computational geometry - korea universityelie.korea.ac.kr/~esen/3dmini.pdf · 2011-12-29 · the...
TRANSCRIPT
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
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
References
J. Oprea, The Mathematics of Soap Films, AMS, (2000).
S. Osher, R. Fedkiw, Level Set Methods and Dynamics Implicit Surfaces, Springer, (2002).