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In mathematics , in the fields of differential geometry and algebraic geometry , the Enneper surface is a 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 in connection with minimal surface theory.
Figure 1. An Enneper surface 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.\ }
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. 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.\ }
Enneper's is a minimal surface . 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.\ }
References
Weisstein, Eric W. "Enneper's Minimal Surface" . MathWorld
