Differential Geometry
for Curves and Surfaces
Dr. Scott Schaefer
1
Intrinsic Properties of Curves
p(t )  (cos(t ), sin( t ))
 1  t 2 2t 2 

q(t )  
,
2
2 
1 t 1 t 
2/57
Intrinsic Properties of Curves
p(t )  (cos(t ), sin( t ))
 1  t 2 2t 2 

q(t )  
,
2
2 
1 t 1 t 
p(0)  q(0)  (1,0)
3/57
Intrinsic Properties of Curves
p(t )  (cos(t ), sin( t ))
 1  t 2 2t 2 

q(t )  
,
2
2 
1 t 1 t 
p(0)  q(0)  (1,0)
p' (0)  (0,1)  (0,2)  q' (0)
Identical curves but different derivatives!!!
4/57
Arc Length
t
s(t )   p' (t ) dt
a
s(t)=t implies arc-length parameterization
 Independent under parameterization!!!

5/57
Frenet Frame
p' (t )
 Unit-length tangent T (t ) 
p' (t )
T (t )
p (t )
6/57
Frenet Frame
p' (t )
 Unit-length tangent T (t ) 
p' (t )
T ' (t )
 Unit-length normal N (t ) 
T ' (t )
N (t )
T (t )
p (t )
7/57
Frenet Frame
p' (t )
 Unit-length tangent T (t ) 
p' (t )
T ' (t )
 Unit-length normal N (t ) 
T ' (t )

Binormal
B(t )  T (t )  N (t )
N (t )
T (t )
p (t )
B (t )
8/57
Frenet Frame
p' (t )
T (t ) 
p' (t )
T ' (t )
N (t ) 
T ' (t )
B(t )  T (t )  N (t )
Provides an orthogonal frame anywhere on
curve
 B(t )  T (t )  B(t )  N (t )  T (t )  N (t )  0

9/57
Frenet Frame
p' (t )
T (t ) 
p' (t )
T ' (t )
N (t ) 
T ' (t )
B(t )  T (t )  N (t )
Provides an orthogonal frame anywhere on
curve
 B(t )  T (t )  B(t )  N (t )  T (t )  N (t )  0

Trivial due to cross-product
10/57
Frenet Frame
p' (t )
T (t ) 
p' (t )
T ' (t )
N (t ) 
T ' (t )
B(t )  T (t )  N (t )
Provides an orthogonal frame anywhere on
curve
 B(t )  T (t )  B(t )  N (t )  T (t )  N (t )  0

T (t )  T (t )  1
11/57
Frenet Frame
p' (t )
T (t ) 
p' (t )
T ' (t )
N (t ) 
T ' (t )
B(t )  T (t )  N (t )
Provides an orthogonal frame anywhere on
curve
 B(t )  T (t )  B(t )  N (t )  T (t )  N (t )  0

T (t )  T (t )  1
T ' (t )  T (t )  T (t )  T ' (t )  0
12/57
Frenet Frame
p' (t )
T (t ) 
p' (t )
T ' (t )
N (t ) 
T ' (t )
B(t )  T (t )  N (t )
Provides an orthogonal frame anywhere on
curve
 B(t )  T (t )  B(t )  N (t )  T (t )  N (t )  0

T (t )  T (t )  1
T ' (t )  T (t )  T (t )  T ' (t )  0
T (t )  N (t )  0
13/57
Uses of Frenet Frames
Animation of a camera
 Extruding a cylinder along a path

14/57
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 T ' (t )  0
p' (t )
T (t ) 
p' (t )
T ' (t )
N (t ) 
T ' (t )
B(t )  T (t )  N (t )
15/57
Osculating Plane
Plane defined by the point p(t) and the
vectors T(t) and N(t)
 Locally the curve resides in this plane

N (t )
T (t )
p (t )
B (t )
16/57
Curvature

Measure of how much the curve bends
N (t )
T (t )
p (t )
17/57
Curvature

Measure of how much the curve bends
T

s
N (t )
T (t )
p (t )
18/57
Curvature

Measure of how much the curve bends
N (t )
T

s
T T s

t
s t
T (t )
p (t )
19/57
Curvature

Measure of how much the curve bends
 (t ) 
T ' (t )
p' (t )
N (t )
T (t )
p (t )
20/57
Curvature

Measure of how much the curve bends
 (t ) 
T ' (t )
p' (t )

p' (t )  p' ' (t )
p' (t )
3
N (t )
T (t )
p (t )
21/57
Curvature

Measure of how much the curve bends
 (t ) 
T ' (t )
p' (t )

p' (t )  p' ' (t )
p' (t )
3
N (t )
T (t )
1
r
 (t )
p (t )
22/57
Torsion

Measure of how much the curve twists or
how quickly the curve leaves the osculating
plane
 (s)  B' (s)
N (t )
T (t )
p (t )
B (t )
23/57
Frenet Equations



T ' ( s)   ( s) N ( s)
N ' ( s)   ( s) B( s)   ( s)T ( s)
B' ( s)   ( s) N ( s)
N (t )
T (t )
p (t )
B (t )
24/57
Frenet Frames
p' (t )
 Unit-length tangent T (t ) 
p' (t )
T ' (t )
 Unit-length normal N (t ) 
T ' (t )

Binormal
Problem!
B(t )  T (t )  N (t )
N (t )
T (t )
p (t )
B (t )
25/57
Rotation Minimizing Frames
p (t )
T (t )
N (t )
B (t )
26/57
Rotation Minimizing Frames
p (t )
T (t )
N (t )
B (t )
27/57
Rotation Minimizing Frames

Build minimal rotation to next tangent
p (t )
T (t )
N (t )
B (t )
28/57
Rotation Minimizing Frames

Build minimal rotation to next tangent
N (t )
T (t )
p (t )
B (t )
29/57
Rotation Minimizing Frames
Frenet Frame
Image taken from “Computation of Rotation Minimizing Frames”
Rotation Minimizing Frame
30/57
Rotation Minimizing Frames
Image taken from “Computation of Rotation Minimizing Frames”
31/57
Surfaces

Consider a curve r(t)=(u(t),v(t))
v
r (t )
u
32/57
Surfaces
Consider a curve r(t)=(u(t),v(t))
 p(r(t)) is a curve on the surface

v
r (t )
p(r (t ))
u
33/57
Surfaces
Consider a curve r(t)=(u(t),v(t))
 p(r(t)) is a curve on the surface

t
s(t )   p' (r (t )) dt
t0
p(r (t ))
34/57
Surfaces
Consider a curve r(t)=(u(t),v(t))
 p(r(t)) is a curve on the surface

t
s(t )   p' (r (t )) dt
t0
p u p v
p ' (r (t )) 

u t v t
p(r (t ))
35/57
Surfaces
Consider a curve r(t)=(u(t),v(t))
 p(r(t)) is a curve on the surface

t
s(t )   p' (r (t )) dt
t0
p' (r (t )) 
pu  pu  ut   2 pu  pv
2
u v
t t
 pv  pv  vt 
2
p(r (t ))
36/57
Surfaces
Consider a curve r(t)=(u(t),v(t))
 p(r(t)) is a curve on the surface

t
s(t )   p' (r (t )) dt
t0
p' (r (t )) 
pu  pu  ut   2 pu  pv
2
u v
t t
 pv  pv  vt 
2
E  pu  pu
p(r (t ))
F  pu  pv
G  pv  pv
First fundamental form
37/57
First Fundamental Form
E  pu  pu F  pu  pv G  pv  pv

Given any curve in parameter space
r(t)=(u(t),v(t)), arc length of curve on surface is
t
s(t )   E
u 2
t
 2F
u v
t t
G
v 2
t
dt
t0
p(r (t ))
38/57
First Fundamental Form
E  pu  pu F  pu  pv G  pv  pv

The infinitesimal surface area at u, v is given
by pu  pv
pu
pv
39/57
First Fundamental Form
E  pu  pu F  pu  pv G  pv  pv

The infinitesimal surface area at u, v is given
by pu  pv
a  b  a b sin(  ) 2
2
2
2
pu
pv
40/57
First Fundamental Form
E  pu  pu F  pu  pv G  pv  pv

The infinitesimal surface area at u, v is given
by pu  pv

a  b  a b 1  cos( ) 2
2
2
2

pu
pv
41/57
First Fundamental Form
E  pu  pu F  pu  pv G  pv  pv

The infinitesimal surface area at u, v is given
by pu  pv
a  b  a b  a  b 
2
2
2
2
pu
pv
42/57
First Fundamental Form
E  pu  pu F  pu  pv G  pv  pv

The infinitesimal surface area at u, v is given
by pu  pv
a  b  a b  a  b 
2
pu  pv 
2
pu
2
2
2
pv   pu  pv 
2
2
pu
pv
43/57
First Fundamental Form
E  pu  pu F  pu  pv G  pv  pv

The infinitesimal surface area at u, v is given
by pu  pv
a  b  a b  a  b 
2
2
2
2
pu  pv  EG  F 2
pu
pv
44/57
First Fundamental Form
E  pu  pu F  pu  pv G  pv  pv

Surface area over U is given by

EG  F 2
U
U
45/57
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 T ' ( s )  p ' ' (r ( s ))   ( s ) M ( s )

p (r ( s ))
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 T ' ( s )  p ' ' (r ( s ))   ( s ) M ( s )

T ' ( s)   ( s) M ( s)  p
u 2
uu s
 2p
u v
uv s s
p
v 2
vv s
p
 2v
v s 2
p
 2u
u s 2
p (r ( s ))
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 T ' ( s )  p ' ' (r ( s ))   ( s ) M ( s )

T ' ( s)   ( s) M ( s)  p
u 2
uu s

 2p
u v
uv s s
p
v 2
vv s
p
 2v
v s 2
p
 2u
u s 2
Let n be the normal of p(u,v)
n  M ( s)  cos( )
p (r ( s ))
48/57
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 T ' ( s )  p ' ' (r ( s ))   ( s ) M ( s )

T ' ( s)   ( s) M ( s)  p
u 2
uu s

 2p
u v
uv s s
p
v 2
vv s
p
 2v
v s 2
p
 2u
u s 2
Let n be the normal of p(u,v)
n  M ( s)  cos( )
n  T ' ( s)   ( s) cos( )
2
2
 (s) cos( )  n  puu us  2n  puv us vs  n  pvv vs
p (r ( s ))
49/57
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 T ' ( s )  p ' ' (r ( s ))   ( s ) M ( s )

T ' ( s)   ( s) M ( s)  p
u 2
uu s

 2p
u v
uv s s
p
v 2
vv s
p
 2v
v s 2
p
 2u
u s 2
Let n be the normal of p(u,v)
n  M ( s)  cos( )
n  T ' ( s)   ( s) cos( )
2
2
 (s) cos( )  n  puu us  2n  puv us vs  n  pvv vs
L  n  puu
p (r ( s ))
M  n  puv
N  n  pvv
Second Fundamental Form
50/57
Meusnier’s Theorem
Assume n  M ( s)  1,  (s) 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

51/57
Lines of Curvature

We can parameterize all tangents through x
using a single parameter 
L  2M  N2
 ( ) 
2
E  2 F  G
dv
  du
 tan( )
52/57
Principle Curvatures
1  min  ( )
 2  max  ( )
53/57
Principle Curvatures
1  min  ( )  2  max  ( )

E2 F G  M  N ( L2 M  N )  F G 
 '( )  0 
E2 FG 
2
2
2 2
54/57
Principle Curvatures
1  min  ( )  2  max  ( )

E2 F G  M  N ( L2 M  N )  F G 
 '( )  0 
E2 FG 
2
2
2 2
EM  FL  EN  GL  FN  GM   0
2
55/57
Gaussian and Mean Curvature
LN  M 2
 Gaussian Curvature: K  1 2 
2
EG  F
1   2 NE  2MF  LG

 Mean Curvature: H 
2
2( EG  F 2 )
56/57
Gaussian and Mean Curvature
LN  M 2
 Gaussian Curvature: K  1 2 
2
EG  F
1   2 NE  2MF  LG

 Mean Curvature: H 
2
2( EG  F 2 )
K  0 : elliptic
 K  0 : hyperbolic
 1  0   2  0 : parabolic
 1  0   2  0 : flat

57/57