Delaunay triangulation in $mathbb R^d$: Empty sphere property works for all $k$-faces?
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 yesterday
Dahn Jahn
2,34811837
add a comment |
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 yesterday
Dahn Jahn
2,34811837
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.
– toric_actions
yesterday
@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?
– Dahn Jahn
yesterday
add a comment |
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 yesterday
Dahn Jahn
2,34811837
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 yesterday
Dahn Jahn
2,34811837
asked yesterday
Dahn Jahn
2,34811837
edited yesterday
asked yesterday
Dahn Jahn
2,34811837
asked yesterday
Dahn Jahn
2,34811837
asked yesterday
Dahn Jahn
2,34811837
2,34811837
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.
– toric_actions
yesterday
@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?
– Dahn Jahn
yesterday
add a comment |
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.
– toric_actions
yesterday
@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?
– Dahn Jahn
yesterday
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.
– toric_actions
yesterday
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.
– toric_actions
yesterday
@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?
– Dahn Jahn
yesterday
@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?
– Dahn Jahn
yesterday
add a comment |
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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.
– toric_actions
yesterday
@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?
– Dahn Jahn
yesterday