Mixing
How to pick a “center” of a concave polygon?
$begingroup$
I asked a question on how to scale concave polygons and a couple of people suggested some very clever solutions.
The issue is that these solutions rely on picking an appropriate point $C$ in the interior of the polygon.
The problem, more clearly, is to find a point $C$ in the interior of a polygon, such that every half segment connecting that point and the vertices of the polygon never intersects the Boundary of the polygon.
Examples:
In the first picture the point is a perfect candidate, in the second, the point is not because the red edge passes through a section outside the control polygon.
For now we can assume such a point exists, although for some shapes such point will not exist and so a more complex curve is needed.
linear-algebra geometry polygons homothety
asked Jan 31 at 16:43
MakoganMakogan
786218
$endgroup$
add a comment |
$begingroup$
I asked a question on how to scale concave polygons and a couple of people suggested some very clever solutions.
The issue is that these solutions rely on picking an appropriate point $C$ in the interior of the polygon.
The problem, more clearly, is to find a point $C$ in the interior of a polygon, such that every half segment connecting that point and the vertices of the polygon never intersects the Boundary of the polygon.
Examples:
In the first picture the point is a perfect candidate, in the second, the point is not because the red edge passes through a section outside the control polygon.
For now we can assume such a point exists, although for some shapes such point will not exist and so a more complex curve is needed.
linear-algebra geometry polygons homothety
asked Jan 31 at 16:43
MakoganMakogan
786218
$endgroup$
6
$begingroup$
What you're talking about are called star-shaped polygons; they're the simplest category of concave polygons and there's a fair amount of literature about them (including how to find a star point if one exists, I believe) that you should be able to find from that name.
$endgroup$
– Steven Stadnicki
Jan 31 at 16:47
1
$begingroup$
The concerned domain is "computational geometry" and you are dealing with the "art gallery problem" brilliant.org/wiki/guarding-a-museum
$endgroup$
– Jean Marie
Feb 1 at 20:17
add a comment |
$begingroup$
I asked a question on how to scale concave polygons and a couple of people suggested some very clever solutions.
The issue is that these solutions rely on picking an appropriate point $C$ in the interior of the polygon.
The problem, more clearly, is to find a point $C$ in the interior of a polygon, such that every half segment connecting that point and the vertices of the polygon never intersects the Boundary of the polygon.
Examples:
In the first picture the point is a perfect candidate, in the second, the point is not because the red edge passes through a section outside the control polygon.
For now we can assume such a point exists, although for some shapes such point will not exist and so a more complex curve is needed.
linear-algebra geometry polygons homothety
asked Jan 31 at 16:43
MakoganMakogan
786218
$endgroup$
I asked a question on how to scale concave polygons and a couple of people suggested some very clever solutions.
The issue is that these solutions rely on picking an appropriate point $C$ in the interior of the polygon.
The problem, more clearly, is to find a point $C$ in the interior of a polygon, such that every half segment connecting that point and the vertices of the polygon never intersects the Boundary of the polygon.
Examples:
In the first picture the point is a perfect candidate, in the second, the point is not because the red edge passes through a section outside the control polygon.
For now we can assume such a point exists, although for some shapes such point will not exist and so a more complex curve is needed.
linear-algebra geometry polygons homothety
linear-algebra geometry polygons homothety
asked Jan 31 at 16:43
MakoganMakogan
786218
asked Jan 31 at 16:43
MakoganMakogan
786218
asked Jan 31 at 16:43
MakoganMakogan
786218
asked Jan 31 at 16:43
MakoganMakogan
786218
asked Jan 31 at 16:43
MakoganMakogan
786218
786218
6
$begingroup$
What you're talking about are called star-shaped polygons; they're the simplest category of concave polygons and there's a fair amount of literature about them (including how to find a star point if one exists, I believe) that you should be able to find from that name.
$endgroup$
– Steven Stadnicki
Jan 31 at 16:47
1
$begingroup$
The concerned domain is "computational geometry" and you are dealing with the "art gallery problem" brilliant.org/wiki/guarding-a-museum
$endgroup$
– Jean Marie
Feb 1 at 20:17
add a comment |
6
$begingroup$
What you're talking about are called star-shaped polygons; they're the simplest category of concave polygons and there's a fair amount of literature about them (including how to find a star point if one exists, I believe) that you should be able to find from that name.
$endgroup$
– Steven Stadnicki
Jan 31 at 16:47
1
$begingroup$
The concerned domain is "computational geometry" and you are dealing with the "art gallery problem" brilliant.org/wiki/guarding-a-museum
$endgroup$
– Jean Marie
Feb 1 at 20:17
6
6
$begingroup$
What you're talking about are called star-shaped polygons; they're the simplest category of concave polygons and there's a fair amount of literature about them (including how to find a star point if one exists, I believe) that you should be able to find from that name.
$endgroup$
– Steven Stadnicki
Jan 31 at 16:47
$begingroup$
What you're talking about are called star-shaped polygons; they're the simplest category of concave polygons and there's a fair amount of literature about them (including how to find a star point if one exists, I believe) that you should be able to find from that name.
$endgroup$
– Steven Stadnicki
Jan 31 at 16:47
1
1
$begingroup$
The concerned domain is "computational geometry" and you are dealing with the "art gallery problem" brilliant.org/wiki/guarding-a-museum
$endgroup$
– Jean Marie
Feb 1 at 20:17
$begingroup$
The concerned domain is "computational geometry" and you are dealing with the "art gallery problem" brilliant.org/wiki/guarding-a-museum
$endgroup$
– Jean Marie
Feb 1 at 20:17
add a comment |
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$begingroup$
What you're talking about are called star-shaped polygons; they're the simplest category of concave polygons and there's a fair amount of literature about them (including how to find a star point if one exists, I believe) that you should be able to find from that name.
$endgroup$
– Steven Stadnicki
Jan 31 at 16:47
1
$begingroup$
The concerned domain is "computational geometry" and you are dealing with the "art gallery problem" brilliant.org/wiki/guarding-a-museum
$endgroup$
– Jean Marie
Feb 1 at 20:17