Conformal mappings that preserve angles and areas but not perimeters?

Conformal mappings from $$U$$ to $$V$$, both subsets of $$\mathbb{C}$$, locally preserve angles. But, in general, such mappings neither preserve areas nor preserve perimeters.

Q. Are there examples of analytic conformal mappings that preserve areas but not perimeters? Or vice versa?

The conformal mapping $$w=z^2$$ in rectangular coordinates: (John Mathews.)

• regarding deformations that preserve areas and perimeters, my question Work on “Churning Polygons” and especially the accepted answer may also shed some light on the problem – Manfred Weis Oct 7 '18 at 11:54

Either condition implies $$|\,f'(z)| = 1$$ for all $$z$$ in the domain of $$\,f$$, which in turn implies that $$\,f'(z)$$ is locally constant (for instance using the open mapping theorem) and thus that $$\,f$$ is a Euclidean isometry on each connected component of its domain.
See this handout where Brian Conrad talks about the mapmaker's paradox. A mapmaker would like to draw a map that is area preserving and conformal, however a $$C^{\infty}$$ isomorphism between Riemannian manifolds with corners is conformal and volume preserving if and only if it is an isometry (theorem 2.4)! The case of perimeter preserving conformal maps was discussed in an older question MO172764. Therefore cartographers make do with maps that are conformal but not area preserving, or nonconformal and area preserving, like examples given in the handout.