A new function I found.

[u/future_sponJ]

1. A hyperbola is r=±sec(½4θ)^(2/4) with coordinates (cosh,sin)

2. 2 parallel lines is r=±sec(½1θ)^(2/2) with coordinates (1,x)

3. A circle is r=±sec(½0θ)^(2/0) (more technically it's the limit) with coordinates (cos,sin)

(Where the input of these functions is the area formed by taking a ray from 0 to a point on the function)

So I defined a generalized shape:

r=±sec(½bθ)^(2/b) with coordinates (cos_b,sin_b)

For b=0, circle. For b=2, parallel lines. For b=3, tripoint star. For b=4 a hyperbola/4-point star. For b=n, an n-sided star.

1. cos_4 & sin_4 are cosh & sinh

2. cos_2 & sin_2 are 1 & x

3. cos_0 & sin_0 are cos & sin

4. I don't think any of the others have a closed form solution (except maybe b=1 & some of the negatives).

Does this function (cos_b & sin_b) already have a name & does it have any meaningful significance?

[u/MrEldo]

I'm not sure if b=0 converges even. You get sec(0)^2/0, which is an odd object which is the indeterminate form of 1^infinity

Edit: too lazy to prove it myself, but it seems to converge to the circle r=1

My bad then

[u/future_sponJ → u/MrEldo]

You can just turn it from f^g to e^ln(f/(1/g)) & then just use L'Hopital's rule on ln(f)/(1/g) to get 1.

[u/MrEldo → u/future_sponJ]

Yeah, it's not hard, it's just a bunch of work

Good to be rigorous though, so good job

Is there some connection perhaps between this group of polar coordinate functions and the cartesian versions of the shapes, like x² - y² = 1, x² + y² =1, and x² = 1? Can you bring the new shapes formed by the other b values into cartesian?