Computational Geometry - Korea Universityelie.korea.ac.kr/~esen/3dmini.pdf · 2011-12-29 · The...

27
Construction of Minimal Surface Computational Lab 28DEC11 Computational Geometry π

Transcript of Computational Geometry - Korea Universityelie.korea.ac.kr/~esen/3dmini.pdf · 2011-12-29 · The...

Construction of Minimal Surface

Computational Lab

28DEC11

Computational Geometry

π

Outline

A review on Geometry

Weierstrass Representation

Surface generation

Unsigned level set function

Data conversion

3D printer

References

Geometry: First and second fundamental form

Parametrize a smooth surface in

by a map

is smooth, regular

First fundamental form

Second fundamental form

S 3R

SX :

X

22 2: GdvFdudvEdu

22 2: NdvMdudvLdu

GF

FE

gf

fe

The mean curvature and the Guass curvature are defined by

The mean curvature

The Gauss curvature

Geometry:mean curvature and gauss curvature

,2

2

1)(

2

1:

2

1

FEG

EgfFeGtrH

,)(:2

21

FEG

fegDetK

Thm) If the surface is locally area minimizing, then the mean curvature vanishes identically .

Def) The surface is minimal if its mean curvature vanishes identically.

3RS

H 0H

3RS

H

Geometry: minimal surface

Minimal surface and soap film

Motion by mean curvature flow

The total curvature of is defined to be

finite total curvature if .

It was long conjectured that the only complete, embedded minimal surfaces in of finite type are the plane, catenoid and helicoid.

Geometry: Finite total curvature

sK S

.||:

dAKK s

S sK

3R

Weierstrass-Enneper representation I

If is holomorphic on a domain , is meromorphic on and is holomorphic on D, then a minimal surface is defined by the parametrization ,where

f DD 2fg

g

)),(),,(),,((),( 321 zzxzzxzzxzz

,)1(Re),( 21 dzgfzzx

,)1(iRe),( 22 dzgfzzx

,)(2Re),(3 dzfgzzx

Weierstrass-Enneper representation II

Consider and define . Then,

for any holomorphic function , a minimal surface is defined by the parameterization

, where

)(F

,)()1(Re),( 21 dFfzzx

.)(2Re),(3 dzFzzx

)),(),,(),,((),( 321 zzxzzxzzxzz

,)()1(iRe),( 22 dFfzzx

g '/)( gfF

Weierstrass-Enneper representation example

Let and . After integration of Weierstrass-Enneper representation II,

we have as follows

The resulting parametrization

22

i)(

F

ze

,2

i)1(Re),(

2

21

dzzx ,2

i)1(iRe),(

2

22

dfzzx .2

i2Re),(

2

3 dzzzx

)coshiRe( z )sinh-Re( z z i

.sinsinh vu .cossinh vu .v

),cossinh,sin(sinh vvuvu

Visualization:Mathematica 3D-plot

helicoid=

ParametricPlot3D[{ Sinh[u] Cos[v], Sinh[u] Sin[v], v},{u, -2,2}, {v,0,2p}] catenoid= ParametricPlot3D[{ Cosh[u] Cos[v], Cosh[u] Sin[v], u},{u, -1.5,1.5}, {r,0,2p]

Costa-Hoffman-Meeks surface

Let be the Reimann surface

and

A riemann surface of genus k from which

three points are removed.

Weierstrass data for this surface

, where c is a positive real constant to be determined. To find constant C, we solve the period problem later.

Parametrizing the Surface

For drawing surface, we exploit its symmetries. To get a parametrization of the surface patch adapted to this geometry, we need coordinate lines which look like polar coordinates near the ends. This can be done using

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5 3 3.5 2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

3

Parametrizing the Surface

We use instead of , that’s,

Lopez-Ros parameter

Period problem

where

01

23

4

3

2

1

0

2

1

0

4

3

2

1

0

02

4

6

8

0

2

4

Costa-Hoffman-Meeks surface

Visualization:MATLAB

Unorganized points: catenoid Costa-Hoffman-Meeks genus 3

Surface generation

Unsigned level set method

Visualization:MATLAB

Unorganized points catenoid Costa-Hoffman-Meeks genus 4

P-surface Diamond Gyroid Neovius

Visualization:3D Print

Visualization:3D PRINT Procedure

Preparation setup Post-Processing

Visualization:3D Print

References

J. Oprea, The Mathematics of Soap Films, AMS, (2000).

S. Osher, R. Fedkiw, Level Set Methods and Dynamics Implicit Surfaces, Springer, (2002).

Any Question?