
Dahn JahnDahn Jahn
2,35811837
asked Jan 3 at 14:07
Dahn JahnDahn Jahn
2,35811837
2,35811837
$begingroup$
Can you check the book Triangulations - Structures for Algorithms and Applications by Jesús A. De Loera, Jörg Rambau, and Francisco Santos? I suppose they discuss this.
$endgroup$
– toric_actions
Jan 3 at 19:19
$begingroup$
@toric_actions I can, but the book is not specifically about Delaunay triangulations and only mentions the empty spheres on one page. I'll try looking (since Delaunay is a special case of regular triangulations, which are also in the book). Do you perhaps have a more concrete pointer?
$endgroup$
– Dahn Jahn
Jan 3 at 19:51
add a comment |
$begingroup$
Can you check the book Triangulations - Structures for Algorithms and Applications by Jesús A. De Loera, Jörg Rambau, and Francisco Santos? I suppose they discuss this.
$endgroup$
– toric_actions
Jan 3 at 19:19
$begingroup$
@toric_actions I can, but the book is not specifically about Delaunay triangulations and only mentions the empty spheres on one page. I'll try looking (since Delaunay is a special case of regular triangulations, which are also in the book). Do you perhaps have a more concrete pointer?
$endgroup$
– Dahn Jahn
Jan 3 at 19:51
$begingroup$
Can you check the book Triangulations - Structures for Algorithms and Applications by Jesús A. De Loera, Jörg Rambau, and Francisco Santos? I suppose they discuss this.
$endgroup$
– toric_actions
Jan 3 at 19:19
$begingroup$
Can you check the book Triangulations - Structures for Algorithms and Applications by Jesús A. De Loera, Jörg Rambau, and Francisco Santos? I suppose they discuss this.
$endgroup$
– toric_actions
Jan 3 at 19:19
$begingroup$
@toric_actions I can, but the book is not specifically about Delaunay triangulations and only mentions the empty spheres on one page. I'll try looking (since Delaunay is a special case of regular triangulations, which are also in the book). Do you perhaps have a more concrete pointer?
$endgroup$
– Dahn Jahn
Jan 3 at 19:51
$begingroup$
@toric_actions I can, but the book is not specifically about Delaunay triangulations and only mentions the empty spheres on one page. I'll try looking (since Delaunay is a special case of regular triangulations, which are also in the book). Do you perhaps have a more concrete pointer?
$endgroup$
– Dahn Jahn
Jan 3 at 19:51
add a comment |
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$begingroup$
Can you check the book Triangulations - Structures for Algorithms and Applications by Jesús A. De Loera, Jörg Rambau, and Francisco Santos? I suppose they discuss this.
$endgroup$
– toric_actions
Jan 3 at 19:19
$begingroup$
@toric_actions I can, but the book is not specifically about Delaunay triangulations and only mentions the empty spheres on one page. I'll try looking (since Delaunay is a special case of regular triangulations, which are also in the book). Do you perhaps have a more concrete pointer?
$endgroup$
– Dahn Jahn
Jan 3 at 19:51