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taby

Hexagon shape, even distribution of points over shape in Rn

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I am trying to determine a method of producing an homogenously spaced distribution of points, contained within a shape in Rn. Overlaying a honeycomb pattern over the shape produces a grid of vertices which are added to the final point list. The centre points of all "cells" within the honeycomb pattern are also added to the final point list. This essentially produces a tesselated face set from the honeycomb's hexagonally shaped faces. Does anyone know if this method has a particular name? [Edited by - taby on April 30, 2006 11:34:43 AM]

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Here is a diagram, where the dots are the final points in the distribution over a circle in R2.

The orange dots represent the first part of the honeycomb shape, using the black lines as the cell border guide.

The yellow dots represent the cell centre point vertices. This network forms a second honeycomb pattern using white lines. The position of these points is offset vertically by exactly 1/2 step from the orange honeycomb.

Does anyone know if this method has a particular name?



[Edited by - taby on April 30, 2006 9:54:32 PM]

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Using a square grid is the classic method, no question about that, but the distance between diagonal neighbours is not the same as between left-right / up/down neighbours (Pythagorean theorem).

With a hexagon, there are no left-right neighbours, but those which do exist up/down and diagonally are the same distance length.

It just seems more even of a distribution to me?

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Quote:
Original post by taby
Using a square grid is the classic method, no question about that, but the distance between diagonal neighbours is not the same as between left-right / up/down neighbours (Pythagorean theorem).

With a hexagon, there are no left-right neighbours, but those which do exist up/down and diagonally are the same distance length.

It just seems more even of a distribution to me?


Since you're using the center point of the hexagons it is really a triangular lattice, which is just a skewed square lattice. In other words, you should be able to perform a simple transformation on the Euclidean basis vectors to get a set of basis vectors for your lattice (called lattice vectors in crystallography, which you might find a useful source of information).

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When diffraction occurs at the surface, how does one determine the ratio of reflected to transmitted light? The Fresnel equations?

Do crystals have an index of refraction? Does the crystal alter the path of blue light more than that of red light?

Is it really as simple as redirecting the photons at the incoming and outgoing surface intersection points?



[Edited by - taby on April 30, 2006 9:15:18 PM]

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