# Implicit equation of the limacon

I am studying from the textbook Elementary Differential Geometry by AN Pressley (2nd edition).

At the end of the first chapter the Author discusses the relationship between level curve and parameterised curves, he provides the following theorem on when you can go from a parameteric curve to a level curve:

Let γ be a regular parametrized plane curve, and let $$γ(t_0) = (x_0, y_0)$$ be a point in the image of γ. Then, there is a smooth real-valued function f(x,y), defined for x and y in open intervals containing $$x_0$$ and $$y_0$$, respectively, and satisfying the conditions in Theorem 1.5.1, such that $$γ(t)$$ is contained in the level curve $$f(x,y) = 0$$ for all values of $$t$$ in some open interval containing $$t_0$$.

Where the conditions in Theorem 1.5.1 is that the function is smooth and such that its partial derivatives are not both zero.

My confusion is that in the discussion of these theorems the Author says that "...it is not in general possible to find a single function $$f(x,y)$$ satisfying the conditions in Theorem 1.5.1 such that the image of γ is contained in the level curve $$f(x,y) = 0$$, for γ may have self-intersections like the limacon."

However on wikipedia the following function is presented for the limacon:

$$(x^2 + y^2 - ax)^2 = b^2(x^2 + y^2)$$

So I am a bit confused why the Author seems to imply that no level curve can represent the limacon.

I don't have the book, so I can't look. But I imagine that an additional condition is imposed in Theorem 1.5.1: At every point $$(x_0,y_0)$$ of the level curve $$f(x,y)=0$$, we have $$\nabla f(x_0,y_0)\ne 0$$. What happens at $$(x_0,y_0)=(0,0)$$ with your function?