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Dr. Scott Schaefer
Differential Geometry for Curves and Surfaces
2/57
Intrinsic Properties of Curves
))sin(),(cos()( tttp
2
2
2
2
1
2,
1
1)(
t
t
t
ttq
3/57
Intrinsic Properties of Curves
))sin(),(cos()( tttp
2
2
2
2
1
2,
1
1)(
t
t
t
ttq
)0,1()0()0( qp
4/57
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!!!
5/57
Arc Length
s(t)=t implies arc-length parameterization Independent under parameterization!!!
t
a
dttpts )(')(
6/57
Frenet Frame
Unit-length tangent)('
)(')(
tp
tptT
)(tp)(tT
7/57
Frenet Frame
Unit-length tangent
Unit-length normal
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN
)(tp)(tT
)(tN
8/57
Frenet Frame
Unit-length tangent
Unit-length normal
Binormal
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN
)()()( tNtTtB
)(tB
)(tp)(tT
)(tN
9/57
Frenet Frame
Provides an orthogonal frame anywhere on curve
0)()()()()()( tNtTtNtBtTtB
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN )()()( tNtTtB
10/57
Frenet Frame
Provides an orthogonal frame anywhere on curve
0)()()()()()( tNtTtNtBtTtB
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN )()()( tNtTtB
Trivial due to cross-product
11/57
Frenet Frame
Provides an orthogonal frame anywhere on curve
0)()()()()()( tNtTtNtBtTtB
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN )()()( tNtTtB
1)()( tTtT
12/57
Frenet Frame
Provides an orthogonal frame anywhere on curve
0)()()()()()( tNtTtNtBtTtB
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN )()()( tNtTtB
0)(')()()(' tTtTtTtT1)()( tTtT
13/57
Frenet Frame
Provides an orthogonal frame anywhere on curve
0)()()()()()( tNtTtNtBtTtB
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN )()()( tNtTtB
0)(')()()(' tTtTtTtT1)()( tTtT
0)()( tNtT
14/57
Uses of Frenet Frames
Animation of a camera Extruding a cylinder along a path
15/57
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
16/57
)(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
17/57
Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
18/57
Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
s
T
19/57
Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
s
T
t
s
s
T
t
T
20/57
Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
)('
)(')(
tp
tTt
21/57
Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
3)('
)('')('
)('
)(')(
tp
tptp
tp
tTt
22/57
Curvature
Measure of how much the curve bends
)(tN
)(tp)(tT
3)('
)('')('
)('
)(')(
tp
tptp
tp
tTt
)(
1
tr
23/57
Torsion
Measure of how much the curve twists or how quickly the curve leaves the osculating plane
)(tN
)(tB
)(tp)(tT
)(')( sBs
24/57
Frenet Equations
)()()(' sNssT )()()()()(' sTssBssN
)()()(' sNssB
)(tN
)(tB
)(tp)(tT
25/57
Unit-length tangent
Unit-length normal
Binormal
Frenet Frames
)('
)(')(
tp
tptT
)('
)(')(
tT
tTtN
)()()( tNtTtB
)(tB
)(tp)(tT
)(tN
Problem!
26/57
Rotation Minimizing Frames
)(tN
)(tB
)(tp
)(tT
27/57
Rotation Minimizing Frames
)(tN
)(tB
)(tp
)(tT
28/57
Rotation Minimizing Frames
Build minimal rotation to next tangent
)(tN
)(tB
)(tp
)(tT
29/57
Rotation Minimizing Frames
Build minimal rotation to next tangent
)(tN
)(tB
)(tp
)(tT
30/57
Rotation Minimizing Frames
Frenet Frame Rotation Minimizing Frame
Image taken from “Computation of Rotation Minimizing Frames”
31/57
Rotation Minimizing Frames
Image taken from “Computation of Rotation Minimizing Frames”
32/57
Surfaces
Consider a curve r(t)=(u(t),v(t))
v
u
)(tr
33/57
Surfaces
Consider a curve r(t)=(u(t),v(t)) p(r(t)) is a curve on the surface
))(( trp
v
u
)(tr
34/57
Surfaces
Consider a curve r(t)=(u(t),v(t)) p(r(t)) is a curve on the surface
))(( trp
dttrptst
t0
))((')(
35/57
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
))((')(
36/57
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
))((')(
37/57
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
38/57
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)(
39/57
First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
upvp
40/57
First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
2222)sin(baba
upvp
41/57
First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
2222)cos(1 baba
upvp
42/57
First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
2222bababa
upvp
43/57
First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
2222bababa
upvp
222
vuvuvu pppppp
44/57
First Fundamental Form
The infinitesimal surface area at u, v is given by
vvvuuu ppGppFppE
vu pp
2222bababa
upvp
2FEGpp vu
45/57
First Fundamental Form
Surface area over U is given by
vvvuuu ppGppFppE
U
FEG 2
U
46/57
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
47/57
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
48/57
))(( 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
49/57
))(( 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
50/57
))(( 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
51/57
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
52/57
Lines of Curvature
We can parameterize all tangents through x using a single parameter
2
2
2
2)(
GFE
NML
)tan( dudv
53/57
Principle Curvatures
)(max)(min 21
54/57
Principle Curvatures
)(max)(min 21
22
22
2
)2(20)('
GFE
GFNMLNMGFE
55/57
Principle Curvatures
)(max)(min 21
02 GMFNGLENFLEM
22
22
2
)2(20)('
GFE
GFNMLNMGFE
56/57
Gaussian and Mean Curvature
Gaussian Curvature:
Mean Curvature:
2
2
21 FEG
MLNK
)(2
2
2 221
FEG
LGMFNEH
57/57
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
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