# Can a contour integral be used to detect complex polygon deformations?

• james_1138
In summary: There is an algorithm that can determine which vertex has "self-penetrated" the polygon. However, it is costly to evaluate.
james_1138
Hi, I am working on a simulation code that simulates the deformation of sand grains in 2D. The sand grains are modeled as simple polygons. However, during the simulation the grains can deform to create non-convex vertices. Further more, when deformation becomes extreme, there is a possibility that "self intersection" can occur (the polygon goes from being non-convex to complex).

My question is then, is there an algorithm that can determine which vertex has "self-penetrated" the polygon? Even if I check for intersections between all edges and determine that the polygon is now complex, I do not know which vertex has penetrated.

Here's an example where vertex A has "folded"

#### Attachments

• Polygon.jpg
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Do the "well behaviored" polygons have to be convex?

All of the grains are initially simple and convex (ie. created using a Voronoi tessellation) . During the simulation, many become non-convex (one or more reflex angles are created) due to deformation. And some could even become complex (have self-intersections). This is what I need to check for and correct.

Below is an image sequence of what could happen to a grain. (1) Simple, Convex Grain (2) Non-Convex Grain (3) Complex Grain (self-intersection)

#### Attachments

• Polygons.jpg
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I'm sorry for the persistence, but does anyone have any insight on this problem?

You could try embedding the vertices of your polygon in the complex plane. Choose a function that has pairs of simple poles close to each of the vertices of your polygon such that for each vertex, one simple pole lies in the interior of the polygon and close enough to the vertex, and such that the other simple pole lies in the exterior of the polygon and close enough to the vertex.

So long as your polygon vertices do not self intersect,

then only the poles in the interior of your polygon will contribute to the value of the contour integral of the function, where the contour is along the lines of your polygon.

However, when the vertices intersect, as shown in your diagram, a few more pairs of poles will cross over into the interior region enclosed by the contour of the contour integral in question, and evaluating the contour integral piecewise along each line, will result in a different value, and thereby, in theory at least, detect that a self-intersection has occurred.

The question remains on how costly, numerically speaking, the above integral is to evaluate in your particular simulation.

I hope this helps.

Best Regards,

Edwin G. Schasteen

## 1. What is a complex polygon?

A complex polygon is a two-dimensional shape with three or more sides that is made up of multiple smaller polygons, such as triangles, quadrilaterals, or other polygons. It can also have holes or cutouts within the shape.

## 2. Why is detecting complex polygons important?

Detecting complex polygons is important in various fields, such as computer graphics, image processing, and spatial analysis. It allows for accurate and efficient identification and analysis of different shapes and structures present in a given set of data or image.

## 3. How is complex polygon detection performed?

Complex polygon detection involves using algorithms and mathematical techniques to analyze the coordinates and angles of the vertices of a shape. This information is then compared to known geometric properties of polygons to determine if the shape is a complex polygon.

## 4. What are some challenges in complex polygon detection?

One of the main challenges in complex polygon detection is dealing with irregular shapes and overlapping polygons. This can result in false positives or false negatives in the detection process. Another challenge is the performance and efficiency of the algorithms used, as complex polygons can have a large number of vertices and require complex calculations.

## 5. Can complex polygon detection be automated?

Yes, complex polygon detection can be automated with the use of computer programs and algorithms. However, the accuracy of the detection may depend on the quality of the data and the complexity of the polygons being detected. Human intervention and verification may still be necessary in some cases.

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