Helicoid transformation Catenoid

deformation of helicoid catenoid

because members of same associate family of surfaces, 1 can bend catenoid portion of helicoid without stretching. in other words, 1 can make (mostly) continuous , isometric deformation of catenoid portion of helicoid such every member of deformation family minimal (having mean curvature of zero). parametrization of such deformation given system

x
(
u
,
v
)
=
cos
⁡
θ

sinh
⁡
v

sin
⁡
u
+
sin
⁡
θ

cosh
⁡
v

cos
⁡
u


{\displaystyle x(u,v)=\cos \theta \,\sinh v\,\sin u+\sin \theta \,\cosh v\,\cos u}








y
(
u
,
v
)
=
−
cos
⁡
θ

sinh
⁡
v

cos
⁡
u
+
sin
⁡
θ

cosh
⁡
v

sin
⁡
u


{\displaystyle y(u,v)=-\cos \theta \,\sinh v\,\cos u+\sin \theta \,\cosh v\,\sin u}








z
(
u
,
v
)
=
u
cos
⁡
θ
+
v
sin
⁡
θ


{\displaystyle z(u,v)=u\cos \theta +v\sin \theta }




for



(
u
,
v
)
∈
(
−
π
,
π
]
×
(
−
∞
,
∞
)


{\displaystyle (u,v)\in (-\pi ,\pi ]\times (-\infty ,\infty )}

, deformation parameter



−
π
<
θ
≤
π


{\displaystyle -\pi <\theta \leq \pi }

,

where



θ
=
π


{\displaystyle \theta =\pi }

corresponds right-handed helicoid,



θ
=
±
π

/

2


{\displaystyle \theta =\pm \pi /2}

corresponds catenoid, ,



θ
=
0


{\displaystyle \theta =0}

corresponds left-handed helicoid.