Clarification on How to derive Voronoi diagram from Delaunay triangulation in linear time
$begingroup$
Definitions:
Assume that a set of points $P={p_1,dots,p_n}$ in $mathbb R^d $ is given. For each $p_i in P$, the Voronoi region of $p_i$ is defined as:
$Vor(p_i)={pinmathbb R^d:forall p_jin Pquad p_jneq p_iimplies ||p-p_i||leq||p-p_j||}$
And the Voronoi diagram of $P$ is defined as:
$V(P)=cup_{p_iin P} Vor(p_i)$
Delaunay triangulation of $P$ is a triangulation $DT(P)$ such that no point in $P$ is inside the circumcircle of any triangle in $DT(P)$.
We know that Delaunay triangulation is the dual of Voronoi diagram.
Question:
How can we derive Voronoi diagram from Delaunay triangulation in linear time?
So, as the input, we have some points in $mathbb R^d$ and we know that which points are connected. The output should be the Voronoi diagram of that points.
What I know:
One method was to consider a list in which each element has a point and the triangles having it as a vertex. But it seems that obtaining this list takes more than $O(n)$ time. There was another method which said we have to compute the intersection of perpendicular bisectors of Delaunay edges. The problem with this method is that the bisectors are infinite. I'm looking for a precise algorithm which can be implemented in a computer. So, I know the ideas but it's not enough.
Note: I know that this question has been asked many times. But it seems that the time complexity was not important. Another problem is that the answers were not precise enough.
computational-geometry triangulation voronoi-diagram
asked Dec 13 '18 at 9:08
Arman MalekzadehArman Malekzadeh
1,8221031
$endgroup$
add a comment |
$begingroup$
Definitions:
Assume that a set of points $P={p_1,dots,p_n}$ in $mathbb R^d $ is given. For each $p_i in P$, the Voronoi region of $p_i$ is defined as:
$Vor(p_i)={pinmathbb R^d:forall p_jin Pquad p_jneq p_iimplies ||p-p_i||leq||p-p_j||}$
And the Voronoi diagram of $P$ is defined as:
$V(P)=cup_{p_iin P} Vor(p_i)$
Delaunay triangulation of $P$ is a triangulation $DT(P)$ such that no point in $P$ is inside the circumcircle of any triangle in $DT(P)$.
We know that Delaunay triangulation is the dual of Voronoi diagram.
Question:
How can we derive Voronoi diagram from Delaunay triangulation in linear time?
So, as the input, we have some points in $mathbb R^d$ and we know that which points are connected. The output should be the Voronoi diagram of that points.
What I know:
One method was to consider a list in which each element has a point and the triangles having it as a vertex. But it seems that obtaining this list takes more than $O(n)$ time. There was another method which said we have to compute the intersection of perpendicular bisectors of Delaunay edges. The problem with this method is that the bisectors are infinite. I'm looking for a precise algorithm which can be implemented in a computer. So, I know the ideas but it's not enough.
Note: I know that this question has been asked many times. But it seems that the time complexity was not important. Another problem is that the answers were not precise enough.
computational-geometry triangulation voronoi-diagram
asked Dec 13 '18 at 9:08
Arman MalekzadehArman Malekzadeh
1,8221031
$endgroup$
add a comment |
$begingroup$
Definitions:
Assume that a set of points $P={p_1,dots,p_n}$ in $mathbb R^d $ is given. For each $p_i in P$, the Voronoi region of $p_i$ is defined as:
$Vor(p_i)={pinmathbb R^d:forall p_jin Pquad p_jneq p_iimplies ||p-p_i||leq||p-p_j||}$
And the Voronoi diagram of $P$ is defined as:
$V(P)=cup_{p_iin P} Vor(p_i)$
Delaunay triangulation of $P$ is a triangulation $DT(P)$ such that no point in $P$ is inside the circumcircle of any triangle in $DT(P)$.
We know that Delaunay triangulation is the dual of Voronoi diagram.
Question:
How can we derive Voronoi diagram from Delaunay triangulation in linear time?
So, as the input, we have some points in $mathbb R^d$ and we know that which points are connected. The output should be the Voronoi diagram of that points.
What I know:
One method was to consider a list in which each element has a point and the triangles having it as a vertex. But it seems that obtaining this list takes more than $O(n)$ time. There was another method which said we have to compute the intersection of perpendicular bisectors of Delaunay edges. The problem with this method is that the bisectors are infinite. I'm looking for a precise algorithm which can be implemented in a computer. So, I know the ideas but it's not enough.
Note: I know that this question has been asked many times. But it seems that the time complexity was not important. Another problem is that the answers were not precise enough.
computational-geometry triangulation voronoi-diagram
asked Dec 13 '18 at 9:08
Arman MalekzadehArman Malekzadeh
1,8221031
$endgroup$
Definitions:
Assume that a set of points $P={p_1,dots,p_n}$ in $mathbb R^d $ is given. For each $p_i in P$, the Voronoi region of $p_i$ is defined as:
$Vor(p_i)={pinmathbb R^d:forall p_jin Pquad p_jneq p_iimplies ||p-p_i||leq||p-p_j||}$
And the Voronoi diagram of $P$ is defined as:
$V(P)=cup_{p_iin P} Vor(p_i)$
Delaunay triangulation of $P$ is a triangulation $DT(P)$ such that no point in $P$ is inside the circumcircle of any triangle in $DT(P)$.
We know that Delaunay triangulation is the dual of Voronoi diagram.
Question:
How can we derive Voronoi diagram from Delaunay triangulation in linear time?
So, as the input, we have some points in $mathbb R^d$ and we know that which points are connected. The output should be the Voronoi diagram of that points.
What I know:
One method was to consider a list in which each element has a point and the triangles having it as a vertex. But it seems that obtaining this list takes more than $O(n)$ time. There was another method which said we have to compute the intersection of perpendicular bisectors of Delaunay edges. The problem with this method is that the bisectors are infinite. I'm looking for a precise algorithm which can be implemented in a computer. So, I know the ideas but it's not enough.
Note: I know that this question has been asked many times. But it seems that the time complexity was not important. Another problem is that the answers were not precise enough.
computational-geometry triangulation voronoi-diagram
computational-geometry triangulation voronoi-diagram
asked Dec 13 '18 at 9:08
Arman MalekzadehArman Malekzadeh
1,8221031
asked Dec 13 '18 at 9:08
Arman MalekzadehArman Malekzadeh
1,8221031
asked Dec 13 '18 at 9:08
Arman MalekzadehArman Malekzadeh
1,8221031
asked Dec 13 '18 at 9:08
Arman MalekzadehArman Malekzadeh
1,8221031
asked Dec 13 '18 at 9:08
Arman MalekzadehArman Malekzadeh
1,8221031
1,8221031
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3037786%2fclarification-on-how-to-derive-voronoi-diagram-from-delaunay-triangulation-in-li%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3037786%2fclarification-on-how-to-derive-voronoi-diagram-from-delaunay-triangulation-in-li%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
