Grids

Square Spiral Grid

<TODO>

Given n find (i,j). And inverse.
Given n and direction k={0,1,2,3}, find n' after moving 1 square in k direction.
Investigate whether this can be used for general rectangles. eg are there > and < operators which can bound the edges in this space?

It is called the spiral function and defined differently such that 1 goes up instead of left. I did the problem this way before finding a reference in "Concrete Mathematics" page 99, which is a better definition because the numbers are rotated -90 degrees from mine. The 4 regions which the numbers are partitioned into now are in line with the axes and correspond to the 4 quadrants. This simplifies the formula.

Problem : given (i,j) as normal grid points find (i,j+1) ie move up. Process: (i,j) corresponds to n, n -> n' where (i,j+1) corresponds to n'.

By looking at the grid the patterns can be found by considering horizontal and vertical number sequences. Divide the grid into 4 regions about the origin. Consider the bottom triangle.

$2 = 1 + 1 + 0$
$13 = 9 + 3 + 1$
$32 = 25 + 5 + 2$
$59 = 49 + 7 + 3$
$n = j^{2} + j + k$
where $< n, k >$ forms its own natural coordinate system.

Formula for the other regions can be constructed and joined together for a solution.

Hex Grid

Introducing the hex grid coordinate system.

"C++ Math Class Library - Permutations, Partitions, Calculators & Gaming" by Scott N. Gerard provides a chapter on the hex grid. The midpoint of a hexagon grid is essentially a rectangular array of points.

Taking the midpoint of a hexagon grid produces a grid of triangles which can be bent to a rectangular array. For the reverse take the midpoints of 5 triangles about a vertex and join them to form a hexagon, repeating generates the hexagon grid. The representations are equivalent and I believe are called each others duals.

Scott Gerard explains the importance in games and explains the natural 3D hex coordinate system. The coordinate system is non-unique because $i + j + k = 0$ .

$(u, v, w) = (u + k, v + k, w + k)$

To make the coordinate system unique again an algorithm is employed that simply subtracts the middle value of i,j,k.

I have also found a way of mapping this coordinate system to a rectangular array.

$(i, j, k) \mapsto (i - k + (j - k < - 1) - (j - k > 0), j - k)$

Linked List Grids

The cost of these date structures is obvious - many links per element. This can be offset by hierarchical structures where one of the elements can be a pointer to a regular grid with many points and much more memory efficient data structure.

Their generality is less well known. From one of these elements you can choose a direction or path independent of the underlying memory structure. This allows transversal algorithms to be written. So surfaces can be more easily transversed - an alternative to the maths group theory approach. Its often much harder to analyze a surface than it is to describe it. By describing it you can write something to transverse it....

Writing algorithms using the natural data structure (eg triangle has 3 links for triangulated surface) is good. I hope to explore this much further.

Higher Dimensions