# Grid translation

I have a 6x6 grid, and in its first cell (row 1, column 1), its value is (-3, 2) and on its last cell (row 6, cell 6), its value is (2, -3).

Another values inside this grid are:

$$(x_0, y_0) => (x_1, y_1)$$

(1, 2) => (-2, 2)
(1, 3) => (-1, 2)
(1, 4) => (0, 2)
(1, 5) => (1, 2)
(1, 6) => (2, 2)
(2, 1) => (-3, 1)
(3, 1) => (-3, 0)
(4, 1) => (-3, -1)
(5, 1) => (-3, -2)
(6, 1) => (-3, -3)


$$(x_0, y_0)$$ are the row, column of the grid, and $$(x_1, y_1)$$ are the values on each cell.

Is there any formula to translate $$(x_0, y_0)$$ into $$(x_1, y_1)$$?

Maybe the grid will be bigger than 6x6 (or smaller...). All the grids will be nxn., and the first cell will be always (-n/2, n/2 - 1).

• What are the values at the other cells in the grid? – saulspatz Jul 11 at 20:15
• I don't quite understand what you mean or what you want. I think $(x_0, y_0)$ are the coordinates of a cell, and $(x_1, y_1)$ the value there. If so, then just make the $x$ values constant on the columns and the $y$ values constant on the rows. If that's not what you want, please edit to clarify. Perhaps tell us where the problem comes from. – Ethan Bolker Jul 11 at 20:20
• Can $n$ be odd? If so, do we round $n/2$ up, round it down, use the exact value (not an integer), or something else? – David K Jul 12 at 4:06
• No, $n$ can not be odd. – VansFannel Jul 12 at 5:13

## 1 Answer

Based on the data you provided, for the $$6\times 6$$ grid the formulas would be \begin{align} x_1 &= y_0 - 4, \\ y_1 &= 3 - x_0. \end{align}

Based on the information that the grid is always square and that for an $$n\times n$$ grid, the cell at $$x_0=y_0=1$$ has $$x_1 = -\frac n2$$ and $$y_1 = \frac n2 - 1,$$ the general formulas are \begin{align} x_1 &= y_0 - \left(\frac n2 + 1\right), \\ y_1 &= \frac n2 - x_0. \end{align}

Note that $$\frac n2$$ is not an integer if $$n$$ is odd. If $$n$$ is always even, however, then $$x_1$$ and $$y_1$$ will be integers.

The clues that lead to these formulas are that $$x_1$$ is constant when $$y_0$$ is constant, $$y_1$$ is constant when $$x_0$$ is constant, $$x_1$$ increases when $$y_0$$ increases, and $$y_1$$ decreases when $$x_0$$ increases. This tells us we need formulas $$x_1 = y_0 + a$$ and $$y_1 = b - x_0$$ for some constants $$a$$ and $$b.$$ Then use the values of $$x_1$$ and $$y_1$$ when $$x_0=y_0=1$$ to figure out what $$a$$ and $$b$$ must be.

• All the grids will be squares. – VansFannel Jul 11 at 20:31
• Thanks a lot for answer and explanations. Maybe, it could be generalized if we know the value inside the first cell, by the way the first cell is (1, 1); and we know the number of rows and columns of the grid (the number of rows doesn't have to be the same as the number of columns). But I think this is another answer. – VansFannel Jul 12 at 12:30