Define boundary conditions of a polygon in a unit square cell

In summary, the speaker is asking how to define the boundary condition for a shape in a unit cell, specifically for complex polygons like hexagons. The suggested method is to use simple linear equations for convex polygons and a piecewise defined curve for more complex shapes.
  • #1
tomallan
20
0
Hi,

I am wondering as to how to define the boundary condition for a shape in a unit cell. Just imagine that the shape is the hole for the unit cell. Hence, for a constant thickness on the untextured boundary, thickness is, let's say C, then for the circle, it's C+depth.


For example for a unit cell with width 2r1 by 2r1 with a circle inside the unit cell, the boundary condition is:

h(x,y) = C+depth, if x^2+y^2<r^2, and
h(x,y) = C, if x^2+y^2>r^2.

I am wondering how to do this with complicated polygons, like hexagon and stuff or if there is a formula or principle that i don't know of.

Thank you heaps.
 

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  • #2
Each of the sides of a polygon is definable by a simple linear equation in x and y. The interior of a convex polygon is the intersection of the half-planes defined by the corresponding inequalities.
 
Last edited:
  • #3
For convex polygons, the most economic way would probably be to restrict the boundary conditions to particular angles. For more complex polygons, a piecewise defined curve is probably the best way.
 

1. What is a polygon?

A polygon is a two-dimensional shape with straight sides that are connected to form a closed figure. Examples of polygons include triangles, squares, and hexagons.

2. What are boundary conditions?

Boundary conditions refer to the constraints or rules that define the behavior of a system at its edges or boundaries. In the case of polygons in a unit square cell, boundary conditions determine how the polygon interacts with the edges of the square.

3. How are boundary conditions defined for a polygon in a unit square cell?

Boundary conditions for a polygon in a unit square cell are typically defined by specifying the coordinates of the polygon's vertices. These vertices must lie on the edges of the square and can be connected in any order to form the polygon.

4. Can there be multiple boundary conditions for a polygon in a unit square cell?

Yes, there can be multiple boundary conditions for a polygon in a unit square cell. For example, a polygon can have different boundary conditions at each of its vertices, such as different angles or lengths for each side.

5. How do boundary conditions affect the behavior of a polygon in a unit square cell?

Boundary conditions determine how a polygon interacts with the edges of the unit square cell. They can affect the polygon's shape, size, and orientation within the cell. Different boundary conditions can result in different behaviors, such as the polygon being able to move freely within the cell or being constrained to certain movements.

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