Generalization of a metric space

Question

I am considering the complex plane $\mathbb{C}$ equipped with a smooth function $f:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{R}$ such that

• $f(x,y) = f(y,x)$
• $f(x,y)\geq 0$
• $f(x,x)\neq 0$

for all $ x,y\in\mathbb{C} .$

Now I consider the curve that contains the points equidistant from given two points $x$ and $y$, i.e., $$\mathcal{C} = \{z: z\in\mathbb{C},\ f(x,z)=f(y,z)\}.$$

I need find (if it exist) a transformation $g:\mathbb{C}\rightarrow\mathbb{C}$ such that $g(\mathcal{C})$ is a straight line.

Any idea, what keywords to use to search for the appropriate math literature?

I understand that it is some kind of generalization of a metric space, but none of the generalizations that I could find exactly fits.