I've been studying Hough transform. Basically, let's say we have a line
$$y = mx+b$$
We can change our view to a parametric view (e.g. parameter space of $m,b$ while $(x,y)$ is constant). This would give us all possible lines that will intercept with that point constant. Having two lines in the parameter space intercept, their point of intersection is the line ($x$-$y$ space) that is collinear to the two points (represent as lines in the parameter space).
To avoid having infinite slope, polar coordinate representation was used such that a line is defined by $(\rho, \theta)$. The line in the $x$-$y$ space is perpendicular to this vector.
Now, the mathematical definition of their relation is defined below. I got two sources. Both which contradict each other. What is the right one? Please provide also proof. Thanks.
Best Answer
$(\cos \theta, \sin \theta)$ is a point on a unit circle. $(r \cos \theta, r \sin \theta)$ is a point on circle of radius $r$. Notice that if $X = r \cos \theta$ and $Y = r \sin \theta$, then $$X \cos \theta + Y \sin \theta = r \cos ^2 \theta + r \sin^2 \theta = r(\sin^2 \theta + \cos^2 \theta) = r$$