# how to determine which cell in a grid a point belongs to

I have a square area which is divided into an N X N grid.I need to insert a point (x,y) into this area.I tried to find out if there is a relation between the value of N and x,y coordinates sothat I can say this particular point belongs in the cell (0,3) or some other.

I tried plotting on a graph paper, assuming origin at left bottom corner,and tried to find which cell the point P(5.5,4.5) belongs to. When N=2 , it seems that the point P should be in cell(1,1) .If N=6, the point P belongs in cell(5,4) of grid. I could not make out a relation between the values of N,coordinates of point and grid cell indices.Can someone please point me in the right direction?

It depends on the size of the square. If the square $S$ consists of the points having $0 \le x,y < a$ (that is $S = [0,a)^2$), then the cell $C(i,j)$ (with $0 \le i,j < N$) has $$C(i,j) = \left\{(x,y) \biggm| i\frac aN \le x < (i+1)\frac aN, \quad j \frac aN \le y < (j+1) \frac aN \right\}$$ That gives that $$i = \def\fl#1{\left\lfloor#1\right\rfloor} \fl{\frac{xN}a},\; j = \fl{\frac{yN}a}$$ gives the cell of the point $(x,y) \in S$.
So given a sample size $\Delta x$ and $\Delta y$ you take $(\lfloor\frac{x}{\Delta x}\rfloor+1,\lfloor\frac{y}{\Delta y}\rfloor+1)$, the +1 is because I do not think you are counting from 0. As an example take the unit square with points at (0,0) and (1,1) and give it 2 divisions in the x-direction and 3 divisions in y-direction. Take two points (0.6,0.4) and (0.21,0.798). Then $\Delta x=0.5$ and $\Delta y=\frac{1}{3}$ and your cell calculation becomes $(\lfloor\frac{0.6}{0.5}\rfloor+1,\lfloor\frac{0.4}{\Delta y}\rfloor+1)=(2,2)$. As for the second point $(\lfloor\frac{0.21}{0.5}\rfloor+1,\lfloor\frac{0.798}{\Delta y}\rfloor+1)=(1,3)$.