CURVES ON A PLANE

Mikhail Smilovic

A thesis submitted to McGill University in partial fulfilment of the degree ofMasters of Science

Department of Mathematics and Statistics

McGill University

Montreal, Quebec

2011-10-13

Copyright c© 2011,Mikhail Smilovic

ABSTRACT

In this thesis, we study the space of immersions from the circle to the plane

Imm(S1,R2), modulo the group of diffeomorphisms on S1. We discuss

various Riemannian metrics and find surprisingly that the L2-metric fails to

separate points. We show two methods of strengthening this metric, one to

obtain a non-vanishing metric, and the other to stabilize the minimizing

energy flow. We give the formulas for geodesics, energy and give an example

of computed geodesics in the case of concentric circles. We then carry our

results over to the larger spaces of immersions from a compact manifold M

to a Riemannian manifold (N, g), modulo the group of diffeomorphisms on

M .

ii

ABREGE

Dans cette these, nous etudierons l’espace d’immersions d’un cercle au plan

Imm(S1,R2), modulo le groupe de diffeomorphisme sur S1. Nous discuterons

de divers mtriques riemanniennes et monterons la surprenante impossibilite

de separer des points dans la metrique L2. Nous presenterons deux

methodes de renforcer cette metrique, une pour obtenir une metrique

non-nulle, et une autre pour stabiliser le flot d’energie. Nous donnerons les

formules pour les geodesiques et l’energie, et donnerons un exemple de calcul

de geodesiques dans le cas des cercles concentriques. Nous etendrons alors

nos resultats sur la plus grande espace d’immersion d’une variete M

compacte a une variete riemannienne (N, g), modulo le groupe de

diffeomorphisme sur M .

iii

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

ABREGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Spaces of plane curves and metrics . . . . . . . . . . . . . . . . . 3

2.1 Spaces of plane curves . . . . . . . . . . . . . . . . . . . . 32.1.1 Bcont

i (S1,R2) := Cont(S1,R2)/ ∼ . . . . . . . . . . . 32.1.2 Frechet Spaces and C∞(S1,R2) . . . . . . . . . . . 62.1.3 Bi(S

1,R2) := Imm(S1,R2)/Diff(S1) . . . . . . . . . 72.2 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 “We raised them well.” . . . . . . . . . . . . . . . . 172.2.2 Example: horizontal reparametrization . . . . . . . 19

2.3 Energy and geodesics . . . . . . . . . . . . . . . . . . . . . 202.3.1 Minimal geodesics . . . . . . . . . . . . . . . . . . . 222.3.2 Geodesics on Imm(S1,R2) . . . . . . . . . . . . . . 232.3.3 Example: circles with a common center . . . . . . . 25

3 Conformal metric - geometric heat flow . . . . . . . . . . . . . . . 27

3.1 Geometric heat flow . . . . . . . . . . . . . . . . . . . . . . 273.2 A smooth transition: keeping geometric heat flow in the

metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Conformal versions of H0 . . . . . . . . . . . . . . . . . . . 29

4 Spaces of immersions of compact manifolds . . . . . . . . . . . . . 33

4.1 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Appendix A - Getting from (2.6) to (2.7) . . . . . . . . . . . . . . . . . 37

Appendix B - The H0-distance on Bi(S2,R2) vanishes! . . . . . . . . . 41

Appendix C - Degree of an immersion . . . . . . . . . . . . . . . . . . 46

iv

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

v

LIST OF FIGURESFigure page

2–1 Monotone correspondence as a map onto the box S1 × S1 . . . 4

2–2 Monotone correspondence as a map onto the torus S1 × S1 . . 5

2–3 Translation of the unit circle with a natural parametrization . 19

2–4 Translation of the unit circle with a horizontal parametrization 20

5–1 Continuous function with cusps . . . . . . . . . . . . . . . . . 48

vi

CHAPTER 1Introduction

One is not puzzled when asked if two images, or perhaps the outlines of

images, are similar. Clustering similar images and distinguishing between

different images are skills that seem intrinsically natural. But, how does one

define this idea of similar mathematically? Such a question, of interest to

scientists, engineers, imaging and medicine, has lead to the study of shape

space beginning with the simplest example of shapes, two-dimensional

curves in the plane. The thesis title’s pop-culture reference to the film

Snakes on a Plane is a reference to the beginning study of active contours in

computer vision with the use of “snakes” as defined in [3]. From there,

computer vision research saw a transition from parametrization dependent

models to the evolving curves as independent of parametrization, and now

to incorporating global shape priors. We are then lead to the following

question, how can we determine the distance between closed curves in the

plane?

We present the ideas brought forth by Michor and Mumford [5] in

solving this question and the exploration of finding the simplest Riemannian

metric on the space of curves. In this we find, surprisingly, that the L2-

metric fails to separate points on our space, and we are forced to patch over

this problem by weighing the metric with a curvature term. We compute

the energy and geodesic formulas, and provide an example of the geodesic

running through concentric circes

Although the L2-metric fails to separate to points, its gradient flow

is equal to the natural geometric heat flow, appreciated for its smoothing

1

properties. In moving from this metric to the weighted metric, we lose this

natural gradient flow. In an attempt to keep the gradient flow the same as

the natural geometric heat flow, we present the approach taken by Yezzi and

Mennucci in [7]. In this paper they choose to add a conformal transforma-

tion to the L2-metric instead of weighing the metric, and although this does

not necessarily fix the crux of the identically-zero L2-metric, it does work to

stabilize the minimizing flow and keep the gradient flow equal to that of our

L2-metric.

The results from this simple version of shape space as curves in the

plane are then extended by Michor and Mumford [4] quite generally to

immersions of compact manifolds M into Riemannian manifolds (N, g)

modulo the group of diffeomorphisms on M . It is found once again in this

larger space that the L2-metric fails to separate points, and the weighted

metric is shown to once again be a suitable substitute. The curvature

on this space (including that of our simpler space of curves in the plane)

is shown, and in the case of the L2-metric we see that the space wraps

arbitrarily tightly onto itself, allowing the distance between different curves

to be zero.

2

CHAPTER 2Spaces of plane curves and metrics

2.1 Spaces of plane curves

We begin with an introduction to the spaces we will be working with, first

with the most inclusive and then extracting the core spaces. Fear not the

notational subsection titles, they should be seen as foreshadowing of the

space constructions.

2.1.1 Bconti (S1,R2) := Cont(S1,R2)/ ∼

We denote Cont(S1,R2) the space of all continuous maps c : S1 → R2, whose

elements we call plane curves.1

Example 1. Let c1, c2 ∈ Cont(S1,R2) with

c1 = (cos(θ), sin(θ)), and

c2 = c1 h(θ) where h(θ) =

2θ if θ ∈ [0, 2π

3],

θ2

+ π if θ ∈ [2π3, 2π].

Both are parametrizations of S1, one at constant speed and the other

with a quick start and slow finish. They are however, visually the same

curve and of the same degree2 . We wish to create an equivalence between

such curves, and thus the motivation for the following.

1 All curves we discuss in this paper are closed plane curves, i.e. map-pings from S1 to the plane. We therefore use the term “curves” in place of“plane curves”. The term “curve” is the generic term used for the elementsof the spaces we discuss.

2 Refer to Appendix C for a discussion on the degree of immersions.

3

R ⊂ S1 × S1 is defined to be a monotone correspondence if it is

the image of a map

R→ S1 × S1; x 7→ (h(x) mod 2π, k(x) mod 2π)

where h, k : R → R are monotone non-decreasing continuous

functions such that h(x + 2π) = h(x) + 2π, and k(x +

2π) = k(x) + 2π. These R’s are simply orientation preserving

homeomorphisms from S1 to itself.

Figure 2–1: Visual examples of monotone correspondences, where boxes rep-resent S1 × S1, with the horizontal and vertical lines representing each of thetwo circles accordingly.

Perhaps a more natural way to view a monotone correspondence is as

a map from R onto the Torus, whose image could be called a (1,1)-torus

knot3 ; traveling around the torus “horizontally” completely once and

“vertically” completely once as can be seen in Figure 2–2.

We now use the above monotone correspondence to create the following

equivalence relation:

For c, d ∈ Cont(S1,R2), c ∼ d if and only if there exists a

monotone correspondence R such that for all (θ, φ) ∈ R, c(θ) =

d(φ).

3 This is simply the trivial knot, but the (1,1) is clean notation for theway we want R to map onto the torus.

4

Figure 2–2: An example of a monotone correspondence viewed as a maponto the torus. The middle being a view from above the torus, letting us seeour traversing the torus completely once in the “horizontal” direction, andthe third being a side view allowing us to view our map traversing the toruscompletely once in the “vertical” direction.

We then define Bconti (S1,R2) := Cont(S1,R2)/ ∼, where the objects in this

space are called Frechet curves.

Then the quotient metric on our space is called the Frechet metric,

defined as

d∞(c, d) = inf ( sup(θ,φ)∈R

|c(θ)− d(φ)|)

where the infimum is taken over all monotone correspondences R.

An intuitive definition can be stated as follows:

For the Frechet distance between two curves we introduce a dog and his

owner. The dog walks along one of the curves, and the owner walks along

the other holding the dog by a (infinitely stretchy) leash (perhaps destroying

the purpose of the leash). The two walk independently along their respective

curves, without backtracking but each being able to stop for an indefinite

amount of time. When both dog and owner have completely traversed their

respective circles, we note the minimum leash stretch required for the

specific walk. The infimum of all minimum leash lengths over all such walks

is then defined to be the Frechet distance of the two curves.

5

Example 2. We discuss the Frechet distance between two concentric circles

of radius r1 and r2 respectively. The longest leash is required when the

owner stands still until the dog has completely traversed the circle (r1 + r2),

and the shortest leash when both owner and dog walk at a constant speed

around the circle (|r1 − r2|). The Frechet distance is therefore |r1 − r2|.

With respect to this metric, Bconti (S1,R2) forms a complete metric

space. Before moving to our next space, C∞(S1,R2), an introduction to

Frechet spaces (foreshadowing: C∞(S1,R2) is a Frechet space).

2.1.2 Frechet Spaces and C∞(S1,R2)

Frechet spaces are generalizations of Banach spaces. The generalization

comes from lifting the norm property of positive definiteness, i.e. the

seminorm of a non-zero element is not necessarily non-zero. As this removes

the norm, one instead works with a countable family of seminorms (norms

without the positive definiteness property) to induce a topology (called

either the locally convex topology or C∞-toplogy, the two coincide on

Frechet spaces) and metric.

To construct a Frechet space we begin with a topological vector space

V and a family of seminorms ρi : V → Ri∈I , where I is an indexing set.

We create a base around x0 ∈ V in the following way: for all finite subsets

F ⊆ I and all ε > 0, we have an open set UF,ε(x0) = x ∈ V|ρα(x − x0) <

ε, α ∈ F. We have so far only constructed a locally convex vector space.

We must insure the characteristics of Hausdorff (which is taken care of by

insuring the family of seminorms is countable) and completeness to have

our Frechet space. We can now begin to understand our second space,

C∞(S1,R2), and we show that this space is indeed Frechet.

We define a family of seminorms ρα : C∞(S1,R2) → Rα∈N where

||f ||ρα := supz∈S1|f (α)(z)|. Our family of seminorms is countable and so

6

our space is Hausdorff. A local base around each point is constructed as

above, and now we show completeness.

Lemma 1. C∞(S1,R2) is complete

Proof. Let fnn∈N be a Cauchy sequence and ε > 0. Let ρα be in our family

of seminorms. This implies there exists an N ∈ N such that for all n,m ≥

N , ||fn − fm||ρα < ε. This further implies supz∈S1|fn(α)(z)− fm(α)(z)| < ε.

In particular, for all z ∈ S1, |fn(α)(z) − fm(α)(z)| < ε. Then a Cauchy

sequence of real numbers gives us pointwise convergence to some fα(z) for

each z ∈ S1. Then for z ∈ S1, there exists an mz such that for all n ≥ mz,

|fn(α)(z)− f (α)(z)| < ε

Then,

|fN (α)(z)− f (α)(z)| ≤ |fN (α)(z)− fMz

(α)(z)|+ |fMz

(α)(z)− f (α)(z)|

where Mz := maxN,mz.

Then, for n ≥ N and all z ∈ S1, |fn(α)(z) − f (α)(z)| < ε implies

supz∈S1|fn(α)(z)− f (α)(z)| < ε and ||fn − f ||α < ε.

2.1.3 Bi(S1,R2) := Imm(S1,R2)/Diff(S1)

The next most natural space to examine is Imm(S1,R2), the space of

immersions from S1 → R2, an open subspace of our previous C∞(S1,R2)

making it a manifold in its own right. Similarly, we let Emb(S1,R2) be the

space of embeddings from S1 → R2.

We lay the groundwork with some definitions:

A curve c ∈ Imm(S1,R2) is such that |cθ| > 0, the set of smooth regular

curves in the plane.

7

The volume form on S1 induced by c is given by

vol : Emb(S1,R2)→ Ω1(S1); vol(c) = |cθ|dθ

and its derivative is

dvol(c)(h) =1

2√〈cθ, cθ〉

(〈cθ, hθ〉+ 〈hθ, cθ〉

)=〈hθ, cθ〉|cθ|

where h ∈ C∞(S1,R2) is a tangent vector with foot point c.

The length function of a curve c is given by

len : Imm(S1,R2)→ R; len(c) =

∫S1

|cθ|dθ

and its differential is

dlen(c)(h) =

∫S1

〈hθ, cθ〉|cθ|

dθ

=〈h, cθ〉|cθ|

|S1 −∫S1

⟨h,( cθ|cθ|

)θ

⟩(integration by parts)

= −∫S1

⟨h,( cθ|cθ|

)θ

⟩. (2.1)

By abuse of notation we let(cθ|cθ|

)θ

=(( (c1)θ|cθ|

)θ,( (c2)θ|cθ|

)θ

)Note that

((c1)θ|cθ|

)θ

=(c1)θθ|cθ| − ((c1)θ(c1)θθ+(c2)θ(c2)θθ)(c1)θ

|cθ|

|cθ|2

=(c1)θθ|cθ|2 − ((c1)θ(c1)θθ + (c2)θ(c2)θθ)(c1)θ

|cθ|3

=(c1)θθ|cθ|

− ((c1)θ(c1)θθ + (c2)θ(c2)θθ)(c1)θ|cθ|3

=(c1)θθ|cθ|2 − ((c1)θ(c1)θθ + (c2)θ(c2)θθ)(c1)θ

|cθ|3

=(c1)θθ|cθ|

− 〈cθθ, cθ〉(c1)θ|cθ|3

. (2.2)

8

Using (2.2), (2.1) becomes

−∫S1

⟨h,cθθ|cθ|− 〈cθθ, cθ〉cθ

|cθ|3⟩. (2.3)

We simplify this further by introducing the normal unit field

nc =icθ|cθ|

and curvature mapping

κ : Imm(S1,R2)→ C∞(S1,R); κ(c) =〈cθθ, nc〉|cθ|2

.

Using our new definitions we have that (2.3) is

= −∫S1

1

|cθ|

⟨h, cθθ −

〈cθθ, cθ〉cθ|cθ|2

⟩= −

∫S1

1

|cθ|〈h, 〈cθθ, nc〉nc〉

= −∫S1

〈h, κ(c)icθ〉

= −∫S1

〈h, icθ〉κ(c)

= −∫S1

〈h, nc〉κ(c)|cθ|.

In summary then, we have shown

dlen(c)(h) = −∫S1

〈h, nc〉κ(c)|cθ|. (2.4)

It is sometimes convenient to take the derivative with respect to arclength s

instead of θ, and for such we can use the following equalities:

ds = |cθ|dθ ∂s =1

|cθ|∂θ.

We then notice that in the process of computing (2.4) we showed

csθ =( cθ|cθ|

)θ

= nκ|cθ|

9

or equivalently,

css = nκ. (2.5)

Returning to the curvature mapping and its derivative

dκ(c)(h) =〈ihθ, cθθ〉|cθ|3

+〈icθ, hθθ〉|cθ|3

− 3〈icθ, cθθ〉|cθ|4

〈hθ, cθ〉|cθ|

=〈ihθ, cθθ〉|cθ|3

+〈icθ, hθθ〉|cθ|3

− 3κ〈hθ, cθ〉|cθ|2

(2.6)

=〈h, cθ〉|cθ|2

κθ +〈h, icθ〉|cθ|

κ2 +1

|cθ|

(1

|cθ|

(〈h, icθ〉|cθ|

)θ

)θ

(2.7)

where we compute the calculations showing (2.6) = (2.7) in Appendix A. We

further define Imma(S1,R2) as the mainfold of immersions parametrized by

scaled arc length from S1 → R2, i.e. constant speed.

We define the degree of an immersion4 of c : S1 → R2 as the winding

number of the tangent c′ : S1 → R2 with respect to 0, invariant under

isotopies of immersions. This implies that Imm(S1,R2) decomposes into the

disjoint union of open submanifolds Immk(S1,R2), k ∈ Z, immersions of

degree k from S1 → R2.

The question begs to be asked, “Is there is a relationship between the

newly introduced spaces, and possibly with S1 itself?”. “Yes”, we answer.

We will show the following, where arrows represent (strong) deforma-

tion retractions

Immk(S1,R2) −→ Immka(S

1,R2) −→ S1.

Then in fact, our space Imm(S1,R2) deformation retracts to a count-

able disjoint union of circles.

4 Refer to Appendix C for a discussion on the degree of an immersion.

10

A (strong) deformation retract of a topological space X onto a topologi-

cal space A is a family of maps ft : X → X, 0 ≤ t ≤ 1 such that

f0 = 1X ,

f1(X) = A,

ft|A = 1A ∀t.

When we have such a relation between spaces, we say that they

are homotopy equivalent. It is a convenient way of reducing larger more

complicated spaces to something easier to handle, without losing their

topological properties.

The first deformation retract follows from the following diffeomorphism

where Imm(S1,R2) = Imma(S1,R2) × Diff+

1 (S1), where Diff+1 (S1) are the

orientation preserving diffeomorphisms S1 → S1 fixing 1. The latter is

contractible, and gives our result.

For the second deformation, we must first embed S1 into Immka(S

1,R2).

Let k 6= 0. Let α ∈ S1 ⊂ R2 and let eα(θ) := α · eikθik

We have then

associated to each point in S1 a circle of winding degree k, constant speed 1,

radius 1/|k|, length 2π and orientation the sign of k with starting point π/2

radians clockwise the angle of α.

We now create a deformation retract from Immka(S

1,R2)→ Immk1,0(S

1,R2)

where the latter is the space of degree k immersions of constant speed 1 and

center of mass 0.

The center of mass of a curve c,

C(c) :=1

len(c)

∫ 2π

0

c(u)|c′(u)|du ∈ R2

Notice that

C(eα) =1

2π

∫ 2π

0

(α · e

ikθ

ik

)(1)dθ = 0.

11

We then define Sc(v) :=∫ v

0|c′(u)|du, the arclength function of c, and we now

have all the tools for our isotopy:

A : Immka(S

1,R2)× [0, 1]→ Immk1,0(S

1,R2),

where

A(c, t, u) =(

1− t+ t2π

len(c)

)(c

((1− t)u+ t · S−1

c

(ulen(c)

2π

))− tC(c)

)which takes a constant speed immersion of degree k and deformation

retracts to a curve of unit speed and center of mass 0.

We then form an isotopy H1 : Immk1,0(S

1,R2) × [0, 1] → S1 between c

and a suitable curve eα. In the process however, we lose the property of unit

speed. Where the general form is H1(c, t, θ) (where c is a dummy variable

representing the curve, 0 ≤ t ≤ 1, and θ runs around the curve), we in

general have that H1(c, t,−) is no longer of unit speed. We do however,

maintain to have our deformation retract and thus gives the desired result.

We wish to improve on the space Imm(S1,R2) by creating the natural

equivalence between two curves if they are reparametrizations of each other.

We do this by quotienting by the space of diffeomorphisms on the circle,

Diff(S1), and coin our new space Bi(S1,R2) := Imm(S1,R2)/Diff(S1).

We then notice the similarity in the definitions of Bi(S1,R2) and

Bconti (S1,R2), and remark that the the completion of the first is contained in

the latter:

Bi(S1,R2) ⊆ Bconti (S1,R2).

2.2 Metrics

Let h, k ∈ C∞(S1,R2) be two tangent vectors with foot point

c ∈ Imm(S1,R2). We wish to find the simplest Riemannian metric on our

space Bi(S1,R2), and so we begin our journey. A journey that will at first

12

force us to abandon complete simplicity in favour of invariance under

reparametrization. A journey that will then guide us to a deceptive, and

identically zero metric shown by Michor and Mumford in [5]. A journey that

will then lead us to H0-metric with curvature weight A, our light at the end

the tunnel.

In working with spaces of curves, we must differentiate between an

element in the space, a curve, and the curves between such elements, a curve

of curves. To avoid confusion, we will define a path to be such a curve of

curves, a curve in our space of curves, and thus reserve the word curve for

elements of our space.

If we wish to find the simplest metric, we should start with the ab-

solute simplest, namely the pointwise metric on the space of immersions

Imm(S1,R2). This, however, does not induce a sensible metric on Bi(S1,R2)

as it is not invariant under reparametrization. In fact, given any two curves

C0, C1 ∈ Bi(S1,R2), the infimum of arc lengths of paths in Imm(S1,R2)

connecting embeddings c0, c1 ∈ Imm(S1,R2) with π(Ci) = ci is zero.

So, we need our metric to at least be invariant under reparametrization.

We consider then the simple H0-weak Riemannian metric (which we call the

H0-metric, the L2-Riemannian metric) on Imm(S1,R2):

Gc(h, k) :=

∫S1

〈h(θ), k(θ)〉|cθ|dθ.

We now have invariance under reparametrization, and thus the map π :

Imm(S1,R2) → Bi(S1,R2) is Riemannian submersion (off the singularities

of Bi(S1,R2)).

13

We define N → Imm(S1,R2) as the bundle of tangent vectors which are

normal to the Diff(S1)-orbits, and specific to a curve c we have

Nc = h ∈ C∞(S1,R2) : 〈h, cθ〉 = 0

= aicθ ∈ C∞(S1,R2) : a ∈ C∞(S1,R2)

= bnc ∈ C∞(S1,R2) : b ∈ C∞(S1,R2).

Given a tangent vector h ∈ TcImm(S1,R2) = C∞(S1,R2), we have an

orthonormal decomposition

h = h> + h⊥ ∈ Tc(c Diff+(S1))⊕Nc

where

h> =〈h, cθ〉|cθ|2

cθ, h⊥ =〈h, icθ〉|cθ|2

icθ

and Tc(c Diff+(S1)) = g.cθ : g ∈ C∞(S1,R) are the tangent vectors to the

Diff(S1)-orbits .

We can now show how the metric above induces a metric on our

quotient space Bi(S1,R2). Given curves C0, C1 ∈ Bi(S

1,R2), we consider all

liftings c0, c1 ∈ Imm(S1,R2), where π(ci) = Ci, and all smooth paths c(t,−)

in Imm(S1,R2) where c(0,−) = c0 and c(1,−) = c1. We take the infimum of

the arc lengths of all such paths and all lifts c0, c1 and define this to be our

14

metric. Then, the arclength of a path c(t,−) in Bi(S1,R2) is given by

LhorG (c) = LG(π(c(t,−)))

=

∫ 1

0

√Gπ(c)(Tcπ.ct, Tcπ.ct)dt

=

∫ 1

0

√Gc(c⊥t , c

⊥t )dt

=

∫ 1

0

(∫S1

⟨〈h, icθ〉|cθ|2

icθ,〈h, icθ〉|cθ|2

icθ

⟩|cθ|dθ

) 12

dt

=

∫ 1

0

(∫S1

〈ct, nc〉2|cθ|dθ) 1

2

dt

=

∫ 1

0

(∫S1

〈ct, icθ〉2dθ

|cθ|

) 12

dt

and as mentioned above, we define the distance between two curves

distBiG (C1, C2) = infcLhorG (c).

where infimum runs through all paths and all such lifts.

However, this metric works out to be identically zero (as shown in

Appendix B)! Moving on. The natural way of strengthening our H0-

metric is to move to an H1-metric5 , but we want to keep our “nice”

properties, namely keep it local (local is good both for mathematics and

the environment!) and leave out derivatives. So, in moving to H1 we simply

remove “not nice” terms and get our H1-weak Riemannian metric on

Imm(S1,R2):

G1c(h, k) :=

∫S1

(〈h(θ), k(θ)〉+ A

〈hθ, kθ〉|cθ|2

)|cθ|dθ.

where A ≥ 0 is a constant.

5 Refer to Chapter 3 for an alternative approach.

15

We play around with the metric, and get the metric coined the H0κ-

metric with curvature weight A

GAc (h, k) :=

∫S1

(1 + Aκc(θ)2)〈h(θ), k(θ)〉|c′(θ)|dθ.

or more specifically, when h = aics for the derivative is with respect to

arc length,

GAc (h, h) :=

∫c

(1 + A2κ)a

2ds

We have that the above metric is invariant under reparametrization and

have that the map π : Imm(S1,R2) → Bi(S1,R2), again a Riemannian

submersion (off the singularities of Bi(S1,R2).

Using this metric we have the an associated arc length similar to that

defined above:

LhorGA(c) := LGA(π(c(t,−)))

=

∫ 1

0

√GAπ(c)(Tcπ.ct, Tcπ.ct)dt

=

∫ 1

0

√GAc (c⊥t , c

⊥t )dt

=

∫ 1

0

(∫S1

(1 + Aκ2c)〈ct, nc〉

2|cθ|dθ) 1

2

dt

=

∫ 1

0

(∫S1

(1 + Aκ2c)〈ct, icθ〉

2 dθ

|cθ|

) 12

dt

and again reminiscent from above, the metric on Bi(S1,R2) is defined by

taking the infimum of all paths between all lifts to get

distBiGA

(C0, C1) = infcLhorGA(c).

Michor and Mumford then proceed to show the following about the

H0κ-metric with curvature weight A:

Theorem 2.1. For any A > 0, distGA is a separating metric on Bi(S1,R2).

16

2.2.1 “We raised them well.”

Assume we are given a path π(c) in Bi(S1,R2) and we lift it to a path c in

Imm(S1,R2). We are able to reparametrize our lift c to have certain

desirable properties. We present three ways in which we can reparametrize

our path.

1. Let c be a path in Imm(S1,R2). There exists a smooth path ϕ in

Diff(S1) such that ϕ(0,−) = IdS1 depending smoothly on c such that

at all points t, the curve c(t, θ) = c(t, ϕ(t, θ)) is traversed at constant

speed (namely, the path is in Imma(S1,R2)), and the path c(t, θ) is

horizontal at a point θh, namely 〈ct, cθ〉(t, θh) = 0 for all t.6

Proof. Let c be a path in Imm(S1,R2), and c(t, θ) = c(t, ϕ−1(t, θ)),

where ϕ is defined as follows:

ϕ(t, θ) =2π

len(c)

∫ θ

0

|cθ(t, u)|du, ϕθ(t, θ) =2π

len(c)|cθ(t, θ)|

where

len(c)7 : [0, 1]→ R+; len(c)(t) =

∫S1

|cθ(t, θ)|dθ.

We have the equality

c(t, ϕ(θ)) = c(t, (t, θ)) (2.8)

6 notice the importance of the the distinction between the terms curveand path in the statement, the two are not interchangeable.

7 len(c) has been defined both for a curve and a path, but in context thisshould not case any confusion.

17

allowing us to use ϕ versus its inverse. Taking the derivative of both

sides of (2.8) yields

cθ(t, ϕ(t, θ))ϕθ(t, θ) = cθ(t, (t, θ)).

Then,

|cθ(t, ϕ(t, θ))| =

|cθ(t, (t, θ))|len(c)

2π

|cθ(t, (t, θ))|

=len(c)

2π

which is constant in θ given a time t, as len(c) is a function of t.

2. Let c be a path in Imm(S1,R2). There exists a smooth path ϕ in

Diff(S1) such that ϕ(0,−) = IdS1 depending smoothly on c such that

the path c(t, θ) = c(t, ϕ(t, θ)) is horizontal, namely 〈ct, cθ〉 = 0, and the

path is constant at time zero, namely |cθ(0,−)| = len(c(0,−))2π

.

Proof. Let c(t, θ) = c(t, ϕ(t, θ)). We want 〈ct, cθ〉 = 0.

〈ct, cθ〉 = 〈∂tc(t, ϕ) + (∂θc(t, ϕ))∂tϕ, (∂θc(t, ϕ))∂θϕ〉

= 〈∂tc(t, ϕ), ∂θc(t, ϕ)〉∂θϕ+ 〈∂θc(t, ϕ), ∂θc(t, ϕ)〉∂θϕ∂tϕ

Now, if ∂tϕ = −〈∂tc(t,ϕ),∂θc(t,ϕ)〉|∂θc|2

, we have 〈ct, cθ〉 = 0.

3. We are also able to reparametrize the variable t so that our path is

traversed at constant speed, namely for a path c of arclength L∫S1 〈ct, icθ〉2 dθ

|cθ|≡ L2 .

Unfortunately, the reparametrizations 1 and 2 presented above are

mutually exclusive, and cannot both be applied to a path c.

18

2.2.2 Example: horizontal reparametrization

Consider the path c given by translating the unit circle along the x-axis

at unit speed, namely

c(t, θ) = (t+ cos(θ), sin(θ)).

Figure 2–3: Translating the unit circle along the x-axis at unit speed, lettingtime be the third variable. The bright green circle represents our path attime 0, and the bright purple circle our path at time 1. The first is a viewfrom the side with the second image a view from above. We can see that ourpath is not horizontal, and the tangent vectors all move in the direction ofthe positive x-axis.

We reparametrize our curve as follows:

c(t, θ) = (t+(1− e2t) + (1 + e2t)cos(θ)

(1 + e2t) + (1− e2t)cos(θ),

2etsin(θ)

(1 + e2t) + (1− e2t)cos(θ))

and show that c is horizontal.

Proof.

ct =(((1− e2t) + cos(θ)(1 + e2v))2, 2etsin(θ)(cos(θ)(1 + e2t) + (1− e2t)))

((1 + e2t) + (1− e2t)cos(θ))2

cθ =(−4e2tsin(θ), 2et(θ)(cos(θ)(1 + e2t) + (1− e2t)))

((1 + e2t) + (1− e2t)cos(θ))2

Then, 〈ct, cθ〉 = 0

Our path c traces out the same path as c above, but is now horizontal

as can be seen in Figure 2–4.

19

Figure 2–4: The horizontal reparametrization of Figure 2–3, with time asthe third variable. Notice that now the tangent vectors have a spiral motion(except the vectors at θ = 0 and θ = π).

2.3 Energy and geodesics

Let c(t,−) be a path in Imm(S1,R2). The energy of its projection π c in

Bi(S1,R2) is given by

EGA(π c) =1

2

∫ b

a

GAπ(c)(Tcπ.ct, Tcπ.ct)dt

=1

2

∫ b

a

GAc (c⊥t , c

⊥t )dt

=1

2

∫ b

a

∫S1

(1 + Aκ2c)〈ct, nc〉

2|cθ|dθdt

=1

2

∫ b

a

∫S1

(1 + Aκ2c)〈ct, icθ〉

2 1

|cθ|dθdt.

If the path c is horizontal, i.e. 〈ct, cθ〉 = 0, then

|〈ct, icθ〉| = 8 |ct||cθ|. (2.9)

Using, (2.9), for a horizontal path c we have

8 0 = 〈ct, cθ〉 = 〈(ct1, ct2), (cθ1, cθ2)〉0 = 〈ct, cθ〉2 = (ct1cθ1)

2 + (ct2cθ2)2 + 2ct1ct2cθ1cθ2

(ct1cθ2)2− (ct1cθ2)

2 + (ct2cθ1)2− (ct2cθ1)

2 = (ct1cθ1)2 + (ct2cθ2)

2 + 2ct1ct2cθ1cθ2(ct1cθ2)

2 + (ct2cθ1)2− 2ct1ct2cθ1cθ2 = (ct1cθ1)

2 + (ct2cθ2)2 + (ct1cθ2)

2 + (ct2cθ1)2

(−ct1cθ2 + ct2cθ1)2 = (ct1

2 + ct22)(cθ1

2 + cθ22)

|〈ct, icθ〉| = |ct||cθ|

20

EhorGA(π c) =

1

2

∫ b

a

∫S1

(1 + Aκ2c)|ct|2|cθ|dθdt. (2.10)

Let c(t, θ) = (x(t, θ), y(y, θ)) be horizontal, and consider the graph in R3

given by,

Φ(t, θ) = (t, x(t, θ), y(y, θ)).

We have Φt × Φθ = (xtyθ − xθyt,−yθ, xθ), and following

|Φt × Φθ|2 = (xtyθ − xθyt)2 + (yθ)2 + (xθ)

2

= 〈ict, cθ〉2 + |cθ|2

= (−〈ct, icθ〉)2 + |cθ|2

= (|ct||cθ|)2 + |cθ|2

= |cθ|2(|ct|2 + 1).

We now wish to express Ehor(c) as an integral over the immersed

surface S ⊂ R3 parametrized by Φ in terms of the surface area

dµS = |Φt × Φθ|dθdt.

If that were not enough, we wish to express the integrand as a function of

the unit normal ns = (Φt × Φθ)/|Φt × Φθ|. Let e0 = (1, 0, 0)

|n0S| := |〈e0, nS〉| =

|〈ict, cθ〉||cθ|√|ct|2 + 1

=|ct|√|ct|2 + 1

(2.11)

21

|ct| = |ct|√|ct|2 + 1√|ct|2 + 1

=|ct|√|ct|2 + 1

11√|ct|2+1

=|ct|√|ct|2 + 1

1√1− |ct|2

|ct|2+1

=|n0S|√

1− |n0S|2

. (2.12)

Then combining (2.11) and (2.12), we get

|ct|2√|ct|2 + 1

=|n0S|2√

1− |n0S|2

.

Now with all our work above,

EhorGA(c) =

1

2

∫ b

a

∫S1

(1 + Aκ2c)|ct|2|cθ|

|Φt × Φθ||Φt × Φθ|

dθdt

=1

2

∫[a,b]×S1

(1 + Aκ2c)

|ct|2√|ct|2 + 1

dµS

=1

2

∫[a,b]×S1

(1 + Aκ2c)

|n0S|2√

1− |n0S|2

dµS (2.13)

referred to as the Horizontal Energy as Anisotropic Area.

2.3.1 Minimal geodesics

To prove geodesics exist between arbitrary curves (of the same degree)

in Bi(S1,R2) we must show that that the horizontal energy given as (2.13)

can be minimized. This however remains an open problem.

Conjecture Fix two curves c0 and c1. The energy EGA(c) admits a

minimum in the class C of homotopies connecting c0 and c1, where C is the

class of all homotopies c : [0, 1] × S1 → R2 continuous on [0, 1] × S1 and

locally Lipschitz in (0, 1) × S1. An approach to proving the conjecture is in

answering the following question:

22

Question For immersions c0, c1 : S1 → R2 does there exist an immersed

surface S = (ins[0,1], c) : [0, 1] × S1 → R × R2 such that the functional given

in (2.13) is critical at S?

Michor and Mumford give a first step to proving this:

For any path [a, b] 3 t 7→ c(t,−), we have that the area of the graph

surface S=S(c) is bounded, namely

Area(S) =

∫[a,b]×S1

dµS ≤ EhorGA(c) + max

t(len(c(t,−)))(b− a)

Further insight on solving the conjecture is presented by Yezzi and

Mennucci. They show the existence of minimal geodesics in a subspace S,

where curves in this space are of unit length and bounded curvature.

Using the H0 Riemannian Metric, they show this distance admits

minimal geodesics. We can think of S as a “submanifold with boarder” of

the manifold M of Imm1(S1,R2), closed unit-length immersions. It is for

this reason that it is noted that the minimal geodesic will, in general, not

satisfy the Euler-Lagrange ODE defined by the energy functional.

2.3.2 Geodesics on Imm(S1,R2)

The energy of a path C in Bi(S1,R2), (with the reparametrization making C

horizontal) is given as

EhorGA(π c) =

1

2

∫ b

a

∫S1

(1 + Aκ2)|ct|2|cθ|dθdt (2.14)

We calculate its first variation and get the equation for a geodesic as

((1 + Aκ2)|cθ|ct)t =

(−1 + Aκ2

2

|ct|2

|cθ|cθ + A

(κ|ct|2)θ|cθ|2

icθ

)θ

(2.15)

23

Proof.

∂s|0EhorGA(π c) =

1

2∂s|0

∫ b

a

∫S1

(1 + Aκ2)|ct|2|cθ|dθdt

=1

2

∫ b

a

∫S1

2(Aκκs)|ct|2|cθ|+ 2(1 + Aκ2)〈cst, ct〉|cθ|

+ (1 + Aκ2)|ct|2〈csθ, cθ〉|cθ|

dθdt (2.16)

We remind ourselves of (2.6)

κs =〈icsθ, cθθ〉|cθ|3

+〈icθ, csθθ〉|cθ|3

− 3κ(c)〈csθ, cθ〉|cθ|2

Then, we get that (2.16) is

=1

2

∫ b

a

∫S1

(〈icsθ, cθθ〉|cθ|3

+〈icθ, csθθ〉|cθ|3

− 3κ(c)〈csθ, cθ〉|cθ|2

)(2Aκ)|ct|2|cθ|

+ 2(1 + Aκ2)〈cst, ct〉|cθ|+ (1 + Aκ2)|ct|2〈csθ, cθ〉|cθ|

dθdt

=1

2

∫ b

a

∫S1

(−〈csθ, icθθ〉|cθ|3

− 〈csθθ, icθ〉|cθ|3

− 3κ(c)〈csθ, cθ〉|cθ|2

)(2Aκ)|ct|2|cθ|

+ 2(1 + Aκ2)〈cst, ct〉|cθ|+ (1 + Aκ2)|ct|2〈csθ, cθ〉|cθ|

dθdt

=

∫ b

a

∫S1

−⟨csθ,

icθθ(Aκ|ct|)2

|cθ|2⟩−⟨csθθ,

icθ(Aκ|ct|2)|cθ|2

⟩−⟨csθ,

cθ(3Aκ2|ct|2)|cθ|

⟩+ 〈cst, ct(1 + Aκ2)|cθ|〉+

⟨csθ, (

cθ(1 + Aκ2)|ct|2

2|cθ|)⟩dθdt

=

∫ b

a

∫S1

⟨cs,(icθθ(Aκ|ct|2)

|cθ|2)θ

⟩+⟨cs,(icθ(Aκ|ct|2)

|cθ|2)θθ

⟩+⟨cs,(cθ(3Aκ2|ct|2)

|cθ|

)θ

⟩− 〈cs, (ct(1 + Aκ2)|cθ|)t〉+

⟨cs,(cθ(1 + Aκ2)|ct|2

2|cθ|

)θ

⟩dθdt (2.17)

Some quick side calculations:(icθ(Aκ|ct|2)|cθ|2

)θθ

= A

(κ|ct|2

|cθ|2icθθ + icθ

( 1

|cθ|2(κ|ct|2)θ − (κ|ct|2)

2|cθ|θ|cθ|3

))θ

(2.18)

cθθ =〈cθθ, cθ〉|cθ|2

cθ +〈cθθ, icθ〉|cθ|2

icθ =|cθ|θ|cθ|

cθ + κ|cθ|icθ (2.19)

24

Let,

F =icθθ(Aκ|ct|2)|cθ|2

+ A

(κ|ct|2

|cθ|2icθθ + icθ

( 1

|cθ|2(κ|ct|2)θ − (κ|ct|2)

2|cθ|θ|cθ|3

))+cθ(3Aκ

2|ct|2)|cθ|

− cθ(1 + Aκ2)|ct|2

2|cθ|

and using (2.19), we get that the simplification

F =(−1 + Aκ2)|ct|2

|cθ|cθ +

A(κ|ct|2)θ|cθ|2

icθ

Then,

(2.17) =

∫ b

a

∫S1

〈cs,−(ct(1 + Aκ2)|cθ|)t + Fθ〉dθdt

and we have our result.

2.3.3 Example: circles with a common center

Here we look at the geodesic given by the set of all circles with a common

center. Let Cr be a circle centered at the origin with radius r. We consider

the path of circles given by Cr(t), where r(t) is a smooth increasing function

r : [0, 1]→,R>0, more explicitly as c(t, θ) = r(t)eiθ, and we then have

Kc(t, θ) = 1r(t)

. This gives us the energy and variation equations as follows:

EhorGA(c) =

1

2

∫ 1

0

∫S1

(1 +

A

r2

)r2t rdθdt

∂s|0EhorGA(c) =

∫ 1

0

∫S1

(1 +

A

r2

)rs

(− rtt −

(1− Ar2

)

2(r + Ar)r2t

)rdθdt

and we only have that our path c is a geodesic, if

rtt +(1− A

r2)

2(r + Ar)r2t = 0

25

We look at the extremes of r and see what occurs there, first as r → 0

and then as r →∞. As r → 0,

rtt +(1− A

r2)

2(Ar)r2t = rtt +

( rA− r

AAr2

)

2r2t → rtt −

r2t

2rr2t = 0

which has the general solution r(t) = C(t− t0)2 for constants C, t0.

As r →∞,

rtt +(1− A

r2)

2(Ar)r2t → rtt +

r2t

2r= 0

which has the general solution r(t) = C(t− t0)2/3 for constants C, t0.

Then at the “zero” end of the geodesic, the path ends in finite time

with the circles imploding at their common center. At the other end of the

geodesic, the circles expand forever but with decreasing speed.

26

CHAPTER 3Conformal metric - geometric heat flow

3.1 Geometric heat flow

One may wish to shorten a curve c ∈ Imm(S1,R2), and the literature in

computer science often makes use of the geometric heat flow (ct = css) for

reasons of its smoothing effect. This process has been coined, all quite

appropriately, “curve shortening”, “flow by curvature” and “heat flow on

isomertric immersions”. Under this flow, embedded curves become convex

without developing singularities and then shrink to a point, becoming round

in the limit. We have the following theorem from [2]:

Theorem 3.1. Let c ∈ Emb(S1,R2). Then there exists a path c : S1 ×

[0, 1)→ R2 in the space Emb(S1,R2) such that

∂C

∂t= nκ

and our path converges to a point as t → 1, and its limiting shape as t → 1

is a round circle, with convergence in the C∞ norm.

The geometric heat flow can be visualized as the evolution of an elastic band

in honey, if the tension is kept constant then the behaviour is approximately

that of the equation satisfied above.

3.2 A smooth transition: keeping geometric heat flow in themetric

The common reference to curve evolution models in the literature as

“gradient flows” gives rise to a unique metric that we have already seen, H0,

and we see this in the following relationship between the metric and

geometric heat flow:

27

We remind ourselves of the length function len(c) for a path c(t, θ),

where,

len(c) : Imm(S1,R2)→ R+ ; len(c)(t) =

∫S1

|cθ|dθ

represents the time-varying arclength of the evolving curve.

(2.4) gives us the derivative at time t as

len′(c)(t) = −∫S1

〈ct, nc〉κ(c)|cθ|dθ

= −∫S1

〈ct, nc〉κ(c)ds

= −∫S1

〈ct, css〉ds (by (2.5))

= −〈ct, css〉H0

giving us the gradient flow for arclength as

ct = css

When we move away from the H0 metric, the inner-product above no longer

corresponds to our new metric. We compute the similar calculations for our

H0κ-metric with curvature weight A

len′(t) = −∫S1

〈ct, css〉ds

= −∫S1

(1 + Aκ2)⟨ct,

css1 + Aκ2

⟩= −

∫S1

(1 + Aκ2)⟨ct,

nκ

1 + Aκ2

⟩= −

⟨ct,

nκ

1 + Aκ2

⟩H0κ

giving us the gradient flow for arclength as

ct =nκ

1 + Aκ2

28

When A > 0 our curvature flow does not correspond with the geometric

heat flow as with H0. But, there are reasons we would want to keep this

relationship between curvature flow and geometric heat flow. We have

already noted that the H0 metric proves useless in Imm(S1,R2), but we

propose an alternative to the H0κ-metric with curvature weight A. We

present the construction of a metric by Yezzi and Mennucci in [7] and [8]

whose gradient structure is as similar as possible to that of the H0 metric,

but dispute the idea that it solves the “identically zero” crux of H0.

3.3 Conformal versions of H0

We search for a metric such that given any energy functional

E : Imm(S1,R2)→ R, the gradient flow of E with respect to our new metric

has a time-reparametrization relationship with our old metric. That is to

say,

c(t) = c(f(t))

where c and c are the gradient flow trajectories according to our proposed

new metric and H0 respectively, and f(t) is our positive time reparametriza-

tion, f ′ > 0. Deriving, we get

ct = f ′ct

We wish to find a metric that satisfies this, and this leads us directly into

finding a conformal factor φ : Imm(S1,R2) → R where φ(c) > 0 and may

depend on the curve c. We will denote our new metric H0φ. We have the

following relationship between the inner products:

〈h1, h2〉H0φ

= φ(c)〈h1, h2〉H0 (3.1)

By definition we have

⟨∂c∂t,∇E(c)

⟩H0

=d

dtE(c(t)) =

⟨∂c∂t,∇φE(c)

⟩H0φ

29

and using (3.1) we have

⟨∂c∂t,∇φE(c)

⟩H0φ

= φ⟨∂c∂t,∇φE(c)

⟩H0

(3.2)

Then (3.1) and (3.2) together give us

⟨∂c∂t,∇E(c)

⟩H0

= φ⟨∂c∂t,∇φE(c)

⟩H0

which implies ∇φE =1

φ∇E

and the conformal gradient flow with respect to H0φ differs only in speed

from that of the flow with respect to H0

∂c

∂t= −∇φE(c) = − 1

φ(c)∇E(c) which gives us f ′ =

1

φ(c)

Exactly what we wanted, our gradient flows related by a time-reparametrization.

Now the work is to choose an appropriate φ. We first work to satisfy the

following:

Theorem 3.2. Assume there exists an a ∈ R>0 such that

minc

( φ(c)

len(c)

)= a > 0 (3.3)

Then for curves c1 6= c2 ∈ Imm(S1,R2), and there does not exist a

family of homotopies connecting c0, c1 with arbitrarily small area1 , then

distBiG0φ(c1, c2) > 0.

1 Yezzi and Mennucci [7] say instead “does not exist a homotopy connect-ing c0, c1 with zero area”. However, as the distance between two curves isdefined using the infimum of arclength, we only need to show the existenceof a family of homotopies with arbitrarily small arclength, and not necessar-ily a homotopy of arclength zero, to have distance zero. We relate area andarclength further in Theorem 3.3.

30

Proof. We consider a homotopy c such that c(0,−) = c0 and c(1,−) = c1,

and its H0φ-energy ∫ 1

0

(φ(c)

∫S1

(〈ct, icθ〉2

dθ

|cθ|

))dt (3.4)

We reparametrize our path c so that at each time t, the curve has

constant speed, namely

|c(t,−)| = len(c(t,−))

2π

Then,

(3.4) =

∫ 1

0

(2π

len(c(t,−))φ(c)

∫S1

(〈ct, icθ〉2dθ

))dt

=

∫ 1

0

(2π

len(c(t,−))φ(c)

∫S1

(|det dc(t, θ)|2dθ

))dt

≥ 2πa

∫ 1

0

∫S1

|det dc(t, θ)|2dθdt

≥ a

(∫ 1

0

∫S1

|det dc(t, θ)|dθdt)2

where the rightmost term is the square of the area swept by the homotopy.

We immediately note that following theorem, proved similarly:

Theorem 3.3. If c is any path from C0, C1 ∈ Bi(S1,R2), then there exists a

constant m > 0 such that(∫ 1

0

∫S1

|det dc(t, θ)|dθdt)≤ mLhor

G0 (c) ≤ mLhorGA(c)

which lets us state the following theorem:

Theorem 3.4. For curves c0 6= c1 ∈ Imm(S1,R2) such that there does not

exist a family of homotopies connecting c0, c1 with arbitrarily small area,

then for A ≥ 0, distBiGA

(c1, c2) > 0.

31

For A > 0, we have show by Theorem 2.1 that the H0κ-metric with

curvature weight A separates curves. However, quite the opposite was

shown for the H0-metric in which every pair of curves was shown to have

a homotopy with arbitrarily small arclength, i.e. there exists a path c

connecting any two curves such that LhorG0 (c) is arbitrarily small. Then

Theorem 3.3 tells us the area traversed by this path is arbitrarily small.

Then, the above theorems are be viewed as simply upper bounds on the

area swept by the homotopies, as there are evidently always homotopies

sweeping out an arbitrarily small area (as shown in Appendix B), and

should not be interpreted as lower bounds on arclength. We then require

such a theorem as Theorem 2.1 to make such a “separates-points” claim.

However, no such theorem is shown by Yezzi and Mennucci for H0φ, and in

fact unless φ is related to curvature (which would then make the H0φ-metric

more-or-less equivalent to the H0κ-metric with curvature weight A described

by Michor and Mumford), this most likely cannot be shown. Then contrary

to what they suggest, such a φ satisfying (3.3) does not necessarily induces

a “non-degenerate distance of curves”. Yezzi and Mennucci however use the

conformal factor φ to stabilize the minimizing flow of H0. This, although

not improving on the crux provided by H0, perhaps provide more useful

tools for computer vision in application. Conformal transformations as

functions of len(c) of a path are discussed in [7] and [8], and is brought into

particular detail in [6] introducing conformal transformations as a function

of both len(c) and κ.

32

CHAPTER 4Spaces of immersions of compact manifolds

4.1 Metrics

Up until this point we have only concerned ourselves with the space of

immersions from the circle to the plane, modulo diffeomorphisms of the

circle. But, what if we were to completely generalize these mappings, and

say investigate the space of immersions from a compact manifold M to a

Riemannian manifold (N, g) with dim N > dim M ; coining this space

Imm(M,N)/Diff(M). It turns out that our results from the simpler case

translate to this more generalized space of immersions.

Before we define the extension of the H0κ-metric with curvature weight

A as follows, we must update some of our definitions to fit the new space of

immersions.

For an immersion f ∈ Imm(M,N), we have the normal bundle

N(f) = Tf⊥ ⊂ f ∗TN → M and that every vector field h : M → TN along

f splits as h = Tf · h> + h⊥.

We have Sf as the shape operator or second fundamental form, and we

define Trf∗g(Sf ) ∈ N(f) as the mean curvature, and ‖Trf

∗g(Sf )‖gN(f) the

norm. Finally, we have the volume density

volg(f) = vol(f ∗g) ∈ Vol(M)

on M, and for any chart (U, u : U → Rm) of M , we have the local formula as

volg(f)|U =√

det((f ∗g)ij)|du1 ∧ . . . ∧ dum for said chart.

33

Let h, k ∈ C∞f (M,TN) be two tangent vectors with foot point f ∈

Imm(M,N). For a constant A ≥ 0 we have

GAf (h, k) :=

∫M

(1 + A‖Trf∗g(Sf )‖2gN(f))g(h, k)vol(f ∗g).

Then the bundle of tangent vectors which are normal to Diff(M)-orbits

is

Nf = h ∈ C∞(M,TN) : g(h, Tf) = 0

= Γ(N(f))

which gives us the space of sections of the normal bundle.

Length and distance are defined similarly to the Imm(S1,R2) case, and

so we state the following:

Theorem 4.1. Let A = 0. For f0, f1 ∈ Imm(M,N) there exists a path

f(t,−) in Imm(M,N) with f(0,−) = f0, and f(1,−) = f1 such that LhorG0 (f)

is arbitrarily small.

The H0 metric then fails to separate points even in the generalized case,

and is proved using a similar technique (refer to Appendix B on the proof of

the vanishing distance in the Imm(S1,R2) case).

We have as well that for A > 0, our metric separates points and the

same open-question is asked if whether there exists a minimal geodesic

between any two f0, f1 ∈ Imm(M,N). Michor and Mumford [4] suggest

trying the same technique as above, finding an immersed surface which is

critical for the functional EGA when viewed as anisotropic volume1

EGA(π f) =1

2

∫[a,b]×M

(1 + A‖Trf∗g(Sf )‖2gN(f))

‖f⊥t ‖2√1 + ‖f⊥t ‖2

vol(γ∗f (dt2 + g))

1 Refer to (2.13) for the Imm(S1,R2) case.

34

where

γf : [a, b]×M → [a, b]×N ; (t, x) 7→ (t, f(x))

is the graph of the path f .

4.2 Curvature

Michor and Mumford wished to explore the sectional curvature of the spaces

of immersions to further understand the vanishing distance of the H0 metric.

For Bi(S1,R2), it was found that for A = 0, namely using the H0 metric,

that all sectional curvature was non-negative. For A > 0, in general,

sectional curvature was strictly negative only when the curve C had large

curvature or the plane section had high frequency. It was later then found

that for Bi(M,N), where M and N are of codimension one (as in the case of

Imm(S1,R2)) and our metric employs A=0, that all sectional curvatures are

non-negative. These sectional curvatures are not only non-negative, but

unbounded in certain directions causing the spaces to wrap arbitrarily

tightly onto themselves, allowing for the distance between curves to be zero.

When A = 0 and the codimension is not one, there are conflicting terms in

the computation for sectional curvature, some giving negative curvature, and

others positive.

35

CHAPTER 5Conclusion

We have presented a suitable Riemannian metric on Bi(S1,R2) that

separates curves C0 6= C1 first given in [5]. The formula for energy and

geodesics have been computed, and we have provided the example of

concentric circles. The question of whether minimal geodesics exist between

arbitrary curves in Bi(S1,R2) is left as an open question, and we remark

that in the completion of this space, Bi(S1,R2), this is not the case.

We have introduced the ideas of Yezzi and Mennucci in [7] and com-

mented that the suggested conformal transformation being solely a function

of the length function is not strong enough to separate points. The confor-

mal transformation has another function however, in the stabilizing of the

minimal energy flow.

We have stated that the results from Bi(S1,R2) also prove true in the

more general space Imm(M,N)/Diff(M). The open question presented

for the case of curves in the plane remains a conjecture for the more

general space, and we have presented the beginning steps in proving this

as per the advice of Michor and Mumford in [5] and [4]. The curvature

of Imm(M,N)/Diff(M) for the weighted metric was stated, and shown to

provide a useful tool in understanding the vanishing nature of the L2-metric.

36

Appendix A - Getting from (2.6) to (2.7)

We note first that our curvature mapping is equivariant, namely that

κ(c f) = ±κ(c) f for f ∈ Diff±(S1). It then suffices to check the equality

for constant speed parametrizations, namely

• |cθ| is constant

• cθθ = κ|cθ|icθ

Then by linearity, it is enough to take two cases:

1. h = aicθ, and

2. h = bcθ.

Case 1: h = aicθ

(2.6) =〈i(aicθ)θ, cθθ〉|cθ|3

+〈icθ, (aicθ)θθ〉|cθ|3

− 3κ(c)〈(aicθ)θ, cθ〉|cθ|2

=〈−acθθ, cθθ〉|cθ|3

+〈icθ, aicθθθ〉|cθ|3

− 3κ(c)〈(aicθθ, cθ〉|cθ|2

Now, working with each of the three summands separately

〈−acθθ, cθθ〉|cθ|3

=−a〈cθθ, cθθ〉|cθ|3

=−aκ2〈icθ, icθ〉

|cθ|= −aκ2|cθ| (5.1)

37

cθθθ = (κ|cθ|icθ)θ = |cθ|(κθicθ + κicθθ), then

〈icθ, aicθθθ〉|cθ|3

=a〈cθ, cθθθ〉|cθ|3

=a〈cθ, (κθicθ + κicθθ)〉

|cθ|2

=a(〈cθ, κθicθ〉+ 〈cθ, κicθθ)〉

|cθ|2

=a(

κθ〈cθ, icθ〉+ κ〈cθ, icθθ)〉|cθ|2

=aκ〈cθ, i(κ|cθ|icθ)〉

|cθ|2

=−aκ2〈cθ, cθ〉|cθ|

= −aκ2|cθ| (5.2)

−3κ〈(aicθθ, cθ〉|cθ|2

= −3κ〈(ai(κ|cθ|icθ), cθ〉|cθ|2

=3aκ2〈cθ, cθ〉|cθ|

= 3aκ2|cθ| (5.3)

Then all the king’s horses and all the king’s men bring the three parts

together to reveal (2.6) = −aκ2|cθ|+−aκ2|cθ|+ 3aκ2|cθ| = aκ2|cθ|

(2.7) =〈(aicθ), cθ〉|cθ|2

κθ +〈(aicθ), icθ〉|cθ|

κ2 +1

|cθ|(

1

|cθ|(〈(aicθ), icθ〉|cθ|

)θ)θ

=

a〈icθ, cθ〉

|cθ|2κθ +

a〈icθ, icθ〉|cθ|

κ2 +a

|cθ|3〈icθ, icθ〉θθ

= a|cθ|κ2 +

a

|cθ|3(|cθ|2)θθ

= (2.6)

38

Case 2: h = bcθ

(2.6) =〈i(bcθ)θ, cθθ〉|cθ|3

+〈icθ, (bcθ)θθ〉|cθ|3

− 3κ〈(bcθ)θ, cθ〉|cθ|2

=

b〈icθθ, cθθ〉

|cθ|3+b〈icθ, cθθθ〉|cθ|3

− 3bκ〈cθθ, cθ〉|cθ|2

=b〈icθ, cθθθ〉|cθ|3

− 3bκ〈(κ|cθ|icθ), cθ〉|cθ|2

=b〈icθ, cθθθ〉|cθ|3

−

3bκ2〈icθ, cθ〉|cθ|

Now, we solve the remaining part in two ways:

•

b

|cθ|3〈icθ, |cθ|(κθicθ + κicθθ)〉 =

b

|cθ|2(〈icθ, κθicθ〉+ 〈icθ, κicθθ〉)

=b

|cθ|2(κθ〈icθ, icθ〉+ κ〈icθ, icθθ〉)

=b

|cθ|2(κθ|cθ|2 −(((((

((κ2|cθ|〈icθ, cθ〉)

= bκθ

•

〈icθ, cθθθ〉 = 〈((−c2)θ, (c1)θ), ((c1)θθθ, (c2)θθθ)〉

= (−c2)θ(c1)θθθ + (c1)θ(c2)θθθ

= −(c2)θ(c1)θθθ − (c2)θθ(c1)θθ + (c2)θθ(c1)θθ + (c1)θ(c2)θθθ

= (−(c2)θ(c1)θθ + (c1)θ(c2)θθ)θ

= 〈((−c2)θ, (c1)θ), ((c1)θθ, (c2)θθ)〉θ

= 〈icθ, cθθ〉θ

which tells us that

b〈icθ, cθθθ〉|cθ|3

=b〈icθ, cθθ〉θ|cθ|3

= bκθ

39

Therefore, (2.6)= bκθ

(2.7) =〈bcθ, cθ〉|cθ|2

κθ +〈bcθ, icθ〉|cθ|

κ2 +1

|cθ|

(1

|cθ|

(〈bcθ, icθ〉|cθ|

)θ

)θ

= bκθ +

b〈cθ, icθ〉

|cθ|κ2 +

b

|cθ|3〈cθ, icθ〉θθ

= bκθ

= (2.6)

40

Appendix B - The H0-distance on Bi(S2,R2) vanishes!

The distance between any two curves C0, C1 ∈ Bi is identically zero. Given

any two curves c0, c1 ∈ Imm(S1,R2), we show there exists a

c(t, θ) ∈ Imm(S1,R2) such that c(0,−) = c0, c(1,−) = c1, and LhorG (c) is

arbitrarily small.

Let c(t, θ) be a path in Imm(S1,R2) from c0 to c1. We can reparametrize

to have our path horizontal, namely 〈ct, cθ〉 = 0. As discussed above, this

doesn’t affect the parametrization of c0, but we cannot say the same for c1.

We keep the name c for this reparametrization. We will now view our path

c as a smooth mapping c : [0, 1] × [0, 1] → R2, and we shall reparametrize

our curve (again), c(t, θ) = c(ϕn(t, θ), θ) with the piecewise linear function

ϕn(t, θ) defined below:

ϕn(t, θ) =

2t(2nθ − 2k) for 0 ≤ t ≤ 12, kn≤ θ ≤ 2k+1

2n,

2t(2k + 2− 2nθ) for 0 ≤ t ≤ 12, 2k+1

2n≤ θ ≤ k+1

n,

2t− 1 + 2(1− t)(2nθ − 2k) for 12≤ t ≤ 1, k

n≤ θ ≤ 2k+1

2n,

2t− 1 + 2(1− t)(2k + 2− 2nθ) for 12≤ t ≤ 1, 2k+1

2n≤ θ ≤ k+1

n.

Given our reparametrization, we have

∂θC(t, θ) = Ct(ϕn(t, θ), θ)ϕnθ(t, θ)+Cθ(ϕn(t, θ), ∂tC(t, θ) = Ct(ϕn(t, θ), θ)ϕnt(t, θ)

and,

(ϕn)θ =

4nt

−4nt

4n(1− t)

−4n(1− t)

(ϕn)t =

4nθ − 4k

4k + 4− 4nθ

2− 4nθ + 4k

−(2− 4nθ + 4k)

41

Because our path is horizontal, we can show the following:

|〈ct, icθ〉| = |ϕnt||ct||cθ|, |cθ| = |cθ|

√1 + ϕn2

θ

( |ct||cθ|

)2

|〈ct, icθ〉| = |〈(ct1ϕnt1 , ct2ϕnt2), (−(ϕnθct2 + cθ2), ϕnθct1 + cθ1)〉|

= |ϕnt(((((((−ϕnθct1ct2 − ct1cθ2 +(((((ϕnθct1ct2 + ct2cθ1)|

= |ϕnt||〈cθ, ict〉|

= |ϕnt||cθ||ct|

|cθ| = |(ϕnθct1 + cθ1 , ϕnθct2 + cθ2)|

=√

((ϕnθct1 + cθ1)2 + (ϕnθct2 + cθ2)

2

=√

(ϕnθct1)2 + (ϕnθct2)

2 + cθ12 + cθ2

2 + 2cθ1ct1 + 2cθ2ct2

=√

(|ϕnθct|2 + |cθ|2 +2〈ct, cθ〉

= |cθ|

√1 + |ϕnθ|2

( |ct||cθ|

)2

42

And we can now calculate the arc length of our curve:

LhorG (c) = Lhor

G0(c)

=

∫ 1

0

(∫ 1

0

〈ct, icθ〉2dθ

|cθ|

) 12

dt

=

∫ 1

0

(∫ 1

0

|ϕnt|2|ct|2|cθ|√1 + |ϕnθ|2

( |ct||cθ|

)2dθ) 1

2

dt

=

∫ 12

0

( n−1∑k=0

(∫ 2k+12k

kn

(4nθ − 4k)2|ct|2|cθ|√1 + (4nt)2

( |ct||cθ|

)2 dθ+

∫ k+1n

2k+12n

(4k + 4− 4nθ)2|ct|2|cθ|√1 + (4nt)2

( |ct||cθ|

)2 dθ

)) 12

dt

+

∫ 1

12

( n−1∑k=0

(∫ 2k+12k

kn

(2− 4nθ + 4k)2|ct|2|cθ|√1 + |ϕnθ|2

( |ct||cθ|

)2 dθ

+

∫ k+1n

2k+12n

(2− 4nθ + 4k)2|ct|2|cθ|√1 + (4nt)2

( |ct||cθ|

)2 dθ

)) 12

dt

We have that |cθ(ϕn, θ)| is uniformly bounded above, and away from 0

because by definition a curve c is an immersion when |cθ(θ)| > 0 at all

points. Then there exists d ∈ N such that 1d< |cθ(θ)|. We also have that

|ct(ϕn, θ)| is uniformly bounded. Now a little proof, Let a, b ∈ R and assume

there exists d ∈ N such that 0 < 1d< b. We show there exists a K ∈ R such

that

1√1 + a2

b2

<K√

1 + a2(5.4)

43

Proof.

1√1 + a2

b2

=a√

b2 + a2

<|a|√1d2

+ a2

=|a|d√

1 + a2d2

≤ |a|d√1 + a2

Then by (5.4) we have the following inequality:

n−1∑k=0

∫ 2k+12k

kn

(4nθ − 4k)2|ct|2|cθ|√1 + (4nt)2

( |ct||cθ|

)2 dθ < n−1∑k=0

∫ 2k+12k

kn

K(4nθ − 4k)2|ct|2√1 + (4nt)2|ct|2

dθ (5.5)

where K ∈ R. Then, by simplifying the boundaries of the integral and

subsequently substituting θ with θ + kn, we get that (5.5) is

= Kn−1∑k=0

∫ 12n

0

16n2θ2|ct(ϕn(t, kn

+ θ), kn

+ θ)|2√1 + (4nt)2|ct(ϕn(t, k

n+ θ), k

n+ θ)|2

dθ (5.6)

Now, let ε < 0. We split the integral∫ 1

2

t=0into

∫ εt=0

+∫ 1

2n

ε, the first of which

is O(ε) uniformly in n (i.e. regardless of n). We now split the integral∫ 1

2n

0

as∫D1,k

+∫D2,k

where D1,k := θ ∈ (0, 12n

)]||ct(ϕn(t, kn

+ θ), kn

+ θ)| < ε (a

countable union of open sets) and D2,k its complement in (0, 12n

). Then,∫ 12

0

(5.6)dt ≤ O(ε) +

∫ 12

ε

(K( n−1∑k=0

∫D1,k

+n−1∑k=0

∫D2,k

))dt (5.7)

44

We can now estimate as follows:

n−1∑k=0

∫D1,k

=n−1∑k=0

∫D1,k

16n2θ2|ct(ϕn(t, kn

+ θ), kn

+ θ)|2√1 + (4nt)2|ct(ϕn(t, k

n+ θ), k

n+ θ)|2

dθ

≤n−1∑k=0

∫D1,k

16n2θ2ε2√1 + 0

dθ

≤ n16n2ε2(θ3

3|

12n0

)≤ O(ε2) (5.8)

In the following estimate we use that t ≥ ε and |ct| is uniformly

bounded:

n−1∑k=0

∫D2,k

=n−1∑k=0

∫D2,k

16n2θ2|ct(ϕn(t, kn

+ θ), kn

+ θ)|2√1 + (4nt)2|ct(ϕn(t, k

n+ θ), k

n+ θ)|2

dθ

≤ O(1)n−1∑k=0

∫D1,k

16n2θ2√(4nε)2ε2

dθ

= O(1)n−1∑k=0

∫D1,k

4nθ2

ε2dθ

≤ O(1)n4nθ3

3ε2|

12n0

≤ O( 1

nε2

)(5.9)

Then using (5.8) and (5.9), we get (5.7) tends towards 0 as n → ∞. The

other integrals can be estimated similarly, and we get the above vanishes.

We can then approximate ϕn by a smooth function without changing the

estimates essentially, and we have our result.

45

Appendix C - Degree of an immersion

The degree (turning number) of a curve c c ∈ Imm(S1,R2), degree(c) is the

winding number respect to the origin of the tangent vector cθ, a multiple of

2π; positive in the counter-clockwise direction, and negative in the clockwise

direction. An equivalent definition which perhaps more useful in practice is

degree(c) =

∫S1 Kds

2π=

∫S1 K|cθ|dθ

2π(5.10)

where the numerator of (5.10) is known as the total curvature of c;

the integral of curvature with respect to arclength. The total curvature

is always a multiple of 2π. We note that the requirement for c to be an

immersion is necessary, since the first definition would be undefined if

cθ = 0. However, we wished to use a similar notion when it came to

monotone correspondence. To reiterate the definition, we have

R ⊂ S1 × S1 is defined to be a monotone correspondence if it is

the image of a map

R→ S1 × S1

x 7→ (h(x) mod 2π, k(x) mod 2π)

where h, k : R → R are monotone non-decreasing continuous

functions such that h(x + 2π) = h(x) + 2π, and k(x + 2π) =

k(x) + 2π.

The fact that our functions are required to be continuous and satisfy the

sort of periodic nature induced by h(x + 2π) = h(x) + 2π insures that the

slopes are bounded. The latter part of the definition is required as we do not

46

to wish to relate, say the curves c1, c2 : S1 → R2 where

c1 = (cos(θ), sin(θ)) and c2 = (cos(2θ), sin(2θ))

as

degree(c1) =

∫S1 K(c1)ds

2π=

∫S1 1ds

2π=

∫S1 1dθ

2π= 1

whereas

degree(c2) =

∫S1 K(c2)ds

2π=

∫S1 1ds

2π=

∫S1 2dθ

2π= 2

Removing the latter part of the definition would let allow the function set

h(x) = x, g(x) = 2x and would thus create an equivalence between c1 and

c2. This part of the definition is perhaps better illustrated by saying the

average speed of the function is one: A Caution! This does not make sense.∫ 2π

0

h′(x)dx = 2π

Unfortunately, our functions are only continuous, and nowhere guaranteed

to be differentiable. We can however, take note of the nice continuous

functions we employ, namely monotone non-decreasing functions with

bounded-slope, and we are able to tiptoe around the fact that they are

possibly not everywhere differentiable. 1 Having nice functions means we

only need to concern ourselves with bends and cusps. If there are finitely

many of these points, then we simply split up the integral at these points.

Since our functions are monotone, they are differentiable almost everywhere

1 A brief history note: Weierstrass suspected that a continuous monotonicnowhere differentiable function could exist. He was however proved incorrectin 1903 when Henri Lebesgue showed that such a function would have to bedifferentiable everywhere except on a set of measure zero.

47

Figure 5–1: f4(x)

and we are able to “make sense” of the above integral. We illustrate a case

of a continuous function with infinitely many cusps in the following example.

Example 3. Define

fn(x) = −√( π

2n−1

)2

−(x− kπ

2n−1

)2

+k + 1

2n−1for x ∈

[kπ

2n−1,(k + 1)π

2n−1

]where 0 ≤ k ≤ 2n − 1. We then define f = limn→∞ fn. Then∫ 2π

0

f ′(x)dx = limn→∞

2n−1∑k=0

∫ (k+1)π

2n−1

kπ2n−1

−√( π

2n−1

)2

−(x− kπ

2n−1

)2

+k + 1

2n−1dx

= limn→∞

2n−1∑k=0

[−√( π

2n−1

)2

−(x− kπ

2n−1

)2] (k+1)π

2n−1

kπ2n−1

= limn→∞

2n−1∑k=0

√( π

2n−1

)2

= limn→∞

(2n)π

2n−1

= 2π

48

References

[1] Martin Bauer. Almost local metrics on shape space. PhD thesis,Universitat Wien, 2010.

[2] Matthew A. Grayson. The heat equation shrinks embedded plane curvesto round points. J. Differential Geometry, 26:285–314, 1987.

[3] Michael Kass, Andrew Witkin, and Demetri Terzopoulos. Snakes: activecontour models. International Journal of Computer Vision, 1:321–331,1988.

[4] Peter W. Michor and David Mumford. Vanishing geodesic distance onspaces of submanifolds and diffeomorphisms. Documenta Mathematica10, 10:217–245, 2005.

[5] Peter W. Michor and David Mumford. Riemannian geometries on spacesof plane curves. J. Eur. Math. Soc. (JEMS), 8:1–48, 2006.

[6] Jayant Shah. H0-type Riemannian metrics on the space of planar curves.arXiv:math.DG/0510192, v1, 2008.

[7] A. Yezzi and A. C. G. Mennucci. Metrics in the space of curves.arXiv:math.DG/0412454, v2, 2005.

[8] A. Yezzi and A. C. G. Menucci. Conformal metrics and true gradientflow for curves. In ICCV, pages 913–919, 2005.

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