Vrftsjtryk

I'm implementing Hough Transform in MatLab to detect straight lines in an Image. It uses normal form of a straight line equation
$$x cos(theta) + y sin(theta) = rho$$

I'm interested in the lines which pass through the 1st quadrant only, that too in the rectangle with vertices $(0,0), (M,0), (0,N), (M,N)$ i.e. any line that doesn't pass through this rectangle, I'm not interested in finding the equation of that line. Orientation of the line can be anything. What is the range of values for the parameters $rho$ and $theta$?

If I assume $theta$ varies from $-90^text{o}$ to $90^text{o}$, what is the range of values for $rho$? Can $rho$ be negative?

As depicted in the graph we have better split for negative anf for positive values of $theta$.

Thus we must have
$$
left{ {matrix{
{left{ matrix{
- rho /cos theta le M hfill cr
- rho /sin theta le N hfill cr} right.} hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
{left{ matrix{
rho /cos theta - Ntan theta le M hfill cr
0 le rho hfill cr} right.} hfill & {left| {;0 < theta < pi /2} right.} hfill cr
} } right.
$$
that is
$$
left{ {matrix{
{ - Mcos theta le rho le - Nsin theta } hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
{0 le rho le Mcos theta + Nsin theta } hfill & {left| {;0 < theta < pi /2} right.} hfill cr
} } right.
$$