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Define boundary conditions of a polygon in a unit square cell

  1. Aug 24, 2014 #1

    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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  3. Aug 24, 2014 #2


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    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
  4. Aug 24, 2014 #3


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    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.
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