In full isometry we have differential lengths and angles between them both conserved for 2D surfaces in $\ \mathbb R^3$.
In partial isometry like in case of Chebychev Net "rhombus" lengths are conserved in a "rhombic" net parts but angles can change or distort, keeping Gauss double curvature K constant. Sine-Gordon Equation is obeyed that describes the changing angle and arc length relation.
In the remaining possibility can we in any mapping having angles conserved (conformal) but lengths varying? Are there any examples? Is there any differential equation (pde or ode) of the mapping that describes the arc length variation and differential angle?
Thanks!