Polar coordinates of a point P

In the plane we choose a fixed point O, and we call it the pole.
Additionally we choose an axis x through the pole and call it the polar axis.
On that x-axis, there is just 1 vector E such that abs(E)=1.
The pole and the polar axis constitute the basis of the polar coordinate system.

Now, we take a point P.
On the line OP we choose an axis u.
The number t is a value of the angle from the x-axis to the u-axis.
The number r is such that P = r.U
The numbers r and t define unambiguous the point P.
We say that (r,t) is a pair of polar coordinates of P.

The polar coordinates of the pole O are by definition (0,t) with t perfectly arbitrary.

Consider a connection between the polar coordinates of a point and suppose, that connection can be expressed in the form
F(r,t)=0 or maybe in the explicit form r = f(t).
Such equation is a polar equation of a curve.
With each solution (ro,to) of the polar equation, corresponds a point with polar coordinates (ro,to). Generally the equation has
an infinity number of such solutions and so, we have an infinity number of points. The set of all these points is the curve of the
equation.

Each point P of that curve has at least one pair of polar coordinates who satisfy the equation. Note that, in general, not all pairs
of polar coordinates of P are solutions of the equation.

Note that one curve can have different polar equations.

Click here for the examples

Let's investigate .

How about when a=b? The following graph shows the equations when k varies from 1 to 3.

How about the graph of various a, b(a=b)? The following graph shows the equations when k=1.

We can see that the bigger a and b are, the wider the width. If so how about if a and b are 10, 50 or 100. You can click here to match your thought of the graph.

Let's think about the curve which has the shape of a petalled flower. Compare the following two equation .

If k is odd, the rose is k-petalled. If k is even, the rose is 2k-petalled. If k is irrational, then there are an infinite number of petals.

Now investigate the orginal starting equation .

When a and b are equal, and k is an integer, this becomes the "n-petalled rose." Click here to check the graph of the variou k.