Winding number for a point that lies over a closed curve

In summary: This is because the winding number is defined as the number of times the path crosses the point. If the path does not cross the point, then the winding number is undefined.
  • #1
pigna
12
1
The definitions of the winding number, that I have found, do not consider the case in which the point lies over the curve. Is there the winding number undefined ? I'm interested in this issue because I'm writing an algorihm for polygon offsetting that as first step creates a row offset polygon ( generally a non simple polygon with self intersections). Than i have to classify these self-intersection to remove the invalid loops of the row offset curve. I have successfully reach this purpose implementing an algorithm that gives me a value for the winding numbers of these self-intersections ( simply using the summ ω(point,Poly)=∑[θ][/i]), but actually I don't know if this operation is consistent with the theory... I need some clarification about this point and eventually some reference to some papers that address this issue...
 
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  • #2
I don't know what you mean by saying "the point lies under the curve". The usual definition of winding number is for curves in 2 dimensions, and you compute the winding number about a particular point in the plane. So what does "under" mean in that context? Are you talking "under" in the sense of 3 dimensional space?
 
  • #3
pigna said:
The definitions of the winding number, that I have found, do not consider the case in which the point lies over the curve. Is there the winding number undefined ? I'm interested in this issue because I'm writing an algorihm for polygon offsetting that as first step creates a row offset polygon ( generally a non simple polygon with self intersections). Than i have to classify these self-intersection to remove the invalid loops of the row offset curve. I have successfully reach this purpose implementing an algorithm that gives me a value for the winding numbers of these self-intersections ( simply using the summ ω(point,Poly)=∑[θ][/i]), but actually I don't know if this operation is consistent with the theory... I need some clarification about this point and eventually some reference to some papers that address this issue...
what is an invalid loop? example?
 
  • #4
thanks for the replies... looks at the pdf in which I clarify what I'm interested in...
 

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  • #5
If a point is on the path, the winding number of the path around the point is not defined. You may be able to define it appropriately for your particular use, but that would be up to you and how you are going to use it. If you want to rely on established definitions of "winding number", you will need to avoid that situation by changing either the path or the point position.
For many applications, such as determining if a point is inside or outside a simple closed path, it is just a matter of you stating whether points on the path will be considered inside or outside.
 
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1. What is the definition of winding number for a point over a closed curve?

The winding number for a point over a closed curve is a measure of how many times the curve wraps around the point in a counterclockwise direction. It is a numerical value that can be positive, negative, or zero.

2. How is the winding number for a point over a closed curve calculated?

The winding number is calculated by drawing a line from the point to any point outside of the curve and counting the number of times the curve is crossed in a counterclockwise direction. The winding number is the number of times the curve is crossed minus the number of times the curve is crossed in a clockwise direction.

3. What does a winding number of zero mean for a point over a closed curve?

A winding number of zero means that the point is not enclosed by the closed curve and the curve does not wrap around the point in any direction.

4. Can the winding number for a point over a closed curve be a decimal or fraction?

No, the winding number for a point over a closed curve is always an integer. This is because it represents the number of times the curve wraps around the point, and fractions or decimals do not make sense in this context.

5. How is the winding number used in mathematics and science?

The winding number is used in various fields of mathematics and science, such as complex analysis, topology, and physics. It is particularly useful in the study of curves and their behavior around points. It also has applications in image processing, computer graphics, and robotics.

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