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Figure 1. The Enneper surface Figure 2. The Enneper surface in Figure 1 has been rotated 30° around the +z axis. Figure 3. The Enneper surface in Figure 1 has been rotated 60° around the +z axis. In mathematics , in the fields of differential geometry and algebraic geometry , the Enneper surface is a self-intersecting surface that can be described parametrically by:
x
=
u
(
1
−
u
2
/
3
+
v
2
)
/
3
,
{\displaystyle x=u(1-u^{2}/3+v^{2})/3,\ }
y
=
−
v
(
1
−
v
2
/
3
+
u
2
)
/
3
,
{\displaystyle y=-v(1-v^{2}/3+u^{2})/3,\ }
z
=
(
u
2
−
v
2
)
/
3.
{\displaystyle z=(u^{2}-v^{2})/3.\ }
It was introduced by Alfred Enneper 1864 in connection with minimal surface theory. [ 1] [ 2] [ 3] [ 4]
The Weierstrass–Enneper parameterization is very simple,
f
(
z
)
=
1
,
g
(
z
)
=
z
{\displaystyle f(z)=1,g(z)=z}
conjugate to itself.
Implicitization methods of algebraic geometry can be used to find out that the points in the Enneper surface given above satisfy the degree-9 polynomial equation
64
z
9
−
128
z
7
+
64
z
5
−
702
x
2
y
2
z
3
−
18
x
2
y
2
z
+
144
(
y
2
z
6
−
x
2
z
6
)
{\displaystyle 64z^{9}-128z^{7}+64z^{5}-702x^{2}y^{2}z^{3}-18x^{2}y^{2}z+144(y^{2}z^{6}-x^{2}z^{6})\ }
+
162
(
y
4
z
2
−
x
4
z
2
)
+
27
(
y
6
−
x
6
)
+
9
(
x
4
z
+
y
4
z
)
+
48
(
x
2
z
3
+
y
2
z
3
)
{\displaystyle {}+162(y^{4}z^{2}-x^{4}z^{2})+27(y^{6}-x^{6})+9(x^{4}z+y^{4}z)+48(x^{2}z^{3}+y^{2}z^{3})\ }
−
432
(
x
2
z
5
+
y
2
z
5
)
+
81
(
x
4
y
2
−
x
2
y
4
)
+
240
(
y
2
z
4
−
x
2
z
4
)
−
135
(
x
4
z
3
+
y
4
z
3
)
=
0.
{\displaystyle {}-432(x^{2}z^{5}+y^{2}z^{5})+81(x^{4}y^{2}-x^{2}y^{4})+240(y^{2}z^{4}-x^{2}z^{4})-135(x^{4}z^{3}+y^{4}z^{3})=0.\ }
Dually, the tangent plane at the point with given parameters is
a
+
b
x
+
c
y
+
d
z
=
0
,
{\displaystyle a+bx+cy+dz=0,\ }
a
=
−
(
u
2
−
v
2
)
(
1
+
u
2
/
3
+
v
2
/
3
)
,
{\displaystyle a=-(u^{2}-v^{2})(1+u^{2}/3+v^{2}/3),\ }
b
=
6
u
,
{\displaystyle b=6u,\ }
c
=
6
v
,
{\displaystyle c=6v,\ }
d
=
−
3
(
1
−
u
2
−
v
2
)
.
{\displaystyle d=-3(1-u^{2}-v^{2}).\ }
Its coefficients satisfy the implicit degree-6 polynomial equation
162
a
2
b
2
c
2
+
6
b
2
c
2
d
2
−
4
(
b
6
+
c
6
)
+
54
(
a
b
4
d
−
a
c
4
d
)
+
81
(
a
2
b
4
+
a
2
c
4
)
{\displaystyle 162a^{2}b^{2}c^{2}+6b^{2}c^{2}d^{2}-4(b^{6}+c^{6})+54(ab^{4}d-ac^{4}d)+81(a^{2}b^{4}+a^{2}c^{4})\ }
+
4
(
b
4
c
2
+
b
2
c
4
)
−
3
(
b
4
d
2
+
c
4
d
2
)
+
36
(
a
b
2
d
3
−
a
c
2
d
3
)
=
0.
{\displaystyle {}+4(b^{4}c^{2}+b^{2}c^{4})-3(b^{4}d^{2}+c^{4}d^{2})+36(ab^{2}d^{3}-ac^{2}d^{3})=0.\ }
The Jacobian , Gaussian curvature and mean curvature are
J
=
(
1
+
u
2
+
v
2
)
4
/
81
,
{\displaystyle J=(1+u^{2}+v^{2})^{4}/81,\ }
K
=
−
(
4
/
9
)
/
J
,
{\displaystyle K=-(4/9)/J,\ }
H
=
0.
{\displaystyle H=0.\ }
The total curvature is
−
4
π
{\displaystyle -4\pi }
R
3
{\displaystyle \mathbb {R} ^{3}}
−
4
π
{\displaystyle -4\pi }
catenoid or the Enneper surface.[ 5]
Another property is that all bicubical minimal Bezier surfaces are, up to an affine transformation , pieces of the surface. [ 6]
It can be generalized to higher order rotational symmetries by using the Weierstrass–Enneper parameterization
f
(
z
)
=
1
,
g
(
z
)
=
z
k
{\displaystyle f(z)=1,g(z)=z^{k}}
[ 3]
R
n
{\displaystyle \mathbb {R} ^{n}}
[ 7]
References
^ J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975)
^ Francisco J. López, Francisco Martín, Complete minimal surfaces in R3. [1]
^ a b Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny (2010). Minimal Surfaces. Berlin Heidelberg: Springer. ISBN 978-3-642-11697-1.
^ Weisstein, Eric W. "Enneper's Minimal Surface" . MathWorld ^ R. Osserman, A survey of Minimal Surfaces. Vol. 1, Cambridge Univ. Press, New York (1989).
^ Cosín, C., Monterde, Bézier surfaces of minimal area. In Computational Science — ICCS 2002, eds. J., Sloot, Peter, Hoekstra, Alfons, Tan, C., Dongarra, Jack. Lecture Notes in Computer Science 2330, Springer Berlin / Heidelberg, 2002. pp. 72-81 Isbn: 978-3-540-43593-8
^ Jaigyoung Choe, On the existence of higher dimensional Enneper's surface, Commentarii Mathematici Helvetici 1996, Volume 71, Issue 1, pp 556-569
External links
