# I Winding number for a point that lies over a closed curve

1. Feb 16, 2017

### pigna

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...

2. Feb 16, 2017

### stevendaryl

Staff Emeritus
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. Feb 16, 2017

### lavinia

what is an invalid loop? example?

4. Feb 17, 2017

### pigna

thanks for the replies... looks at the pdf in which I clarify what I'm interested in....

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5. Feb 17, 2017

### FactChecker

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.