DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces
Differential Geometry for Curves and Surfaces
57
1 Dr. Scott Schaefer Differential Geometry for Curves and Surfaces
description
Differential Geometry for Curves and Surfaces. Dr. Scott Schaefer. Intrinsic Properties of Curves. Intrinsic Properties of Curves. Intrinsic Properties of Curves. Identical curves but different derivatives!!!. Arc Length. s ( t )= t implies arc-length parameterization - PowerPoint PPT Presentation
Transcript of Differential Geometry for Curves and Surfaces
1
Dr. Scott Schaefer
Differential Geometry for Curves and Surfaces
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Intrinsic Properties of Curves
))sin(),(cos()( tttp
2
2
2
2
1
2,
1
1)(
t
t
t
ttq
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Intrinsic Properties of Curves
))sin(),(cos()( tttp
2
2
2
2
1
2,
1
1)(
t
t
t
ttq
)0,1()0()0( qp
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Intrinsic Properties of Curves
))sin(),(cos()( tttp
2
2
2
2
1
2,
1
1)(
t
t
t
ttq
)0,1()0()0( qp
)0(')2,0()1,0()0(' qp
Identical curves but different derivatives!!!
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Arc Length
s(t)=t implies arc-length parameterization Independent under parameterization!!!
t
a
dttpts )(')(
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Frenet Frame
Unit-length tangent)('
)(')(
tp
tptT
)(tp)(tT
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Frenet Frame
Unit-length tangent
Unit-length normal
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN
)(tp)(tT
)(tN
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Frenet Frame
Unit-length tangent
Unit-length normal
Binormal
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN
)()()( tNtTtB
)(tB
)(tp)(tT
)(tN
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Frenet Frame
Provides an orthogonal frame anywhere on curve
0)()()()()()( tNtTtNtBtTtB
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN )()()( tNtTtB
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Frenet Frame
Provides an orthogonal frame anywhere on curve
0)()()()()()( tNtTtNtBtTtB
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN )()()( tNtTtB
Trivial due to cross-product
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Frenet Frame
Provides an orthogonal frame anywhere on curve
0)()()()()()( tNtTtNtBtTtB
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN )()()( tNtTtB
1)()( tTtT
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Frenet Frame
Provides an orthogonal frame anywhere on curve
0)()()()()()( tNtTtNtBtTtB
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN )()()( tNtTtB
0)(')()()(' tTtTtTtT1)()( tTtT
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Frenet Frame
Provides an orthogonal frame anywhere on curve
0)()()()()()( tNtTtNtBtTtB
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN )()()( tNtTtB
0)(')()()(' tTtTtTtT1)()( tTtT
0)()( tNtT
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Uses of Frenet Frames
Animation of a camera Extruding a cylinder along a path
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Uses of Frenet Frames
Animation of a camera Extruding a cylinder along a path
Problems: The Frenet frame becomes unstable at inflection points or even undefined when 0)(' tT
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN )()()( tNtTtB
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)(tN
Osculating Plane
Plane defined by the point p(t) and the vectors T(t) and N(t)
Locally the curve resides in this plane
)(tB
)(tp)(tT
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Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
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Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
s
T
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Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
s
T
t
s
s
T
t
T
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Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
)('
)(')(
tp
tTt
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Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
3)('
)('')('
)('
)(')(
tp
tptp
tp
tTt
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Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
3)('
)('')('
)('
)(')(
tp
tptp
tp
tTt
)(
1
tr
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Torsion
Measure of how much the curve twists or how quickly the curve leaves the osculating plane
)(tN
)(tB
)(tp)(tT
)(')( sBs
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Frenet Equations
)()()(' sNssT )()()()()(' sTssBssN
)()()(' sNssB
)(tN
)(tB
)(tp)(tT
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Unit-length tangent
Unit-length normal
Binormal
Frenet Frames
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN
)()()( tNtTtB
)(tB
)(tp)(tT
)(tN
Problem!
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Rotation Minimizing Frames
)(tN
)(tB
)(tp
)(tT
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Rotation Minimizing Frames
)(tN
)(tB
)(tp
)(tT
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Rotation Minimizing Frames
Build minimal rotation to next tangent
)(tN
)(tB
)(tp
)(tT
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Rotation Minimizing Frames
Build minimal rotation to next tangent
)(tN
)(tB
)(tp
)(tT
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Rotation Minimizing Frames
Frenet Frame Rotation Minimizing Frame
Image taken from “Computation of Rotation Minimizing Frames”
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Rotation Minimizing Frames
Image taken from “Computation of Rotation Minimizing Frames”
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Surfaces
Consider a curve r(t)=(u(t),v(t))
v
u
)(tr
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Surfaces
Consider a curve r(t)=(u(t),v(t)) p(r(t)) is a curve on the surface
))(( trp
v
u
)(tr
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Surfaces
Consider a curve r(t)=(u(t),v(t)) p(r(t)) is a curve on the surface
))(( trp
dttrptst
t0
))((')(
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Surfaces
Consider a curve r(t)=(u(t),v(t)) p(r(t)) is a curve on the surface
t
v
v
p
t
u
u
ptrp
))(('
))(( trp
dttrptst
t0
))((')(
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Surfaces
Consider a curve r(t)=(u(t),v(t)) p(r(t)) is a curve on the surface
22 2))((' tv
vvtv
tu
vutu
uu pppppptrp
))(( trp
dttrptst
t0
))((')(
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Surfaces
Consider a curve r(t)=(u(t),v(t)) p(r(t)) is a curve on the surface
22 2))((' tv
vvtv
tu
vutu
uu pppppptrp
))(( trp
dttrptst
t0
))((')(
vv
vu
uu
ppG
ppF
ppE
First fundamental form
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First Fundamental Form
Given any curve in parameter space r(t)=(u(t),v(t)), arc length of curve on surface is
vvvuuu ppGppFppE
))(( trp
dtGFEtst
ttv
tv
tu
tu
0
22 2)(
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First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
upvp
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First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
2222)sin(baba
upvp
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First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
2222)cos(1 baba
upvp
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First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
2222bababa
upvp
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First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
2222bababa
upvp
222
vuvuvu pppppp
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First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
2222bababa
upvp
2FEGpp vu
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First Fundamental Form
Surface area over U is given by
vvvuuu ppGppFppE
U
FEG 2
U
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Second Fundamental Form
Consider a curve p(r(s)) parameterized with respect to arc-length where r(s)=(u(s),v(s))
Curvature is given by )()())(('')(' sMssrpsT
))(( srp
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Second Fundamental Form
Consider a curve p(r(s)) parameterized with respect to arc-length where r(s)=(u(s),v(s))
Curvature is given by )()())(('')(' sMssrpsT
2
2
2
222 2)()()('su
usv
vsv
vvsv
su
uvsu
uu pppppsMssT
))(( srp
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))(( srp
Second Fundamental Form
Consider a curve p(r(s)) parameterized with respect to arc-length where r(s)=(u(s),v(s))
Curvature is given by
Let n be the normal of p(u,v)
)()())(('')(' sMssrpsT
2
2
2
222 2)()()('su
usv
vsv
vvsv
su
uvsu
uu pppppsMssT
)cos()( sMn
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))(( srp
Second Fundamental Form
Consider a curve p(r(s)) parameterized with respect to arc-length where r(s)=(u(s),v(s))
Curvature is given by
Let n be the normal of p(u,v)
)()())(('')(' sMssrpsT
2
2
2
222 2)()()('su
usv
vsv
vvsv
su
uvsu
uu pppppsMssT
)cos()( sMn )cos()()(' ssTn 22 2)cos()( s
vvvs
vsu
uvsu
uu pnpnpns
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))(( srp
Second Fundamental Form
Consider a curve p(r(s)) parameterized with respect to arc-length where r(s)=(u(s),v(s))
Curvature is given by
Let n be the normal of p(u,v)
)()())(('')(' sMssrpsT
2
2
2
222 2)()()('su
usv
vsv
vvsv
su
uvsu
uu pppppsMssT
)cos()( sMn )cos()()(' ssTn 22 2)cos()( s
vvvs
vsu
uvsu
uu pnpnpns
vv
uv
uu
pnN
pnM
pnL
Second Fundamental Form
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Meusnier’s Theorem
Assume , is called the normal curvature
Meusnier’s Theorem states that all curves on p(u,v) passing through a point x having the same tangent, have the same normal curvature
1)( sMn )(s
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Lines of Curvature
We can parameterize all tangents through x using a single parameter
2
2
2
2)(
GFE
NML
)tan( dudv
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Principle Curvatures
)(max)(min 21
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Principle Curvatures
)(max)(min 21
22
22
2
)2(20)('
GFE
GFNMLNMGFE
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Principle Curvatures
)(max)(min 21
02 GMFNGLENFLEM
22
22
2
)2(20)('
GFE
GFNMLNMGFE
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Gaussian and Mean Curvature
Gaussian Curvature:
Mean Curvature:
2
2
21 FEG
MLNK
)(2
2
2 221
FEG
LGMFNEH
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Gaussian and Mean Curvature
Gaussian Curvature:
Mean Curvature:
: elliptic : hyperbolic : parabolic : flat
2
2
21 FEG
MLNK
0K0K
00 21 00 21
)(2
2
2 221
FEG
LGMFNEH
