# Define boundary conditions of a polygon in a unit square cell

1. Aug 24, 2014

### tomallan

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, lets 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 dont know of.

Thank you heaps.

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2. Aug 24, 2014

### jbriggs444

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: Aug 24, 2014
3. Aug 24, 2014

### disregardthat

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.