A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings

Table 1

Images of 𝑓_𝑐,𝜃(|𝑧|=𝑟), 𝑟=0.4,0.6,0.8,0.9999, and the corresponding minimal surfaces.

𝑐=0	𝜃=0	𝜃=𝜋/2	𝜃=𝜋
Surface	Helicoid	Associated surface	Catenoid
𝑐=0.5	𝜃=0	𝜃=𝜋/2	𝜃=𝜋
Surface
𝑐=1	𝜃=0	𝜃=𝜋/2	𝜃=𝜋
Surface	Wavy plane	Associated surface	Wavy plane
𝑐=1.5	𝜃=0	𝜃=𝜋/2	𝜃=3𝜋/4
Surface
𝑐=2	𝜃=0	𝜃=𝜋/20	𝜃=𝜋/2
Surface	Enneper’s surface	Enneper’s surface	Enneper’s surface