Construction for minimal surfaces
In mathematics , the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry .
Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863.
Weierstrass parameterization facilities fabrication of periodic minimal surfaces Let f {\displaystyle f} and g {\displaystyle g} be functions on either the entire complex plane or the unit disk, where g {\displaystyle g} is meromorphic and f {\displaystyle f} is analytic , such that wherever g {\displaystyle g} has a pole of order m {\displaystyle m} , f {\displaystyle f} has a zero of order 2 m {\displaystyle 2m} (or equivalently, such that the product f g 2 {\displaystyle fg^{2}} is holomorphic ), and let c 1 , c 2 , c 3 {\displaystyle c_{1},c_{2},c_{3}} be constants. Then the surface with coordinates ( x 1 , x 2 , x 3 ) {\displaystyle (x_{1},x_{2},x_{3})} is minimal, where the x k {\displaystyle x_{k}} are defined using the real part of a complex integral, as follows:
x k ( ζ ) = ℜ { ∫ 0 ζ φ k ( z ) d z } + c k , k = 1 , 2 , 3 φ 1 = f ( 1 − g 2 ) / 2 φ 2 = i f ( 1 + g 2 ) / 2 φ 3 = f g {\displaystyle {\begin{aligned}x_{k}(\zeta )&{}=\Re \left\{\int _{0}^{\zeta }\varphi _{k}(z)\,dz\right\}+c_{k},\qquad k=1,2,3\\\varphi _{1}&{}=f(1-g^{2})/2\\\varphi _{2}&{}=\mathbf {i} f(1+g^{2})/2\\\varphi _{3}&{}=fg\end{aligned}}} The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.[1]
For example, Enneper's surface has f (z ) = 1 , g (z ) = zm .
Parametric surface of complex variables [ edit ] The Weierstrass-Enneper model defines a minimal surface X {\displaystyle X} ( R 3 {\displaystyle \mathbb {R} ^{3}} ) on a complex plane ( C {\displaystyle \mathbb {C} } ). Let ω = u + v i {\displaystyle \omega =u+vi} (the complex plane as the u v {\displaystyle uv} space), the Jacobian matrix of the surface can be written as a column of complex entries:
J = [ ( 1 − g 2 ( ω ) ) f ( ω ) i ( 1 + g 2 ( ω ) ) f ( ω ) 2 g ( ω ) f ( ω ) ] {\displaystyle \mathbf {J} ={\begin{bmatrix}\left(1-g^{2}(\omega )\right)f(\omega )\\i\left(1+g^{2}(\omega )\right)f(\omega )\\2g(\omega )f(\omega )\end{bmatrix}}} where
f ( ω ) {\displaystyle f(\omega )} and
g ( ω ) {\displaystyle g(\omega )} are holomorphic functions of
ω {\displaystyle \omega } .
The Jacobian J {\displaystyle \mathbf {J} } represents the two orthogonal tangent vectors of the surface:[2]
X u = [ Re J 1 Re J 2 Re J 3 ] X v = [ − Im J 1 − Im J 2 − Im J 3 ] {\displaystyle \mathbf {X_{u}} ={\begin{bmatrix}\operatorname {Re} \mathbf {J} _{1}\\\operatorname {Re} \mathbf {J} _{2}\\\operatorname {Re} \mathbf {J} _{3}\end{bmatrix}}\;\;\;\;\mathbf {X_{v}} ={\begin{bmatrix}-\operatorname {Im} \mathbf {J} _{1}\\-\operatorname {Im} \mathbf {J} _{2}\\-\operatorname {Im} \mathbf {J} _{3}\end{bmatrix}}} The surface normal is given by
n ^ = X u × X v | X u × X v | = 1 | g | 2 + 1 [ 2 Re g 2 Im g | g | 2 − 1 ] {\displaystyle \mathbf {\hat {n}} ={\frac {\mathbf {X_{u}} \times \mathbf {X_{v}} }{|\mathbf {X_{u}} \times \mathbf {X_{v}} |}}={\frac {1}{|g|^{2}+1}}{\begin{bmatrix}2\operatorname {Re} g\\2\operatorname {Im} g\\|g|^{2}-1\end{bmatrix}}} The Jacobian J {\displaystyle \mathbf {J} } leads to a number of important properties: X u ⋅ X v = 0 {\displaystyle \mathbf {X_{u}} \cdot \mathbf {X_{v}} =0} , X u 2 = Re ( J 2 ) {\displaystyle \mathbf {X_{u}} ^{2}=\operatorname {Re} (\mathbf {J} ^{2})} , X v 2 = Im ( J 2 ) {\displaystyle \mathbf {X_{v}} ^{2}=\operatorname {Im} (\mathbf {J} ^{2})} , X u u + X v v = 0 {\displaystyle \mathbf {X_{uu}} +\mathbf {X_{vv}} =0} . The proofs can be found in Sharma's essay: The Weierstrass representation always gives a minimal surface.[3] The derivatives can be used to construct the first fundamental form matrix:
[ X u ⋅ X u X u ⋅ X v X v ⋅ X u X v ⋅ X v ] = [ 1 0 0 1 ] {\displaystyle {\begin{bmatrix}\mathbf {X_{u}} \cdot \mathbf {X_{u}} &\;\;\mathbf {X_{u}} \cdot \mathbf {X_{v}} \\\mathbf {X_{v}} \cdot \mathbf {X_{u}} &\;\;\mathbf {X_{v}} \cdot \mathbf {X_{v}} \end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}} and the second fundamental form matrix
[ X u u ⋅ n ^ X u v ⋅ n ^ X v u ⋅ n ^ X v v ⋅ n ^ ] {\displaystyle {\begin{bmatrix}\mathbf {X_{uu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{uv}} \cdot \mathbf {\hat {n}} \\\mathbf {X_{vu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{vv}} \cdot \mathbf {\hat {n}} \end{bmatrix}}} Finally, a point ω t {\displaystyle \omega _{t}} on the complex plane maps to a point X {\displaystyle \mathbf {X} } on the minimal surface in R 3 {\displaystyle \mathbb {R} ^{3}} by
X = [ Re ∫ ω 0 ω t J 1 d ω Re ∫ ω 0 ω t J 2 d ω Re ∫ ω 0 ω t J 3 d ω ] {\displaystyle \mathbf {X} ={\begin{bmatrix}\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{1}d\omega \\\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{2}d\omega \\\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{3}d\omega \end{bmatrix}}} where
ω 0 = 0 {\displaystyle \omega _{0}=0} for all minimal surfaces throughout this paper except for
Costa's minimal surface where
ω 0 = ( 1 + i ) / 2 {\displaystyle \omega _{0}=(1+i)/2} .
Embedded minimal surfaces and examples [ edit ] The classical examples of embedded complete minimal surfaces in R 3 {\displaystyle \mathbb {R} ^{3}} with finite topology include the plane, the catenoid , the helicoid , and the Costa's minimal surface . Costa's surface involves Weierstrass's elliptic function ℘ {\displaystyle \wp } :[4]
g ( ω ) = A ℘ ′ ( ω ) {\displaystyle g(\omega )={\frac {A}{\wp '(\omega )}}} f ( ω ) = ℘ ( ω ) {\displaystyle f(\omega )=\wp (\omega )} where
A {\displaystyle A} is a constant.
[5] Helicatenoid [ edit ] Choosing the functions f ( ω ) = e − i α e ω / A {\displaystyle f(\omega )=e^{-i\alpha }e^{\omega /A}} and g ( ω ) = e − ω / A {\displaystyle g(\omega )=e^{-\omega /A}} , a one parameter family of minimal surfaces is obtained.
φ 1 = e − i α sinh ( ω A ) {\displaystyle \varphi _{1}=e^{-i\alpha }\sinh \left({\frac {\omega }{A}}\right)} φ 2 = i e − i α cosh ( ω A ) {\displaystyle \varphi _{2}=ie^{-i\alpha }\cosh \left({\frac {\omega }{A}}\right)} φ 3 = e − i α {\displaystyle \varphi _{3}=e^{-i\alpha }} X ( ω ) = Re [ e − i α A cosh ( ω A ) i e − i α A sinh ( ω A ) e − i α ω ] = cos ( α ) [ A cosh ( Re ( ω ) A ) cos ( Im ( ω ) A ) − A cosh ( Re ( ω ) A ) sin ( Im ( ω ) A ) Re ( ω ) ] + sin ( α ) [ A sinh ( Re ( ω ) A ) sin ( Im ( ω ) A ) A sinh ( Re ( ω ) A ) cos ( Im ( ω ) A ) Im ( ω ) ] {\displaystyle \mathbf {X} (\omega )=\operatorname {Re} {\begin{bmatrix}e^{-i\alpha }A\cosh \left({\frac {\omega }{A}}\right)\\ie^{-i\alpha }A\sinh \left({\frac {\omega }{A}}\right)\\e^{-i\alpha }\omega \\\end{bmatrix}}=\cos(\alpha ){\begin{bmatrix}A\cosh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\cos \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\-A\cosh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\sin \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\\operatorname {Re} (\omega )\\\end{bmatrix}}+\sin(\alpha ){\begin{bmatrix}A\sinh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\sin \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\A\sinh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\cos \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\\operatorname {Im} (\omega )\\\end{bmatrix}}} Choosing the parameters of the surface as ω = s + i ( A ϕ ) {\displaystyle \omega =s+i(A\phi )} :
X ( s , ϕ ) = cos ( α ) [ A cosh ( s A ) cos ( ϕ ) − A cosh ( s A ) sin ( ϕ ) s ] + sin ( α ) [ A sinh ( s A ) sin ( ϕ ) A sinh ( s A ) cos ( ϕ ) A ϕ ] {\displaystyle \mathbf {X} (s,\phi )=\cos(\alpha ){\begin{bmatrix}A\cosh \left({\frac {s}{A}}\right)\cos \left(\phi \right)\\-A\cosh \left({\frac {s}{A}}\right)\sin \left(\phi \right)\\s\\\end{bmatrix}}+\sin(\alpha ){\begin{bmatrix}A\sinh \left({\frac {s}{A}}\right)\sin \left(\phi \right)\\A\sinh \left({\frac {s}{A}}\right)\cos \left(\phi \right)\\A\phi \\\end{bmatrix}}} At the extremes, the surface is a catenoid ( α = 0 ) {\displaystyle (\alpha =0)} or a helicoid ( α = π / 2 ) {\displaystyle (\alpha =\pi /2)} . Otherwise, α {\displaystyle \alpha } represents a mixing angle. The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the X 3 {\displaystyle \mathbf {X} _{3}} axis in a helical fashion.
A catenary that spans periodic points on a helix, subsequently rotated along the helix to produce a minimal surface. The fundamental domain (C) and the 3D surfaces. The continuous surfaces are made of copies of the fundamental patch (R3) Lines of curvature [ edit ] One can rewrite each element of second fundamental matrix as a function of f {\displaystyle f} and g {\displaystyle g} , for example
X u u ⋅ n ^ = 1 | g | 2 + 1 [ Re ( ( 1 − g 2 ) f ′ − 2 g f g ′ ) Re ( ( 1 + g 2 ) f ′ i + 2 g f g ′ i ) Re ( 2 g f ′ + 2 f g ′ ) ] ⋅ [ Re ( 2 g ) Re ( − 2 g i ) Re ( | g | 2 − 1 ) ] = − 2 Re ( f g ′ ) {\displaystyle \mathbf {X_{uu}} \cdot \mathbf {\hat {n}} ={\frac {1}{|g|^{2}+1}}{\begin{bmatrix}\operatorname {Re} \left((1-g^{2})f'-2gfg'\right)\\\operatorname {Re} \left((1+g^{2})f'i+2gfg'i\right)\\\operatorname {Re} \left(2gf'+2fg'\right)\\\end{bmatrix}}\cdot {\begin{bmatrix}\operatorname {Re} \left(2g\right)\\\operatorname {Re} \left(-2gi\right)\\\operatorname {Re} \left(|g|^{2}-1\right)\\\end{bmatrix}}=-2\operatorname {Re} (fg')} And consequently the second fundamental form matrix can be simplified as
[ − Re f g ′ Im f g ′ Im f g ′ Re f g ′ ] {\displaystyle {\begin{bmatrix}-\operatorname {Re} fg'&\;\;\operatorname {Im} fg'\\\operatorname {Im} fg'&\;\;\operatorname {Re} fg'\end{bmatrix}}} Lines of curvature make a quadrangulation of the domain One of its eigenvectors is
f g ′ ¯ {\displaystyle {\overline {\sqrt {fg'}}}} which represents the principal direction in the complex domain.
[6] Therefore, the two principal directions in the
u v {\displaystyle uv} space turn out to be
ϕ = − 1 2 Arg ( f g ′ ) ± k π / 2 {\displaystyle \phi =-{\frac {1}{2}}\operatorname {Arg} (fg')\pm k\pi /2} See also [ edit ] References [ edit ] ^ Dierkes, U.; Hildebrandt, S.; Küster, A.; Wohlrab, O. (1992). Minimal surfaces . Vol. I. Springer. p. 108. ISBN 3-540-53169-6 . ^ Andersson, S.; Hyde, S. T.; Larsson, K.; Lidin, S. (1988). "Minimal Surfaces and Structures: From Inorganic and Metal Crystals to Cell Membranes and Biopolymers". Chem. Rev . 88 (1): 221–242. doi :10.1021/cr00083a011 . ^ Sharma, R. (2012). "The Weierstrass Representation always gives a minimal surface". arXiv :1208.5689 [math.DG ]. ^ Lawden, D. F. (2011). Elliptic Functions and Applications . Applied Mathematical Sciences. Vol. 80. Berlin: Springer. ISBN 978-1-4419-3090-3 . ^ Abbena, E.; Salamon, S.; Gray, A. (2006). "Minimal Surfaces via Complex Variables". Modern Differential Geometry of Curves and Surfaces with Mathematica . Boca Raton: CRC Press. pp. 719–766. ISBN 1-58488-448-7 . ^ Hua, H.; Jia, T. (2018). "Wire cut of double-sided minimal surfaces". The Visual Computer . 34 (6–8): 985–995. doi :10.1007/s00371-018-1548-0 . S2CID 13681681 .