- Source: Henneberg surface
In differential geometry, the Henneberg surface is a non-orientable minimal surface named after Lebrecht Henneberg.
It has parametric equation
x
(
u
,
v
)
=
2
cos
(
v
)
sinh
(
u
)
−
(
2
/
3
)
cos
(
3
v
)
sinh
(
3
u
)
y
(
u
,
v
)
=
2
sin
(
v
)
sinh
(
u
)
+
(
2
/
3
)
sin
(
3
v
)
sinh
(
3
u
)
z
(
u
,
v
)
=
2
cos
(
2
v
)
cosh
(
2
u
)
{\displaystyle {\begin{aligned}x(u,v)&=2\cos(v)\sinh(u)-(2/3)\cos(3v)\sinh(3u)\\y(u,v)&=2\sin(v)\sinh(u)+(2/3)\sin(3v)\sinh(3u)\\z(u,v)&=2\cos(2v)\cosh(2u)\end{aligned}}}
and can be expressed as an order-15 algebraic surface. It can be viewed as an immersion of a punctured projective plane. Up until 1981 it was the only known non-orientable minimal surface.
The surface contains a semicubical parabola ("Neile's parabola") and can be derived from solving the corresponding Björling problem.
References
Further reading
E. Güler; Ö. Kişi; C. Konaxis, Implicit equations of the Henneberg-type minimal surface in the four-dimensional Euclidean space. Mathematics 6(12), (2018) 279. doi:10.3390/math6120279.
E. Güler; V. Zambak, Henneberg's algebraic surfaces in Minkowski 3-space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68(2), (2019) 1761–1773. doi:10.31801/cfsuasmas.444554.
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