Irregular polygons - Equal area under horizontal line

[jamnitzer]

I've been trying to find an algorithm for finding a line passing through any 2-dimensional polygon that will divide the shape horizontally or vertically into two sections that each have the exact same area, or rather exactly half the area of the whole. I've prepared some simple examples below.

My question is if someone has heard of this or done something similar, and/or if anyone knows where to find more information.

Using the shoelace method of measuring area and/or after calculating the center of mass, I want to find a way to find either a horizontal line that will divide the top and bottom of a shape or a vertical line that will divide left and right such that the area of the shape on either side will be equal.

So the half-way line in terms of area, but only from one direction. I'm hoping there's already a name for this somewhere and I just haven't been able to find it. If this is more of a calculus question let me know. Really any information will be appreciated.


Some simple examples:

These are two shapes defined with their coordinates (x,y), repeating the first at the end.
{(4,0),(6,0),(6,2),(8,4),(4,4),(4,6),(2,4),(0,4),(4,0)} Area = 20
{(1,0),(1,1),(7,1),(4,0),(8,0),(8,1),(6,4),(4,2),(2,4),(6,6),(8,8),(0,8),(0,4),(1,0)} Area = 40.5

For the first shape, the vertical line evenly dividing left and right the area is x= 4.
The horizontal line dividing top and bottom is y= 4/sqrt(2).

In the second shape, the vertical line is x= -(sqrt(109)-20)/3
The horizontal line is y= (sqrt(2)sqrt(5.5)+6)/2

 

[jvicens]

Thank you for your interesting question about finding a line that divides a 2-dimensional polygon into two sections with equal area. This type of problem falls under the field of computational geometry and there are several algorithms that can help you achieve this goal.

One approach is to use the centroid of the polygon to find the line of symmetry. The centroid is the point at which the polygon would balance if it were cut out of a sheet of uniform density material. To find the centroid, you can use the shoelace method or any other method for finding the center of mass.

Once you have the centroid, you can draw a line passing through it that divides the polygon into two sections with equal area. This line will be perpendicular to the line connecting the centroid to any vertex of the polygon.

Another approach is to use a binary search algorithm. This involves drawing a line through the polygon and calculating the area of each section on either side of the line. If the areas are unequal, you can adjust the position of the line and recalculate until you find a line that divides the polygon into two sections with equal area.

There are also more complex algorithms, such as the sweep line algorithm, that can be used to find a line of symmetry for a 2-dimensional polygon.

I recommend researching these algorithms and determining which one best suits your needs. You may also find helpful resources by searching for terms such as "symmetry line algorithm" or "equal area division of polygon."

I hope this information helps in your search for a solution. Good luck!