General equation for a grid's internal nodes

Mr.Plough

I wasn't sure whether to put this in programming or mathematics but I guess as It's just finding a general equation it gos here. Anyway


If I have an NxM grid like so (N = 6 and M = 5 in this case)

x x x x x x

x x x x x x

x x x x x x

x x x x x x

x x x x x x


Where N = no. of columns
M = no. of rows

I need an equation of the form Q = f(N,j,i)

Where j is the column number and i is the row number

and Q is the number of an internal node


For this one, if j = 2 and i = 2, Q = 1 as shown below



x x x x x x

x 1 2 3 4 x

x 5 6 7 8 x

x x x x x x

The general equation is

Q = (j-1)*N + i + c

c for 6x5 happens to be = -N - i + 1


What I need to do is find a general equation that will calculate the internal node numbers as a function of j,N,i,M (not sure if you use M) that will still work if I change N and M

I'm pretty sure only c changes as the no. of columns and rows increase so I guess an expression for c is what I need

for example I need an equation which will work for not just a 6x5 but for any combination of rows and columns. (They'll be getting larger)

I need it to work no matter what N and M are. There could be 71 columns and 51 rows


Is it possible to do this? Any help would be great my brain's fried from sitting in front of matlab for hours and can't think straight.

Cheers

MathsManiac

Q=(i-2)*(N-2)+(j-1)

Mr.Plough

Great stuff thanks. As a matter of interest how did you come to that eqn?

MathsManiac

If it weren't for the X's around the edge, then to get to the Qth position you would count (i-1) rows of N and then go across (i.e. add) j. [That would give Q= (i-1)*N + j.] The X's on the left and right of every row means your count has jumped to a new row on every N-2 instead of every N, and the row that you're not counting at the start means you need i-2 instead of i-1. Then, on the row you're actually in, the fact that you've an X at the start means you need j-1 instead of j.

Finally, you check that it works!