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Manhattan Intersections. Number within a Boundary and Number on the Boundary The N ( r ) and B ( r ) Formulas. This is the Manhattan two-way grid with the distance from the origin to each intersection shown. Our task is to count the number of intersections within the boundary .

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manhattan intersections

Manhattan Intersections

Number within a Boundary and Number on the Boundary

The N(r) and B(r) Formulas

Jeff Rose - MAE 6127

slide2

This is the Manhattan two-way grid with the distance from the origin to each intersection shown. Our task is to count the number of intersections within the boundary

Here are the first few counts:

Jeff Rose - MAE 6127

slide3

Each radius forms a rhombus pattern that is rotated 45° about the origin with respect to the horizontal. From a previous solution, the area of this rhombus is 2r 2.

For example, if r = 3, then A = 2(3)2 = 2(9) = 18

Jeff Rose - MAE 6127

slide4

However, when looking for a pattern in the intersection count problem, I find it easier to look at it as four congruent right triangles with one point leftover at the origin.

Each triangle has one row with r intersections and subsequent rows with one fewer intersection all the way to the last row - which has 1 intersection

Jeff Rose - MAE 6127

slide5

With compliments to Gauss, the sum of this arithmetic sequence is

That means the number of intersections in each triangle is r + (r - 1) + (r - 2) + ... 3 + 2 + 1

Jeff Rose - MAE 6127

slide6

So, the number of intersection points enclosed by radius r, N(r), is four times the number in each triangle - plus 1 for the origin

Jeff Rose - MAE 6127

slide7

To count the number of intersections that are exactly r distance from the origin, we go back to the rotated rhombus. Here’s a chart that shows the boundary intersections:

Jeff Rose - MAE 6127

slide8

The formula for B(r) is pretty straight forward from the chart. When r = 0, the number of boundary intersections is 1 - otherwise, it’s 4 * r

B(0) = 1

B(r) = 4r

Jeff Rose - MAE 6127

slide9

The last part of the problem is to investigate what happens to N(r) when r gets very large. This chart shows some values of N(r) and A(r), as well as the ratio of the two:

As r gets larger, the ratio of N(r) to A(r) goes to 1. Only the highest order term of the polynomials will matter for large values of r. Both formulas have 2r2 as the highest order term - which means

r   N(r) = A(r)

Jeff Rose - MAE 6127