Mixing
Find vertices of a Voronoi diagram of convex polygons
$begingroup$
From a set of polygons guaranteed to be :
- Convex
- Full (no holes)
- Non-intersecting (polygons may share edges/points, but not penetrate each other)
How do I find the vertices of the Voronoi diagram where each polygon is a seed?
Precisions :
- Any answer for a Voronoi diagram of lines is also welcomed (I can merge the cells for lines belonging to the same polygon)
- I can work with an approximation of the vertices (within reasonable precision). This is for a simulated pathfinding / navigation system, so exact precision is appreciated but not mandatory.
- My math foundations are rusty, so please do mention anything obvious I should know / look into as an alternative.
I already have a working brute-force discretized approach :
- Draw each polygon in a unique color on an image
- For every blank pixel, find closest polygon and fill the pixel with its color
- List all pixels with adjacent to at least 2 different colors
While it works well enough (given a big enough image resolution) the performance is obviously horrible and it adds quite a few corner cases to handle. So I'd prefer to do without brute-forcing if possible.
geometry convex-geometry voronoi-diagram
edited Jan 11 at 13:28
Bernard
120k740116
asked Jan 11 at 12:45
Milan IrigoyenMilan Irigoyen
11
$endgroup$
add a comment |
$begingroup$
From a set of polygons guaranteed to be :
- Convex
- Full (no holes)
- Non-intersecting (polygons may share edges/points, but not penetrate each other)
How do I find the vertices of the Voronoi diagram where each polygon is a seed?
Precisions :
- Any answer for a Voronoi diagram of lines is also welcomed (I can merge the cells for lines belonging to the same polygon)
- I can work with an approximation of the vertices (within reasonable precision). This is for a simulated pathfinding / navigation system, so exact precision is appreciated but not mandatory.
- My math foundations are rusty, so please do mention anything obvious I should know / look into as an alternative.
I already have a working brute-force discretized approach :
- Draw each polygon in a unique color on an image
- For every blank pixel, find closest polygon and fill the pixel with its color
- List all pixels with adjacent to at least 2 different colors
While it works well enough (given a big enough image resolution) the performance is obviously horrible and it adds quite a few corner cases to handle. So I'd prefer to do without brute-forcing if possible.
geometry convex-geometry voronoi-diagram
edited Jan 11 at 13:28
Bernard
120k740116
asked Jan 11 at 12:45
Milan IrigoyenMilan Irigoyen
11
$endgroup$
add a comment |
$begingroup$
From a set of polygons guaranteed to be :
- Convex
- Full (no holes)
- Non-intersecting (polygons may share edges/points, but not penetrate each other)
How do I find the vertices of the Voronoi diagram where each polygon is a seed?
Precisions :
- Any answer for a Voronoi diagram of lines is also welcomed (I can merge the cells for lines belonging to the same polygon)
- I can work with an approximation of the vertices (within reasonable precision). This is for a simulated pathfinding / navigation system, so exact precision is appreciated but not mandatory.
- My math foundations are rusty, so please do mention anything obvious I should know / look into as an alternative.
I already have a working brute-force discretized approach :
- Draw each polygon in a unique color on an image
- For every blank pixel, find closest polygon and fill the pixel with its color
- List all pixels with adjacent to at least 2 different colors
While it works well enough (given a big enough image resolution) the performance is obviously horrible and it adds quite a few corner cases to handle. So I'd prefer to do without brute-forcing if possible.
geometry convex-geometry voronoi-diagram
edited Jan 11 at 13:28
Bernard
120k740116
asked Jan 11 at 12:45
Milan IrigoyenMilan Irigoyen
11
$endgroup$
From a set of polygons guaranteed to be :
- Convex
- Full (no holes)
- Non-intersecting (polygons may share edges/points, but not penetrate each other)
How do I find the vertices of the Voronoi diagram where each polygon is a seed?
Precisions :
- Any answer for a Voronoi diagram of lines is also welcomed (I can merge the cells for lines belonging to the same polygon)
- I can work with an approximation of the vertices (within reasonable precision). This is for a simulated pathfinding / navigation system, so exact precision is appreciated but not mandatory.
- My math foundations are rusty, so please do mention anything obvious I should know / look into as an alternative.
I already have a working brute-force discretized approach :
- Draw each polygon in a unique color on an image
- For every blank pixel, find closest polygon and fill the pixel with its color
- List all pixels with adjacent to at least 2 different colors
While it works well enough (given a big enough image resolution) the performance is obviously horrible and it adds quite a few corner cases to handle. So I'd prefer to do without brute-forcing if possible.
geometry convex-geometry voronoi-diagram
geometry convex-geometry voronoi-diagram
edited Jan 11 at 13:28
Bernard
120k740116
asked Jan 11 at 12:45
Milan IrigoyenMilan Irigoyen
11
edited Jan 11 at 13:28
Bernard
120k740116
asked Jan 11 at 12:45
Milan IrigoyenMilan Irigoyen
11
edited Jan 11 at 13:28
Bernard
120k740116
edited Jan 11 at 13:28
Bernard
120k740116
edited Jan 11 at 13:28
Bernard
120k740116
120k740116
asked Jan 11 at 12:45
Milan IrigoyenMilan Irigoyen
11
asked Jan 11 at 12:45
Milan IrigoyenMilan Irigoyen
11
asked Jan 11 at 12:45
Milan IrigoyenMilan Irigoyen
11
11
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%2f3069795%2ffind-vertices-of-a-voronoi-diagram-of-convex-polygons%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%2f3069795%2ffind-vertices-of-a-voronoi-diagram-of-convex-polygons%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
