Mixing
Calculate Overlapping Area of $2$-Dimensional Shapes
$begingroup$
I am running a Computer Simulation where 2 Shapes are moving towards each other and will eventually overlap.
I want to calculate the overlapping Area of the shapes - in this example a Circle and a Square in a $2$-Dimensional Room.
Let's describe the Circle as $(x_r, y_r, r)$ and the square as $(x_s, y_s, w, h)$.
Unfortunately I don't find the hint that shows me how to start approaching this problem.
How do I start solving this problem?
geometry circle area
asked Jan 11 at 8:17
ElHombreElHombre
62
$endgroup$
add a comment |
$begingroup$
I am running a Computer Simulation where 2 Shapes are moving towards each other and will eventually overlap.
I want to calculate the overlapping Area of the shapes - in this example a Circle and a Square in a $2$-Dimensional Room.
Let's describe the Circle as $(x_r, y_r, r)$ and the square as $(x_s, y_s, w, h)$.
Unfortunately I don't find the hint that shows me how to start approaching this problem.
How do I start solving this problem?
geometry circle area
asked Jan 11 at 8:17
ElHombreElHombre
62
$endgroup$
$begingroup$
Well there is not much context though
$endgroup$
– ElHombre
Jan 11 at 8:49
$begingroup$
@ElHombre What's your motivation for asking this question?
$endgroup$
– Toby Mak
Jan 11 at 9:45
$begingroup$
I want to use a specific area size as a trigger. As the idea is that The objects and movements can be set randomly The mentioned Parameters are the only that I can determine for that moment. I think I could also get the Points of Intersection, that might make things easier, won't it?
$endgroup$
– ElHombre
Jan 15 at 2:43
add a comment |
$begingroup$
I am running a Computer Simulation where 2 Shapes are moving towards each other and will eventually overlap.
I want to calculate the overlapping Area of the shapes - in this example a Circle and a Square in a $2$-Dimensional Room.
Let's describe the Circle as $(x_r, y_r, r)$ and the square as $(x_s, y_s, w, h)$.
Unfortunately I don't find the hint that shows me how to start approaching this problem.
How do I start solving this problem?
geometry circle area
asked Jan 11 at 8:17
ElHombreElHombre
62
$endgroup$
I am running a Computer Simulation where 2 Shapes are moving towards each other and will eventually overlap.
I want to calculate the overlapping Area of the shapes - in this example a Circle and a Square in a $2$-Dimensional Room.
Let's describe the Circle as $(x_r, y_r, r)$ and the square as $(x_s, y_s, w, h)$.
Unfortunately I don't find the hint that shows me how to start approaching this problem.
How do I start solving this problem?
geometry circle area
geometry circle area
asked Jan 11 at 8:17
ElHombreElHombre
62
asked Jan 11 at 8:17
ElHombreElHombre
62
edited Jan 11 at 8:48
ElHombre
asked Jan 11 at 8:17
ElHombreElHombre
62
asked Jan 11 at 8:17
ElHombreElHombre
62
asked Jan 11 at 8:17
ElHombreElHombre
62
62
$begingroup$
Well there is not much context though
$endgroup$
– ElHombre
Jan 11 at 8:49
$begingroup$
@ElHombre What's your motivation for asking this question?
$endgroup$
– Toby Mak
Jan 11 at 9:45
$begingroup$
I want to use a specific area size as a trigger. As the idea is that The objects and movements can be set randomly The mentioned Parameters are the only that I can determine for that moment. I think I could also get the Points of Intersection, that might make things easier, won't it?
$endgroup$
– ElHombre
Jan 15 at 2:43
add a comment |
$begingroup$
Well there is not much context though
$endgroup$
– ElHombre
Jan 11 at 8:49
$begingroup$
@ElHombre What's your motivation for asking this question?
$endgroup$
– Toby Mak
Jan 11 at 9:45
$begingroup$
I want to use a specific area size as a trigger. As the idea is that The objects and movements can be set randomly The mentioned Parameters are the only that I can determine for that moment. I think I could also get the Points of Intersection, that might make things easier, won't it?
$endgroup$
– ElHombre
Jan 15 at 2:43
$begingroup$
Well there is not much context though
$endgroup$
– ElHombre
Jan 11 at 8:49
$begingroup$
Well there is not much context though
$endgroup$
– ElHombre
Jan 11 at 8:49
$begingroup$
@ElHombre What's your motivation for asking this question?
$endgroup$
– Toby Mak
Jan 11 at 9:45
$begingroup$
@ElHombre What's your motivation for asking this question?
$endgroup$
– Toby Mak
Jan 11 at 9:45
$begingroup$
I want to use a specific area size as a trigger. As the idea is that The objects and movements can be set randomly The mentioned Parameters are the only that I can determine for that moment. I think I could also get the Points of Intersection, that might make things easier, won't it?
$endgroup$
– ElHombre
Jan 15 at 2:43
$begingroup$
I want to use a specific area size as a trigger. As the idea is that The objects and movements can be set randomly The mentioned Parameters are the only that I can determine for that moment. I think I could also get the Points of Intersection, that might make things easier, won't it?
$endgroup$
– ElHombre
Jan 15 at 2:43
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint:
This is a pretty complex problem as there are many possible cases, but you can address it in a manageable way by the sweepline approach.
Draw horizontals by the square corners and by the N and S pole of the circle. The two shapes delimit line segments on these horizontals, and finding the intersection of two segments is trivial. When you go down from one line to the next, if the endpoints switch position, you have an intersection between the circle and the square outlines.
Scanning all horizontals in turn, you can follow the outline of the intersection shape, and as you go you describe curvilinear trapezoids (the oblique sides are circular arcs). Now it "suffices" to accumulate the areas of these trapezoids, using the standard formula, corrected with the areas of circular segments.
Warning:
The criterion "if the endpoints switch position" is not sufficient. You can very well have a large arc which is intersected twice by a vertical side, so that the endpoints switch position twice. So it is safer to write a test that explicitly looks for intersections between an arc and a vertical line in the current vertical span.
answered Jan 11 at 9:08
Yves DaoustYves Daoust
127k673226
$endgroup$
$begingroup$
that sounds like a pretty good solution, it nearly covers all possibilities right?
$endgroup$
– ElHombre
Jan 15 at 5:52
$begingroup$
@ElHombre: if you take care, it covers all possibilities. I have added a warning.
$endgroup$
– Yves Daoust
Jan 15 at 8:41
add a comment |
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1 Answer
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1 Answer
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oldest
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active
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$begingroup$
Hint:
This is a pretty complex problem as there are many possible cases, but you can address it in a manageable way by the sweepline approach.
Draw horizontals by the square corners and by the N and S pole of the circle. The two shapes delimit line segments on these horizontals, and finding the intersection of two segments is trivial. When you go down from one line to the next, if the endpoints switch position, you have an intersection between the circle and the square outlines.
Scanning all horizontals in turn, you can follow the outline of the intersection shape, and as you go you describe curvilinear trapezoids (the oblique sides are circular arcs). Now it "suffices" to accumulate the areas of these trapezoids, using the standard formula, corrected with the areas of circular segments.
Warning:
The criterion "if the endpoints switch position" is not sufficient. You can very well have a large arc which is intersected twice by a vertical side, so that the endpoints switch position twice. So it is safer to write a test that explicitly looks for intersections between an arc and a vertical line in the current vertical span.
answered Jan 11 at 9:08
Yves DaoustYves Daoust
127k673226
$endgroup$
$begingroup$
that sounds like a pretty good solution, it nearly covers all possibilities right?
$endgroup$
– ElHombre
Jan 15 at 5:52
$begingroup$
@ElHombre: if you take care, it covers all possibilities. I have added a warning.
$endgroup$
– Yves Daoust
Jan 15 at 8:41
add a comment |
$begingroup$
Hint:
This is a pretty complex problem as there are many possible cases, but you can address it in a manageable way by the sweepline approach.
Draw horizontals by the square corners and by the N and S pole of the circle. The two shapes delimit line segments on these horizontals, and finding the intersection of two segments is trivial. When you go down from one line to the next, if the endpoints switch position, you have an intersection between the circle and the square outlines.
Scanning all horizontals in turn, you can follow the outline of the intersection shape, and as you go you describe curvilinear trapezoids (the oblique sides are circular arcs). Now it "suffices" to accumulate the areas of these trapezoids, using the standard formula, corrected with the areas of circular segments.
Warning:
The criterion "if the endpoints switch position" is not sufficient. You can very well have a large arc which is intersected twice by a vertical side, so that the endpoints switch position twice. So it is safer to write a test that explicitly looks for intersections between an arc and a vertical line in the current vertical span.
answered Jan 11 at 9:08
Yves DaoustYves Daoust
127k673226
$endgroup$
$begingroup$
that sounds like a pretty good solution, it nearly covers all possibilities right?
$endgroup$
– ElHombre
Jan 15 at 5:52
$begingroup$
@ElHombre: if you take care, it covers all possibilities. I have added a warning.
$endgroup$
– Yves Daoust
Jan 15 at 8:41
add a comment |
$begingroup$
Hint:
This is a pretty complex problem as there are many possible cases, but you can address it in a manageable way by the sweepline approach.
Draw horizontals by the square corners and by the N and S pole of the circle. The two shapes delimit line segments on these horizontals, and finding the intersection of two segments is trivial. When you go down from one line to the next, if the endpoints switch position, you have an intersection between the circle and the square outlines.
Scanning all horizontals in turn, you can follow the outline of the intersection shape, and as you go you describe curvilinear trapezoids (the oblique sides are circular arcs). Now it "suffices" to accumulate the areas of these trapezoids, using the standard formula, corrected with the areas of circular segments.
Warning:
The criterion "if the endpoints switch position" is not sufficient. You can very well have a large arc which is intersected twice by a vertical side, so that the endpoints switch position twice. So it is safer to write a test that explicitly looks for intersections between an arc and a vertical line in the current vertical span.
answered Jan 11 at 9:08
Yves DaoustYves Daoust
127k673226
$endgroup$
Hint:
This is a pretty complex problem as there are many possible cases, but you can address it in a manageable way by the sweepline approach.
Draw horizontals by the square corners and by the N and S pole of the circle. The two shapes delimit line segments on these horizontals, and finding the intersection of two segments is trivial. When you go down from one line to the next, if the endpoints switch position, you have an intersection between the circle and the square outlines.
Scanning all horizontals in turn, you can follow the outline of the intersection shape, and as you go you describe curvilinear trapezoids (the oblique sides are circular arcs). Now it "suffices" to accumulate the areas of these trapezoids, using the standard formula, corrected with the areas of circular segments.
Warning:
The criterion "if the endpoints switch position" is not sufficient. You can very well have a large arc which is intersected twice by a vertical side, so that the endpoints switch position twice. So it is safer to write a test that explicitly looks for intersections between an arc and a vertical line in the current vertical span.
answered Jan 11 at 9:08
Yves DaoustYves Daoust
127k673226
edited Jan 15 at 8:45
answered Jan 11 at 9:08
Yves DaoustYves Daoust
127k673226
answered Jan 11 at 9:08
Yves DaoustYves Daoust
127k673226
answered Jan 11 at 9:08
Yves DaoustYves Daoust
127k673226
127k673226
$begingroup$
that sounds like a pretty good solution, it nearly covers all possibilities right?
$endgroup$
– ElHombre
Jan 15 at 5:52
$begingroup$
@ElHombre: if you take care, it covers all possibilities. I have added a warning.
$endgroup$
– Yves Daoust
Jan 15 at 8:41
add a comment |
$begingroup$
that sounds like a pretty good solution, it nearly covers all possibilities right?
$endgroup$
– ElHombre
Jan 15 at 5:52
$begingroup$
@ElHombre: if you take care, it covers all possibilities. I have added a warning.
$endgroup$
– Yves Daoust
Jan 15 at 8:41
$begingroup$
that sounds like a pretty good solution, it nearly covers all possibilities right?
$endgroup$
– ElHombre
Jan 15 at 5:52
$begingroup$
that sounds like a pretty good solution, it nearly covers all possibilities right?
$endgroup$
– ElHombre
Jan 15 at 5:52
$begingroup$
@ElHombre: if you take care, it covers all possibilities. I have added a warning.
$endgroup$
– Yves Daoust
Jan 15 at 8:41
$begingroup$
@ElHombre: if you take care, it covers all possibilities. I have added a warning.
$endgroup$
– Yves Daoust
Jan 15 at 8:41
add a comment |
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$begingroup$
Well there is not much context though
$endgroup$
– ElHombre
Jan 11 at 8:49
$begingroup$
@ElHombre What's your motivation for asking this question?
$endgroup$
– Toby Mak
Jan 11 at 9:45
$begingroup$
I want to use a specific area size as a trigger. As the idea is that The objects and movements can be set randomly The mentioned Parameters are the only that I can determine for that moment. I think I could also get the Points of Intersection, that might make things easier, won't it?
$endgroup$
– ElHombre
Jan 15 at 2:43