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Maple Minimal Surfaces1.html
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Maple Minimal Surfaces1.html
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<!DOCTYPE html>
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<title>Maple Minimal Surfaces1.html</title>
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<body bgcolor="FFFFFF">
<p align="left"><font color="#680000" size="3" face="Courier New"><strong></strong></font> </p>
<p align="center"><font color="#000000" size="8" face="Times New Roman"><strong>Exploring Minimal Surfaces</strong></font> </p>
<p align="center"><font color="#000000" size="3" face="Times New Roman">Based on Chapter 4.9 of </font><font color="#000000" size="3" face="Times New Roman"><em>Differential Geometry and its Applications</em></font><font color="#000000" size="3" face="Times New Roman">, John Oprea
Emrys Halbertsma
June 2020</font> </p>
<p align="center"><font color="#000000" size="7" face="Times New Roman"><strong></strong></font> </p>
<p align="center"><font color="#000000" size="7" face="Times New Roman"><strong>Overview</strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">The purpose of this project is to explore some key ideas in differential geometry such as curvature and minimal surfaces; and to improve skills in symbolic computation. Most of the Maple code in this dicument comes from the Maple exercises in Oprea's </font><font color="#000000" size="3" face="Times New Roman"><em>Differential Geometry and its Applications, </em></font><font color="#000000" size="3" face="Times New Roman">specifically sections 4.9 and 3.8.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="center"><font color="#000000" size="3" face="Times New Roman"></font><font color="#000000" size="7" face="Times New Roman"><strong>Background</strong></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Curvature</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">The curvature is a geometric notion that describes how much a curve or surface "curves", i.e. how different it is from a straight line. A sharper curve will have a higher curvature.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">The mean curvature is defined as the average of the two principal curvatures (maximum and minimum curvature): <img src="images/Maple Minimal Surfaces_1.gif" width="109" height="44" alt="" align="center" border="0"></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Minimal Surfaces</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">A surface is called a minimal surface if its mean curvature is 0.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">The classic example of a minimal surface is the soap bubble. Soapy water, composed of many slightly polar particles, exhibits intermolecular forces that attract the particles towards each other. Inside the fluid, the forces are balanced throughout, but on the surface of the fluid the forces are unbalanced. The fluid pulls the particles inwards, so that the fluid has a tendency to compactify. However, the fluid cannot attract itself with no bound: the inward force is eventually balanced by the internal pressure pushing back outwards. The surface that is produced on the boundary of this fluid exhibiting a surface tension is an example of a minimal surface: the magnitude of net forces on the particles are minimized.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font><font color="#000000" size="4" face="Times New Roman"><strong><em>Area Minimization</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">Belgian physicist Plateau studied minimal surfaces in the 1800s. He posed: given a boudary curve C, how does one find a minimal surface M satisfying the boundary condition?</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">It turns out that least-area surfaces are minimal, but the converse is not necessarily true.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">An important theorem states that for any given Jordan curve, there exists a least-area disk-like minimal surface that spans the curve.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Weierstrass-Enneper Representation</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">The Weierstrass-Enneper Representation Theorem tells us that any holomorphic function <img src="images/Maple Minimal Surfaces_2.gif" width="62" height="25" alt="F(tau); "_noterminate" align="center" border="0">defines a minimal surface, and gives a parametrization of the surface. This is an important result in geometry and complex analysis.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="center"><font color="#000000" size="3" face="Times New Roman"></font><font color="#000000" size="7" face="Times New Roman"><strong>Maple Analysis</strong></font><font color="#000000" size="3" face="Times New Roman">
</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">Firstly, import the required packages:</font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_3.gif" width="318" height="14" alt="with(plots); -1; with(LinearAlgebra); -1" align="center" border="0"></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="left"><font color="#000000" size="5" face="Times New Roman"><strong>Functions</strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">We next define some functions that will come of use.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>EFG</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font><font color="#000000" size="3" face="Times New Roman"><em>input</em></font><font color="#000000" size="3" face="Times New Roman">: parametrization of a surface in terms of u,v in R^3.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman" style="background-color: #ffffff"></font><font color="#000000" size="3" face="Times New Roman" style="background-color: #ffffff"><em>output</em></font><font color="#000000" size="3" face="Times New Roman" style="background-color: #ffffff">: metric coefficients for curvature</font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_4.gif" width="106" height="13" alt="EFG := proc (X) local Xu, Xv, E, F, G; Xu := `<|>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<|>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); E := LinearAlgebra:-DotProduct(Xu, Xu, conjug..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_5.gif" width="152" height="12" alt="EFG := proc (X) local Xu, Xv, E, F, G; Xu := `<|>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<|>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); E := LinearAlgebra:-DotProduct(Xu, Xu, conjug..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_6.gif" width="238" height="25" alt="EFG := proc (X) local Xu, Xv, E, F, G; Xu := `<|>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<|>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); E := LinearAlgebra:-DotProduct(Xu, Xu, conjug..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_7.gif" width="249" height="25" alt="EFG := proc (X) local Xu, Xv, E, F, G; Xu := `<|>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<|>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); E := LinearAlgebra:-DotProduct(Xu, Xu, conjug..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_8.gif" width="300" height="14" alt="EFG := proc (X) local Xu, Xv, E, F, G; Xu := `<|>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<|>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); E := LinearAlgebra:-DotProduct(Xu, Xu, conjug..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_9.gif" width="300" height="14" alt="EFG := proc (X) local Xu, Xv, E, F, G; Xu := `<|>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<|>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); E := LinearAlgebra:-DotProduct(Xu, Xu, conjug..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_10.gif" width="300" height="14" alt="EFG := proc (X) local Xu, Xv, E, F, G; Xu := `<|>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<|>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); E := LinearAlgebra:-DotProduct(Xu, Xu, conjug..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_11.gif" width="141" height="11" alt="EFG := proc (X) local Xu, Xv, E, F, G; Xu := `<|>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<|>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); E := LinearAlgebra:-DotProduct(Xu, Xu, conjug..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_12.gif" width="41" height="11" alt="EFG := proc (X) local Xu, Xv, E, F, G; Xu := `<|>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<|>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); E := LinearAlgebra:-DotProduct(Xu, Xu, conjug..." align="center" border="0"></strong></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em></em></strong></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Unit Normal</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font><font color="#000000" size="3" face="Times New Roman"><em>input</em></font><font color="#000000" size="3" face="Times New Roman">: parametrization of a surface in terms of u,v in R^3.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font><font color="#000000" size="3" face="Times New Roman"><em>output</em></font><font color="#000000" size="3" face="Times New Roman">: unit outward pointing normal vector</font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_13.gif" width="99" height="13" alt="UN := proc (X) local Xu, Xv, Z, s; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Z := LinearAlgebra:-CrossProduct(Xu, Xv); s := Li..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_14.gif" width="135" height="12" alt="UN := proc (X) local Xu, Xv, Z, s; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Z := LinearAlgebra:-CrossProduct(Xu, Xv); s := Li..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_15.gif" width="252" height="25" alt="UN := proc (X) local Xu, Xv, Z, s; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Z := LinearAlgebra:-CrossProduct(Xu, Xv); s := Li..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_16.gif" width="256" height="25" alt="UN := proc (X) local Xu, Xv, Z, s; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Z := LinearAlgebra:-CrossProduct(Xu, Xv); s := Li..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_17.gif" width="191" height="14" alt="UN := proc (X) local Xu, Xv, Z, s; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Z := LinearAlgebra:-CrossProduct(Xu, Xv); s := Li..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_18.gif" width="343" height="14" alt="UN := proc (X) local Xu, Xv, Z, s; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Z := LinearAlgebra:-CrossProduct(Xu, Xv); s := Li..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_19.gif" width="383" height="13" alt="UN := proc (X) local Xu, Xv, Z, s; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Z := LinearAlgebra:-CrossProduct(Xu, Xv); s := Li..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_20.gif" width="41" height="11" alt="UN := proc (X) local Xu, Xv, Z, s; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Z := LinearAlgebra:-CrossProduct(Xu, Xv); s := Li..." align="center" border="0"></strong></font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong></strong></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>LMN</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman" style="background-color: #ffffff"></font><font color="#000000" size="3" face="Times New Roman" style="background-color: #ffffff"><em>input</em></font><font color="#000000" size="3" face="Times New Roman" style="background-color: #ffffff">: parametrization of a surface in terms of u,v in R^3.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman" style="background-color: #ffffff"></font><font color="#000000" size="3" face="Times New Roman" style="background-color: #ffffff"><em>output</em></font><font color="#000000" size="3" face="Times New Roman" style="background-color: #ffffff">: coefficients of the second fundamental form</font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_21.gif" width="106" height="13" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_22.gif" width="262" height="11" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_23.gif" width="259" height="25" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_24.gif" width="263" height="25" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_25.gif" width="280" height="25" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_26.gif" width="284" height="25" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_27.gif" width="284" height="25" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_28.gif" width="89" height="14" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_29.gif" width="300" height="13" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_30.gif" width="300" height="14" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_31.gif" width="304" height="14" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_32.gif" width="284" height="12" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_33.gif" width="41" height="12" alt="lmn := proc (X) local Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n; Xu := `<,>`(diff(X[1], u), diff(X[2], u), diff(X[3], u)); Xv := `<,>`(diff(X[1], v), diff(X[2], v), diff(X[3], v)); Xuu := `<,>`(diff(Xu[1], u)..." align="center" border="0"></strong></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em></em></strong></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Gaussian Curvature</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font><font color="#000000" size="3" face="Times New Roman"><em>input</em></font><font color="#000000" size="3" face="Times New Roman">: parametrization of a surface in terms of u,v in R^3.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font><font color="#000000" size="3" face="Times New Roman"><em>output</em></font><font color="#000000" size="3" face="Times New Roman">: Gaussian curvature coefficient</font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_34.gif" width="99" height="13" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_35.gif" width="189" height="12" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_36.gif" width="99" height="14" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_37.gif" width="95" height="14" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_38.gif" width="91" height="14" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_39.gif" width="91" height="14" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_40.gif" width="94" height="14" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_41.gif" width="94" height="13" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_42.gif" width="98" height="14" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_43.gif" width="91" height="14" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_44.gif" width="363" height="13" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_45.gif" width="48" height="14" alt="GK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := `*`(S, `*`([3])); l := `*`(T, `*`([1])); m := `*`(T, `*`([2])); n := T[3]; simplify(`/`(`*`(`+`(`*`(n,..." align="center" border="0"></strong></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em></em></strong></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Mean Curvature</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font><font color="#000000" size="3" face="Times New Roman"><em>input</em></font><font color="#000000" size="3" face="Times New Roman">: parametrization of a surface in terms of u,v in R^3.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font><font color="#000000" size="3" face="Times New Roman"><em>output</em></font><font color="#000000" size="3" face="Times New Roman">: mean curvature coefficient</font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_46.gif" width="99" height="13" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_47.gif" width="189" height="11" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_48.gif" width="95" height="14" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_49.gif" width="102" height="14" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_50.gif" width="76" height="14" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_51.gif" width="76" height="13" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_52.gif" width="76" height="14" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_53.gif" width="72" height="13" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_54.gif" width="72" height="13" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_55.gif" width="72" height="13" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_56.gif" width="342" height="26" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_57.gif" width="41" height="12" alt="MK := proc (X) local E, F, G, l, m, n, S, T; S := EFG(X); T := lmn(X); E := S[1]; F := S[2]; G := S[3]; l := T[1]; m := T[2]; n := T[3]; simplify(`/`(`*`(`+`(`*`(l, `*`(G)), `*`(E, `*`(n)), `-`(`*`(2,..." align="center" border="0"></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="left"><font color="#000000" size="5" face="Times New Roman"><strong>Examples of Minimal Surfaces</strong></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Minimal Surface Equation</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">The minimum surface equation can be derived directly form the mean curvature. Consider a Monge parametrization:</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_58.gif" width="183" height="17" alt="monge := `<|>`(u, v, f(u, v)); "_noterminate" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_59.gif" width="194" height="26" alt="Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mi("u", italic = "true", foreground = "[0,0,0]", readonly = "false", mathvariant = "italic"), ..." align="center" border="0"></td>
<td width='5%' align='center'>(1)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">Take the mean curvature and set it to zero to get the equation for a minimal surface:</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_60.gif" width="254" height="14" alt="factor(numer(MK(monge))) = 0; "_noterminate" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_61.gif" width="768" height="35" alt="Typesetting:-mprintslash([`+`(`-`(`*`(2, `*`(diff(f(u, v), u), `*`(diff(f(u, v), v), `*`(diff(f(u, v), u, v)))))), `*`(diff(f(u, v), `$`(v, 2)), `*`(`^`(diff(f(u, v), u), 2))), `*`(diff(f(u, v), `$`(u..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_62.gif" width="768" height="14" alt="Typesetting:-mprintslash([`+`(`-`(`*`(2, `*`(diff(f(u, v), u), `*`(diff(f(u, v), v), `*`(diff(f(u, v), u, v)))))), `*`(diff(f(u, v), `$`(v, 2)), `*`(`^`(diff(f(u, v), u), 2))), `*`(diff(f(u, v), `$`(u..." align="center" border="0"></td>
<td width='5%' align='center'>(2)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Catenoid</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_63.gif" width="398" height="18" alt="catenoid := `<|>`(u, `*`(cosh(u), `*`(cos(v))), `*`(cosh(u), `*`(sin(v)))); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_64.gif" width="344" height="26" alt="Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mi("u", italic = "true", foreground = "[0,0,0]", readonly = "false", mathvariant = "italic"), ..." align="center" border="0"></td>
<td width='5%' align='center'>(3)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong></strong></font><font color="#000000" size="3" face="Times New Roman">We can see the catenoid is a minimal surface, since its mean curvature is zero:</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_65.gif" width="121" height="14" alt="MK(catenoid); "_noterminate" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_66.gif" width="21" height="18" alt="0" align="center" border="0"></td>
<td width='5%' align='center'>(4)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_67.gif" width="768" height="16" alt="plot3d(catenoid, u = -1 .. 1, v = 0 .. `+`(`*`(2, `*`(Pi))), shading = z, scaling = constrained, orientation = [163, 163], lightmodel = light3); 1" align="center" border="0"><br><img src="images/Maple Minimal Surfaces_68.gif" width="768" height="16" alt="plot3d(catenoid, u = -1 .. 1, v = 0 .. `+`(`*`(2, `*`(Pi))), shading = z, scaling = constrained, orientation = [163, 163], lightmodel = light3); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_69.gif" width="400" height="400" alt="Plot_2d" align="center" border="0"></td>
<td width='5%' align='center'></td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Helicoid</em></strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_70.gif" width="343" height="17" alt="helicoid := `<|>`(`*`(u, `*`(cos(v))), `*`(u, `*`(sin(v))), v); "_noterminate" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_71.gif" width="254" height="26" alt="Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mrow(Typesetting:-mi("u", italic = "true", foreground = "[0,0,0]", readonly = "false", mathvar..." align="center" border="0"></td>
<td width='5%' align='center'>(5)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_72.gif" width="768" height="16" alt="plot3d(helicoid, u = 0 .. 1.5, v = 0 .. `+`(`*`(5, `*`(Pi))), shading = z, orientation = [21, 64], lightmodel = light3, grid = [10, 60]); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_73.gif" width="400" height="400" alt="Plot_2d" align="center" border="0"></td>
<td width='5%' align='center'></td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">It may be possible to parametrize one suface into another. For example, the catenoid can be deformed into a helicoid.</font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_74.gif" width="157" height="13" alt="helcatplot := proc (t) local X; X := `<|>`(`+`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v), `*`(sin(u)))), `*`(sin(`*`(t, `*`(Pi))), `*`(cosh(v), `*`(cos(u))))), `+`(`-`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_75.gif" width="70" height="13" alt="helcatplot := proc (t) local X; X := `<|>`(`+`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v), `*`(sin(u)))), `*`(sin(`*`(t, `*`(Pi))), `*`(cosh(v), `*`(cos(u))))), `+`(`-`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_76.gif" width="768" height="16" alt="helcatplot := proc (t) local X; X := `<|>`(`+`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v), `*`(sin(u)))), `*`(sin(`*`(t, `*`(Pi))), `*`(cosh(v), `*`(cos(u))))), `+`(`-`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_77.gif" width="768" height="16" alt="helcatplot := proc (t) local X; X := `<|>`(`+`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v), `*`(sin(u)))), `*`(sin(`*`(t, `*`(Pi))), `*`(cosh(v), `*`(cos(u))))), `+`(`-`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_78.gif" width="768" height="16" alt="helcatplot := proc (t) local X; X := `<|>`(`+`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v), `*`(sin(u)))), `*`(sin(`*`(t, `*`(Pi))), `*`(cosh(v), `*`(cos(u))))), `+`(`-`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_79.gif" width="41" height="13" alt="helcatplot := proc (t) local X; X := `<|>`(`+`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v), `*`(sin(u)))), `*`(sin(`*`(t, `*`(Pi))), `*`(cosh(v), `*`(cos(u))))), `+`(`-`(`*`(cos(`*`(t, `*`(Pi))), `*`(sinh(v..." align="center" border="0"></strong></font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong></strong></font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_80.gif" width="566" height="14" alt="plots:-display3d(seq(helcatplot(`+`(`*`(`/`(1, 40), `*`(t)))), t = 0 .. 40), insequence = true); 1" align="center" border="0"></strong></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_81.gif" width="400" height="400" alt="Plot_2d" align="center" border="0"></td>
<td width='5%' align='center'></td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Scherk's First Surface</em></strong></font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_82.gif" width="311" height="17" alt="scherk1 := `<|>`(u, v, ln(`/`(`*`(cos(v)), `*`(cos(u))))); 1" align="center" border="0"></strong></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_83.gif" width="240" height="47" alt="Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mi("u", italic = "true", foreground = "[0,0,0]", readonly = "false", mathvariant = "italic"), ..." align="center" border="0"></td>
<td width='5%' align='center'>(6)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_84.gif" width="768" height="11" alt="plot3d(scherk1, u = -1.57 .. 1.57, v = -1.57 .. 1.57, shading = xy, lightmodel = light4, orientation = [51, 66], grid = [20, 20]); 1" align="center" border="0"><br><img src="images/Maple Minimal Surfaces_85.gif" width="768" height="11" alt="plot3d(scherk1, u = -1.57 .. 1.57, v = -1.57 .. 1.57, shading = xy, lightmodel = light4, orientation = [51, 66], grid = [20, 20]); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_86.gif" width="400" height="400" alt="Plot_2d" align="center" border="0"></td>
<td width='5%' align='center'></td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Enneper's Surface</em></strong></font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_87.gif" width="447" height="17" alt="enneper := `<|>`(`+`(u, `-`(`*`(`/`(1, 3), `*`(`^`(u, 3)))), `*`(u, `*`(`^`(v, 2)))), `+`(`-`(v), `*`(`/`(1, 3), `*`(`^`(v, 3))), `-`(`*`(v, `*`(`^`(u, 2))))), `+`(`*`(`^`(u, 2)), `-`(`*`(`^`(v, 2))))..." align="center" border="0"></strong></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_88.gif" width="404" height="47" alt="Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mrow(Typesetting:-mi("u", italic = "true", foreground = "[0,0,0]", readonly = "false", mathvar..." align="center" border="0"></td>
<td width='5%' align='center'>(7)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_89.gif" width="768" height="11" alt="plot3d(enneper, u = -2.2 .. 2.2, v = -2.2 .. 2.2, shading = zhue, scaling = constrained, orientation = [94, 42]); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_90.gif" width="400" height="400" alt="Plot_2d" align="center" border="0"></td>
<td width='5%' align='center'></td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="4" face="Times New Roman"><strong><em>Catalan's Surface<img src="images/Maple Minimal Surfaces_91.gif" width="610" height="18" alt="catsurf := `<|>`(`+`(u, `-`(`*`(sin(u), `*`(cosh(v))))), `+`(1, `-`(`*`(cos(u), `*`(cosh(v))))), `+`(`*`(4, `*`(sin(`+`(`*`(`/`(1, 2), `*`(u)))), `*`(sinh(`+`(`*`(`/`(1, 2), `*`(v))))))))); 1" align="center" border="0"></em></strong></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_92.gif" width="519" height="47" alt="Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mrow(Typesetting:-mi("u", italic = "true", foreground = "[0,0,0]", readonly = "false", mathvar..." align="center" border="0"></td>
<td width='5%' align='center'>(8)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_93.gif" width="768" height="16" alt="plot3d(catsurf, u = 0 .. `+`(`*`(8, `*`(Pi))), v = -2.3 .. 2.3, scaling = constrained, shading = zhue, lightmodel = light3, grid = [50, 15], orientation = [-52, 42]); 1" align="center" border="0"><br><img src="images/Maple Minimal Surfaces_94.gif" width="768" height="16" alt="plot3d(catsurf, u = 0 .. `+`(`*`(8, `*`(Pi))), v = -2.3 .. 2.3, scaling = constrained, shading = zhue, lightmodel = light3, grid = [50, 15], orientation = [-52, 42]); 1" align="center" border="0"><br><img src="images/Maple Minimal Surfaces_95.gif" width="84" height="16" alt="plot3d(catsurf, u = 0 .. `+`(`*`(8, `*`(Pi))), v = -2.3 .. 2.3, scaling = constrained, shading = zhue, lightmodel = light3, grid = [50, 15], orientation = [-52, 42]); 1" align="center" border="0"></strong></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_96.gif" width="556" height="398" alt="Plot_2d" align="center" border="0"></td>
<td width='5%' align='center'></td>
</tr>
</table></font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong></strong></font> </p>
<p align="left"><font color="#000000" size="5" face="Times New Roman"><strong>Area Minimization</strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">We can show that a minimal surface does not necessarily minimize the area of a surface satisfying certain boundary conditions. In this example, we choose our boundary condition to be the Jordan curve on Enneper's surface a distance of R=1.5 from the origin. We will then show that a simple cylinder in fact has a lower surface area within the boundary than the Enneper's surface.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">To start, recall Enneper's surface. </font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_97.gif" width="78" height="14" alt="enneper; "_noterminate" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_98.gif" width="326" height="47" alt="Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mrow(Typesetting:-mi("u", italic = "true", foreground = "[0,0,0]", readonly = "false", mathvar..." align="center" border="0"></td>
<td width='5%' align='center'>(9)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">We'll convert this into polar coordinates, as it will simplify the equations for our particular boundary curve.</font> </p>
<p align="left"><font color="#680000" size="3" face="Courier New"><strong><img src="images/Maple Minimal Surfaces_99.gif" width="625" height="17" alt="ennpolar := simplify(subs({u = `*`(r, `*`(cos(theta))), v = `*`(r, `*`(sin(theta)))}, enneper)); "_noterminate" align="center" border="0"></strong></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_100.gif" width="732" height="53" alt="Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mrow(Typesetting:-mo("&uminus0;", foreground = "[0,0,0]", readonly = "false", mathvariant = "n..." align="center" border="0"></td>
<td width='5%' align='center'>(10)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">We now use this procedure to parametrize the Jordan curve on Enneper's surface:</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_101.gif" width="128" height="13" alt="CylEnn := proc (r) local xtheta, ytheta, ztheta, n, X; xtheta := `+`(`*`(r, `*`(cos(theta))), `-`(`*`(`/`(1, 3), `*`(`^`(r, 3), `*`(cos(`+`(`*`(3, `*`(theta))))))))); ytheta := `+`(`-`(`*`(r, `*`(sin(..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_102.gif" width="246" height="12" alt="CylEnn := proc (r) local xtheta, ytheta, ztheta, n, X; xtheta := `+`(`*`(r, `*`(cos(theta))), `-`(`*`(`/`(1, 3), `*`(`^`(r, 3), `*`(cos(`+`(`*`(3, `*`(theta))))))))); ytheta := `+`(`-`(`*`(r, `*`(sin(..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_103.gif" width="319" height="13" alt="CylEnn := proc (r) local xtheta, ytheta, ztheta, n, X; xtheta := `+`(`*`(r, `*`(cos(theta))), `-`(`*`(`/`(1, 3), `*`(`^`(r, 3), `*`(cos(`+`(`*`(3, `*`(theta))))))))); ytheta := `+`(`-`(`*`(r, `*`(sin(..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_104.gif" width="327" height="13" alt="CylEnn := proc (r) local xtheta, ytheta, ztheta, n, X; xtheta := `+`(`*`(r, `*`(cos(theta))), `-`(`*`(`/`(1, 3), `*`(`^`(r, 3), `*`(cos(`+`(`*`(3, `*`(theta))))))))); ytheta := `+`(`-`(`*`(r, `*`(sin(..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_105.gif" width="189" height="13" alt="CylEnn := proc (r) local xtheta, ytheta, ztheta, n, X; xtheta := `+`(`*`(r, `*`(cos(theta))), `-`(`*`(`/`(1, 3), `*`(`^`(r, 3), `*`(cos(`+`(`*`(3, `*`(theta))))))))); ytheta := `+`(`-`(`*`(r, `*`(sin(..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_106.gif" width="131" height="14" alt="CylEnn := proc (r) local xtheta, ytheta, ztheta, n, X; xtheta := `+`(`*`(r, `*`(cos(theta))), `-`(`*`(`/`(1, 3), `*`(`^`(r, 3), `*`(cos(`+`(`*`(3, `*`(theta))))))))); ytheta := `+`(`-`(`*`(r, `*`(sin(..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_107.gif" width="249" height="14" alt="CylEnn := proc (r) local xtheta, ytheta, ztheta, n, X; xtheta := `+`(`*`(r, `*`(cos(theta))), `-`(`*`(`/`(1, 3), `*`(`^`(r, 3), `*`(cos(`+`(`*`(3, `*`(theta))))))))); ytheta := `+`(`-`(`*`(r, `*`(sin(..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_108.gif" width="41" height="12" alt="CylEnn := proc (r) local xtheta, ytheta, ztheta, n, X; xtheta := `+`(`*`(r, `*`(cos(theta))), `-`(`*`(`/`(1, 3), `*`(`^`(r, 3), `*`(cos(`+`(`*`(3, `*`(theta))))))))); ytheta := `+`(`-`(`*`(r, `*`(sin(..." align="center" border="0"></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">Next, we plot the cylinder:</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_109.gif" width="678" height="16" alt="cyl1 := plot3d(CylEnn(1.5), u = 0 .. 2, theta = 0 .. Pi, scaling = constrained, grid = [5, 50], style = patch); -1; cyl2 := plot3d(CylEnn(1.5), u = -2 .. 0, theta = Pi .. `+`(`*`(2, `*`(Pi))), scaling..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_110.gif" width="698" height="16" alt="cyl1 := plot3d(CylEnn(1.5), u = 0 .. 2, theta = 0 .. Pi, scaling = constrained, grid = [5, 50], style = patch); -1; cyl2 := plot3d(CylEnn(1.5), u = -2 .. 0, theta = Pi .. `+`(`*`(2, `*`(Pi))), scaling..." align="center" border="0"></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">Then plot the surface:</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_111.gif" width="768" height="14" alt="enn := plot3d(ennpolar, r = 0 .. 1.5, theta = 0 .. `+`(`*`(2, `*`(Pi))), scaling = constrained, grid = [5, 50], style = patch); -1" align="center" border="0"></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">Then define and plot the Jordan boudary curve:</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_112.gif" width="328" height="17" alt="jorcurve := subs(r = 1.51, ennpolar); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_113.gif" width="768" height="27" alt="Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mrow(Typesetting:-mo("&uminus0;", foreground = "[0,0,0]", readonly = "false", mathvariant = "n..." align="center" border="0"></td>
<td width='5%' align='center'>(11)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_114.gif" width="768" height="20" alt="bound := plots:-tubeplot(convert(jorcurve, list), theta = 0 .. `+`(`*`(2, `*`(Pi))), radius = 0.25e-1, color = black); -1" align="center" border="0"></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">Finally, display the generated plots:</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_115.gif" width="768" height="14" alt="plots:-display({bound, enn}, scaling = constrained, style = wireframe, orientation = [154, -106]); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_116.gif" width="400" height="400" alt="Plot_2d" align="center" border="0"></td>
<td width='5%' align='center'></td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_117.gif" width="653" height="14" alt="plots:-display({bound, cyl2}, scaling = constrained, orientation = [154, -106]); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_118.gif" width="400" height="400" alt="Plot_2d" align="center" border="0"></td>
<td width='5%' align='center'></td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">To prove that the simple cylinder has a lower area than Enneper's surface, we can evaluate the surface area integrals.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_119.gif" width="324" height="17" alt="ytheta := subs(r = 1.5, ennpolar[2]); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_120.gif" width="372" height="25" alt="Typesetting:-mprintslash([ytheta := `+`(`-`(`*`(.5000000000, `*`(`+`(`*`(9.00, `*`(`^`(cos(theta), 2))), .75), `*`(sin(theta))))))], [`+`(`-`(`*`(.5000000000, `*`(`+`(`*`(9.00, `*`(`^`(cos(theta), 2))..." align="center" border="0"></td>
<td width='5%' align='center'>(12)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_121.gif" width="768" height="17" alt="x1 := diff(subs(r = 1.5, ennpolar[1]), theta); 1; z1 := diff(subs(r = 1.5, ennpolar[3]), theta); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_122.gif" width="341" height="25" alt="Typesetting:-mprintslash([x1 := `+`(`*`(13.50000000, `*`(sin(theta), `*`(`^`(cos(theta), 2)))), `-`(`*`(4.875, `*`(sin(theta)))))], [`+`(`*`(13.50000000, `*`(sin(theta), `*`(`^`(cos(theta), 2)))), `-`..." align="center" border="0"></td>
</tr></font> </p>
<p align="center"><tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_123.gif" width="183" height="21" alt="Typesetting:-mprintslash([z1 := `+`(`-`(`*`(9.00, `*`(cos(theta), `*`(sin(theta))))))], [`+`(`-`(`*`(9.00, `*`(cos(theta), `*`(sin(theta))))))])" align="center" border="0"></td>
<td width='5%' align='center'>(13)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">We now evaluate the surface integrals over the Enneper's surface and the cylinder, respectively:</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_124.gif" width="520" height="21" alt="evalf(Int(`+`(`*`(2, `*`(abs(ytheta), `*`(sqrt(`+`(`*`(`^`(x1, 2)), `*`(`^`(z1, 2)))))))), theta = 0 .. Pi)); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_125.gif" width="95" height="18" alt="31.66323514" align="center" border="0"></td>
<td width='5%' align='center'>(14)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_126.gif" width="387" height="21" alt="evalf(subs(r = 1.5, `*`(Pi, `*`(`^`(r, 2), `*`(`+`(1, `*`(`^`(r, 2)), `*`(`/`(1, 3), `*`(`^`(r, 4))))))))); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_127.gif" width="95" height="18" alt="34.90113089" align="center" border="0"></td>
<td width='5%' align='center'>(15)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">Clearly, the cylinder has lower area. So even though the Ennneper's surface is minimal, the cylinder has a lowe surface area within the boundary condition.</font> </p>
<p align="left"><font color="#000000" size="5" face="Times New Roman"><strong>Weierstrass-Enneper Representation</strong></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">Begin by defining the parameters as real values.</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_128.gif" width="768" height="14" alt="assume(u, real); 1; additionally(v, real); 1; additionally(t, real); 1; is(u, real); 1; is(v, real); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_129.gif" width="38" height="18" alt="true" align="center" border="0"></td>
</tr></font> </p>
<p align="center"><tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_130.gif" width="38" height="18" alt="true" align="center" border="0"></td>
<td width='5%' align='center'>(16)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_131.gif" width="176" height="13" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_132.gif" width="218" height="11" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_133.gif" width="170" height="13" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_134.gif" width="186" height="14" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_135.gif" width="150" height="14" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_136.gif" width="100" height="11" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_137.gif" width="178" height="14" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_138.gif" width="178" height="14" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_139.gif" width="192" height="14" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_140.gif" width="107" height="11" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_141.gif" width="220" height="13" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_142.gif" width="216" height="14" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_143.gif" width="237" height="14" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_144.gif" width="748" height="13" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_145.gif" width="748" height="13" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_146.gif" width="748" height="13" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"><br><img src="images/Maple Minimal Surfaces_147.gif" width="163" height="14" alt="Weierfg := proc (f, g, a) local Z1, Z2, X1, X2, X3, Z3, X; Z1 := int(`*`(f, `*`(`+`(1, `-`(`*`(`^`(g, 2)))))), z); Z2 := int(`*`(I, `*`(f, `*`(`+`(1, `*`(`^`(g, 2)))))), z); Z3 := int(`+`(`*`(2, `*`(g..." align="center" border="0"></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_148.gif" width="166" height="16" alt="Weierfg(z, `*`(`^`(z, 3)), 0); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#0000ff" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_149.gif" width="768" height="33" alt="[`+`(`-`(`*`(`/`(1, 8), `*`(`^`(u, 8)))), `*`(`/`(7, 2), `*`(`^`(u, 6), `*`(`^`(v, 2)))), `-`(`*`(`/`(35, 4), `*`(`^`(u, 4), `*`(`^`(v, 4))))), `*`(`/`(1, 8), `*`(`+`(`*`(28, `*`(`^`(v, 6))), 4), `*`(..." align="center" border="0"></td>
<td width='5%' align='center'>(17)</td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">We can now graph the Pope's hat:</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_150.gif" width="768" height="12" alt="plot3d(Weierfg(`*`(`^`(z, 3)), `*`(`^`(z, 2)), z), u = -1 .. 1, v = -1 .. 1, grid = [40, 40], colour = gold, style = patch, scaling = constrained, orientation = [-10, 0, 75], lightmodel = light4); 1" align="center" border="0"><br><img src="images/Maple Minimal Surfaces_151.gif" width="768" height="12" alt="plot3d(Weierfg(`*`(`^`(z, 3)), `*`(`^`(z, 2)), z), u = -1 .. 1, v = -1 .. 1, grid = [40, 40], colour = gold, style = patch, scaling = constrained, orientation = [-10, 0, 75], lightmodel = light4); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_152.gif" width="400" height="400" alt="Plot_2d" align="center" border="0"></td>
<td width='5%' align='center'></td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman">Or a bat:</font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_153.gif" width="768" height="12" alt="plot3d(Weierfg(z, `*`(`^`(z, 3)), 0), u = -1 .. 1, v = -1 .. 1, grid = [40, 40], color = grey, shading = z, style = patch, scaling = constrained, lightmodel = light2, orientation = [0, 53]); 1" align="center" border="0"><br><img src="images/Maple Minimal Surfaces_154.gif" width="768" height="12" alt="plot3d(Weierfg(z, `*`(`^`(z, 3)), 0), u = -1 .. 1, v = -1 .. 1, grid = [40, 40], color = grey, shading = z, style = patch, scaling = constrained, lightmodel = light2, orientation = [0, 53]); 1" align="center" border="0"></font> </p>
<p align="center"><table width='100%'>
<tr>
<td valign="top" align="center"><font color="#000000" size="3" face="Times New Roman"><img src="images/Maple Minimal Surfaces_155.gif" width="400" height="400" alt="Plot_2d" align="center" border="0"></td>
<td width='5%' align='center'></td>
</tr>
</table></font> </p>
<p align="left"><font color="#000000" size="3" face="Times New Roman"></font> </p>
<p align="center"><font color="#000000" size="7" face="Times New Roman"><strong>References</strong></font> </p>
<p align="left"><ul><li><font color="#000000" size="3" face="Times New Roman">Oprea, John. </font><font color="#000000" size="3" face="Times New Roman"><em>Differential Geometry and its Applications. </em></font><font color="#000000" size="3" face="Times New Roman">The Mathematical Association of America, 2007.</font></li></ul> </p>
<p align="left"><ul><li><font color="#000000" size="3" face="Times New Roman">Minimal Surface. (n.d.). In Wikipedia. Retrieved June 15, 2020, from https://en.wikipedia.org/wiki/Minimal_surface.</font></li></ul> </p>
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