Regarding polar substitution meaning in coordinate plane or cartesian plane.

Question

There are two kinds of polar substitution which we generally talk of:-

Type I: $x=r\cos\theta$, $y=r\sin\theta$

Type II: $x=r\sec\theta$, $y=r\tan\theta$

But how do we represent this in cartesian plane like we did for Type-I

Answer 1

The conventional polar coordinate system is, in a sense, based on circles. That is, the sets of points for which $r=c$ for some fixed $c\in\mathbb R\setminus\{0\}$ is a circle of radius $c$. We generally take $r\ge0$ so that every point other than the origin has unique coordinates, but it’s also useful to relax that restriction. Note that there’s still a problem at the origin even if we don’t allow negative values of $r$: $\theta$ can have any value. So, the $r$-coordinate tells you on which circle the point lies, and the $\theta$-coordinate the distance around the circle, i.e., the intersection of the ray that makes an angle of $\theta$ with the $x$ axis with that circle:

Now, let’s look at the curves for which $r$ is constant in your second coordinate system:

These are hyperbolas with asymptotes $x=\pm y$. Now instead of $r$ being the radius of a circle, it’s the semimajor axis length of the hyperbola on which the point lies. It’s a bit harder to come up with a geometric interpretation of the angle $\theta$. If you construct a right triangle with a leg of length $r$ along the $x$-axis and hypotenuse along the ray that makes and angle of $\theta$ with the $x$-axis, then the other leg of the triangle has length $r\tan\theta$, which is the $y$-coordinate of our point. The length of the hypotenuse of this triangle is equal to $r\sec\theta$, our $x$-coordinate. This is also the $x$-intercept of the tangent to the circle of radius $r$ at its intersection with the ray.

Note that this coordinate system has even more problems than the first one. Besides having the same sign ambiguity and issues at the origin, there is no way to represent points for which $\lvert y\rvert\ge\lvert x\rvert$.