Say I have a rectangle, with variable width and height, for example lets use:
width = 20
height = 30
I would like to put n amount of evenly spaced points inside this rectangle:
no of points = 400
How could I calculate the x and y coordinates of each point? Note, that I would like the borders to also have points.
Very rough example, I needed 12 points (but I could have wanted more or less):
Looking for a square grid of size $w \times h$ with $n$ points in total.
$n_x$ points in $x$-direction, spacing $\Delta x$: $$ x_i = i \, \Delta x \quad i \in \{ 0, \ldots, n_x-1 \} $$ and $$ (n_x -1) \, \Delta x = w $$
$n_y$ points in $y$-direction, spacing $\Delta y$: $$ y_j = j \, \Delta y \quad j \in \{ 0, \ldots, n_y-1 \} $$ and $$ (n_y -1) \, \Delta y = h $$
Total number of points: $$ n = n_x \, n_y $$
Assuming $\Delta x = \Delta y$ one gets $$ \Delta x = \frac{w}{n_x - 1} = \frac{h}{n_y - 1} = \Delta y \iff \\ n_y = \frac{h}{w} n_x + 1 - \frac{h}{w} $$ and then $$ n = n_x n_y = \frac{h}{w} n_x^2 + \left( 1 - \frac{h}{w} \right) n_x \iff \\ \frac{w}{h} n + \frac{(w-h)^2}{4h^2} = \left( n_x + \frac{w-h}{2h} \right)^2 $$ which after taking the square root gives the wanted equation for $n_x$ in terms of $n$, $w$ and $h$: $$ n_x = \sqrt{\frac{w}{h} n + \frac{(w-h)^2}{4h^2}} - \frac{w-h}{2h} \quad (*) $$ (the negative solution was dropped). $n_x$ then should be put into $n_y = n / n_x$ and those values can be used to calculate the spacing $\Delta x = \Delta y$.
The interesting bit is that not every integer $n$ in $(*)$ will lead to an integer $n_x$ and then to another integer $n_y$.
Example 1
Given are $n = 12$, $w = 3$, $h = 2$. We put this in equation $(*)$ and get $$ n_x = \sqrt{\frac{3}{2}\cdot 12 + \frac{(3-2)^2}{4\cdot 2^2}} - \frac{3-2}{2\cdot 2} = \sqrt{18 + \frac{1}{16}} - \frac{1}{4} = \frac{17}{4} - \frac{1}{4} = 4 $$ so we have $n_x = 4$ points along the $x$-direction. Further $n_y = n / n_x = 12 / 4 = 3$ points in $y$-direction. The spacing is $\Delta x = w / (n_x - 1) = 3 / (4 - 1) = 1$, a unit spacing and $\Delta y$ is the same.
Example 2
Given are $n = 400$, $w = 20$, $h = 30$. From equation $(*)$ we get $n_x = 16.4974487803453$ and then $n_y = 24.2461731705179$. So this won't work.
But why not try $n = 17 \cdot 25 = 425\,$ with the given $w$ and $h$ values? And this is indeed confirmed by equation $(*)$. Here $\Delta x = \Delta y = 1.25$.
I hope it a got a little bit clear that not all combinations of the problem parameters $n$, $w$ and $h$ have a solution.