L3_Surfaces_II
L3: Surfaces (II)
Hao Su
Machine Learning meets Geometry
Warm Up (Review)
Differential Map
X
f(U)
U
Df (X)
f
Directional Normal Curvature
Df
Df
Note: is not the curvature of κn κ γ
X U
p
f
Dfp(X)
DNp(X)
N
tangent plane
Directional Normal Curvature
Df
Df
Note: is not the curvature of κn κ γ
X U
p
f
Dfp(X)
DNp(X)
N
tangent plane
κn(X) := ⟨T, ⃗κ ⟩ =
⟨Dfp(X), DNp(X)⟩
∥Dfp(X)∥2
Principal Curvatures
Maximal curvature:
Minimal curvature:
κ1 = κmax = max
φ
κn(φ)
κ2 = κmin = min
φ
κn(φ)
N
X1
X2
N X1N X2
Euler’s Theorem: Planes of principal curvature are orthogonal
and independent of parameterization.
Principal Directions
κn(φ) = κ1 cos
2 φ + κ2 sin
2 φ, φ = angle with t1
t1
Tangent plane with principal
directions ( and ) as axest1 t2
t2
φ
Agenda
• Shape Operator
• First Fundamental Form
• Fundamental Theorem of Surfaces
• Gaussian and Mean Curvature
Shape Operator
Shape Operator
• Note that
• is in the tangent plane
• is also in the tangent plane
• So the column space of and are
the same
• In other words,
∀X, DNpX
∀X, DfpX
DNp ∈ ℝ
3×2 Dfp ∈ ℝ
3×2
Shape Operator
• Note that
• is in the tangent plane
• is also in the tangent plane
• So the column space of and are
the same
• In other words, such that
∀X, DNpX
∀X, DfpX
DNp ∈ ℝ
3×2 Dfp ∈ ℝ
3×2
∃S ∈ ℝ2×2 DNp = DfpS
Shape Operator
• Note that
• is in the tangent plane
• is also in the tangent plane
• So the column space of is a subspace of
the column space of
• In other words, such that
• is called the shape operator
∀X, DNpX
∀X, DfpX
DNp ∈ ℝ
3×2
Dfp ∈ ℝ
3×2
∃S ∈ ℝ2×2 DNp = DfpS
S
Dfp(X)
DNp(X)
N
tangent plane
A Linear Map That Tells Us Normal Change
,
• Interpretation:
• When moves along , we want to know the
direction of normal change
• is just along the curve if moves along
• This linear map predicts the normal change when
moves along any direction!
DNp = DfpS
∀X ∈ Tp(ℝ
2), [DNp]X = [Dfp]SX
p X
⃗d ∈ ℝ3
⃗d p SX
S
p
∵
∴
A Linear Map That Tells Us Normal Change
,
• Interpretation:
• When moves along , we want to know the
direction of normal change
• is just along the curve if moves along
• This linear map predicts the normal change when
moves along any direction!
DNp = DfpS
∀X ∈ Tp(ℝ
2), [DNp]X = [Dfp]SX
p X
⃗d ∈ ℝ3
⃗d p SX
S
p
∵
∴
A Linear Map That Tells Us Normal Change
,
• Interpretation:
• When moves along , we want to know the
direction of normal change
• is just along the curve if moves along
• This linear map predicts the normal change when
moves along any direction!
DNp = DfpS
∀X ∈ Tp(ℝ
2), [DNp]X = [Dfp]SX
p X
⃗d ∈ ℝ3
⃗d p SX
S
p
∵
∴
Computation of Principal Directions
• Principal directions are the eigenvectors of
• Principal curvatures are the eigenvalues of
• Note: is not a symmetric matrix! Hence,
eigenvectors are not orthogonal in ; only orthogonal
when mapped to
S
S
S
ℝ2
ℝ3
Consider a nonstandard parameterization of the cylinder
(sheared along ):z
Example
Df =
−sin(u) 0
cos(u) 0
1 1
f(u, v) := [cos(u), sin(u), u + v]T
Df (X2) Df (X1)
N
N =
cos(u)
sin(u)
0
X1 = [01]
κn(X1) = 0
DN =
−sin(u) 0
cos(u) 0
0 0
X2 = [−11 ]
κn(X2) = 1
Verify the eigens of S
DNp = DfpS ⇒ S = [ 1 0−1 0]
Summary of Shape Operator
• A linear map between movement of point and
movement of normal change
• The eigen-decomposition gives the principal curvature
direction and values
First Fundamental Form
Curvature
completely determines
local surface geometry.
First Claim
Does Curvature Uniquely
Determine Global Geometry?
≠
Does Curvature Uniquely
Determine Global Geometry?
f
f*
such that:
(principal) curvature value and directions are the same
for any pair ,
∃f and f*
( f(p), f*(p)) ∀p ∈ U
≠
However,
Does Curvature Uniquely
Determine Global Geometry?
f
f*
such that (principal) curvature value and
directions are the same for any pair ,
∃f and f*
( fp, f*p ) ∀p ∈ U
≠
However,
Curvature is Insufficient to
Determine Surface Globally
Other than measuring how the
surface bends, we should also
measure length and angle!
First Fundamental Form
• Defined as the inner product in :
• : First fundament form, given , we obtain a bilinear
function
• is dependent on both and
Tp(ℝ
3)
Ip(X, Y ) = ⟨DfpX, DfpY⟩
I p
Ip p f
⇒ Ip(X, Y ) = X
T(DfTp Dfp)Y
Arc-length by I(X, Y)
• Suppose a point is moving with velocity
• So:
p ∈ U X(t)
γ(t) = f(p(t)) = f(p0 + ∫
t
0
X(t)dt)
⇒ γ′ (t) = Dfp(t)[X(t)]
s(t) = ∫
t
0
∥γ′ (t)∥dt = ∫
t
0
⟨Dfp(t)X(t), Dfp(t)X(t)⟩dt
= ∫
t
0
Ip(t)(X(t), X(t))dt
Arc-length by I(X, Y)
With , we have completely determined curve length
within the surface without referring to
s(t) = ∫
t
0
Ip(t)(X(t), X(t)) dt
I
f
Local Isometric Surfaces
For two surfaces and ,
• If there exists parameterizations and
• such that ,
• Then the two surfaces are locally isometric
Preserve length between corresponding curves!
M M*
f(U) = M
f*(U) = M*
Ip = I*p ∀p ∈ U
Figure credit: Kenneth Lloyd Patrick RoseReading: P39 of TS
Local Isometric Surfaces
Verify by yourself:
,
on
f(u, v) = [u, v,0]T f*(u, v) = [cos u, sin u, v]T
U = {(u, v) : u ∈ (0,2π), v ∈ (0,1)}
Figure credit: Kenneth Lloyd Patrick RoseReading: P39 of TS
Shape Classification by Isometry
Geodesic Distances
Extrinsically
close
Intrinsically
far
Distance Distribution Descriptor
• Compute distribution of distances for point pairs
randomly picked on the surface
Angle of Curves by I(X, Y)
• Given a vector (e.g.,
maximal principal direction)
• The angle between the tangent and the vector is:
Dfp[Y ] ∈ Tfp(ℝ
3)
φ
cos φ = ⟨
DfpX
∥DfpX∥
,
DfpY
∥DfpY∥
⟩ =
I(X, Y )
I(X, X) I(Y, Y)
fp Dfp[X]
Dfp[Y ]
φ
Angle of Curves by I(X, Y)
With , we have completely determined angles
within the surface without referring to
cos φ =
I(X, Y )
I(X, X) I(Y, Y)
I
f
Summary of First Fundamental Form
• Is a bilinear function over movement directions
(velocities) in the tangent space of
• Induced by the inner product in the tangent space at
surface point
• Completely determines curve lengths and angles
within the surface
Tp(ℝ
2)
f(p)
Fundamental Theorem of
Surfaces
First and Second Fundamental Forms
• First fundamental form (angle and length):
• Second fundamental form (bending):
• Recall the definition of normal curvature:
I(X, Y ) = ⟨DfpX, DfpY⟩
II(X, Y ) = ⟨DNpX, DfpY⟩
κn(X) :=
⟨DNpX, DfpX⟩
⟨DfpX, DfpX⟩
=
II(X, X)
I(X, X)
Uniqueness Result
Theorem:
A smooth surface is determined up to
rigid motion by its first and second
fundamental forms.
Note: compatible first and second fundamental forms
have to satisfy the Gauss-Codazzi condition (just FYI)
Gaussian and Mean Curvature
“developable”
Gaussian and Mean Curvature
• Gaussian and mean curvature also fully describe local
bending:
K := κ1κ2Gaussian:
mean: H :=
1
2
(κ1 + κ2)
“minimal”
Gauss’s Theorema Egregium
The Gaussian curvature of an
embedded smooth surface in is
invariant under the local isometries.
ℝ3
Isometric Invariance
geodesic = intrinsic
isometry = length-preserving transform
End of the Story?
Second derivative quantity
Noisy!
End of the Story?
Non-unique
http://www.integrityware.com/images/MerceedesGaussianCurvature.jpg
Looks the same!
Summary of Gaussian and Mean
Curvatures
• and are Gaussian and mean
curvatures
• Locally isometric surfaces are invariant measured by
Gaussian curvature
• Gaussian curvatures are vulnerable to noises in
practice and not informative
• Stronger shape descriptors are needed
K = κ1κ2 H =
1
2
(κ1 + κ2)