DIFFERENTIAL GEOMETRY OF

MINKOWSKI SPACES

Thomas E. Taylor

SCBMITTED IS PARTIAL FC'LFILLMEST OF TUE

REQUlREhIENTS FOR THE DEGREE OF MASTER OF SCIEXCE

AT

DALHOUSIE UNIVERSITY

HALIFAX, NOVA SCOTIA

SEPTEMBER 1996

@ Copyright by Thomas E. Taylor, 1996

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Contents

List of Tables

List of Figures

Abst ract

vi

vii

X

Acknowledgements xi

List of Symbols xii

1 Introduction 1

1.1 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

. . . . . . . . . . . . . . . 1.2 Finite Dimensional Yormed Vector Spaces :3

1.2.1 T h e Minkowski Metric . . . . . . . . . . . . . . . . . . . . . . 4

1.3 TheDualSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Concepts from Euclidean Differential Geometry . . . . . . . . . . . . 11

1.4.1 TheinnerandCrossProducts . . . . . . . . . . . . . . . . . . 1 2

1.42 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1:3

1.4.3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.4 kfeasurements of Length . Angle and hrea . . . . . . . . . . . 19

2 The Isoperimetric Problern 23

2.1 Introducing the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Other Facets and General Comments . . . . . . . . . . . . . . . . . . 27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 T h e Isoperimetrix 29

. . . . . . . . . . . . . . . . . . . . 2.3.1 The Isoperimetrïx on M2 30

. . . . . . . . . . . . . . . . . . . . 2.4.2 The Isoperimetrix on M~ :35

Two-Dimensional DiEerential Minkowski Geometry 40

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1 Introduction 40

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Curve Length 5:3

3.3 Curvature in the Minkowski Plane with respect to U . . . . . . . . . 54

. . . . . . . . . 3.4 Curvature in the Minkowski Plane with respect to Z, 58

. . . . . 3.5 Studies of the Isoperimetrix under the Busemann Definition 62

. . . . . 5 . Arcs . Curvature. Length and Area of an Isoperimetrix 63

13.6 Studies of the Unit Circle under the Holmes-Thompson Definition . . 67

. . . . . . . . . 3.6.1 Arcs . Curvature . Length and hrea of a Circle 69

Bibliography 71

List of Tables

. . . . . . . . . . . . . . .3.1 The Busemann unit circle and isoperimetrix 41

. . . . . . . . . . 3.2 The Holmes-Thompson unit circle and isoperimetrix -11

List of Figures

1.1 Hermann Minkowski . . . . . . . . . . . . . . . . . . . . . . . . . . . i ) - . . . . . 1.2 The Minkowski geometry generated by the unit 'circle" 2.4. .

r)

1.3 Thebal l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

. . . . . . . . . . . . . . . . . . . . . . 1.4 The vectors 1 of the dual bal1 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The dual bal1 9

1.6 The polar of a hexahedron is an octahedron . . . . . . . . . . . . . . 10

1.7 A simple closed curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.8 A non-simple closed curve . . . . . . . . . . . . . . . . . . . . . . . . 14 4

1.9 The vectors fi_ and b for a circle . . . . . . . . . . . . . . . . . . . . 15

. . . . . . . . . . . . . . . . . . 1 LO Two helices differing by a rigid motion 17

1.11 An example of a regular surface in E~ . . . . . . . . . . . . . . . . . 18

1.E X tubular surface in E~ . . . . . . . . . . . . . . . . . . . . . . . . . I l

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Soap Bubbles 23

Diagram for the Isoperimetric Problem . . . . . . . . . . . . . . . . . 25

The Double Bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Kepler's Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

The unit circle in the Manhattan norm . . . . . . . . . . . . . . . . . 31

The convex curve C . . . . . . . . . . . . . . . . . . . . . . . . . . . . :31

A scalar multiple of the isoperimetrix . . . . . . . . . . . . . . . . . . 34

The plane H translated to H f . . . . . . . . . . . . . . . . . . . . . . 36

The support function . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

vii

3 . 1 The relationship of area to arc length of the isoperirnetrix . . . . . . . 43

3.2 The minimum distance frorn a line to a point . . . . . . . . . . . . . . 49

3.3 bfaxirnizing the determinant on 2, . . . . . . . . . . . . . . . . . . . 50

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Iinear functional h 5.3

5.5 The Frenet representation of the curve C with respect to the unit circle 35

3.6 The Frenet representation of the curve C with respect to the isoperimetrix 39

Abstract

The study of differential geometry in a Minkowski space by O. Biberstein (1937

Ph.D. thesis) has used the unit bal1 as a starting point to define the concept of

curvat ure. This met hod follo~vs nat urally from the Busemann definit ion of area t hat

Biberstein has employed. However, since that tirne another useful definition of area in

a Minkowski space has been developed: Holmes-Thompson area. This thesis explores

the ideas of differential Minkowski geometry in two dimensions iinder both of tliese

defini t ions and compares the t wo streams of results.

Acknowledgement s

1 would like to thank rny supervisor Dr. A C . Thompson for introducing me to this

topic and for al1 of the help and advice he has given me over the past year with matters

both academic and non-academic. Also, 1 would like to thank Dr. K. Johnson and

Dr. K. Dunn for taking the time to be the readers of this thesis.

List of Symbols

Points - a . bt c

Lines and Curves - A, B, C

Functions - f g. h

Minkowski sine - sm

Domain - D

Determinant - det (5. y7

Inner Product - (2: z j )

Euclidean norm - II 11 Euclidean rnetric - dE

Gnit circle - U

Open Ball(center,radius) - B(r. r )

OriQn - O 4

C'nit normal vector - n

Torsion - T

Curvature - k

Length - L Arc - O

Euclidean space (n-dim.) - En

Scalars - a, 13: - - - . Vectors - x. y? z

Sets - S. T.. V

Tangent plane - 7

Region - R

Norm - 11 . II Cross Product 5 x y Minkowski norm - II - Il.ir Minkowski metric - dLLr

Isoperirnetrix - 1

Closed Ball(center.radius) - B(r. r )

Angle - 8

h i t tangent vector

Unit binorrnal vector C .\nt i-curvat ure -

Area - A

Radius - r

Minkowski space (n-dim.) - M n

Chapter 1

Introduction

1.1 Brief History

The first thing one should note about this thesis has to do with the label i1/iinko1uski

geo.metry. Unfortunatel- both rnathematicians and physicists have adopted this name

to refer to different concepts in their respective fields. The latter group, when referring

to Minkowski geometry. is referring to the geometry of special relativity. This version

is used to study four-dimensional space-time and not what will be discussed here.

Therefore to avoid any confusion 1 shall begin with the mathematical clefinition of

the terrn as it appcars in thc "Encyclopacdia of Mathematics'' [9].

Definition 1 (Minkowski geometry) The geornetry of a finite-dimensional normed

space, that is. an afine space with a Minkowski m e t n c - a rnetn'c invariant un-

der parallel translation - in which the role of the unit sphere is played by a gicen

ce n trally-s ymmet nc convex b O dy .

The concept of a convex body is essential to the following discussion so it is

appropriate to define what is meant by convexity here as well:

Definition 2 (Convex set) A set S in s o m e linear space X is said to be convex if it satisfies Val b E S, y a + (1 - y)b E S for O 5 y 5 1. Geornetrically, this means that

Figure 1.1: Hermann Minkowski

for any two points in the sel, the straight line joining them is ais0 contained in the

set,

Minkowski's fame in the general scientific community resulted mainly from set ting

up the mathematical framework for Einstein. However. this was not t h e only signifi-

cant work that Minkowski contributed in his lifetime, his imost original achievement"

[il arose from his work with n-ary quadradic forms. Using a geometrical method intro-

duced by Gauss and Dirichlet for ternary quadradic forms, Minkowski augmented the

theory by introducing the concept of volume. From this association sprang signifiant

results in algebraic number t heory.

The next step for Minkowski was to study the problem of packing ellipsoids into

the tightest volume in a given region of space (which is equivalent t o the sphere

packing problem). As a variant, Minkowski decided to replace the ellipsoid or sphere

with some other convex set. The main class of convex sets that he worked with

was polyhedrons. As this theory developed? Minkowski was led to consider the more

general case of convex sets in n-dimensional space and realized that. for each n- a

symmetric convex body within the space defined an alternate concept of "distance".

Thus Minkowski geometry was born. This theory was essential for t he creation of

normed spaces and funct ional analysis.

1.2 Finite Dimensional Normed Vector Spaces

The settings that are of prïmary importance to this work are En, the n-dimensional

Euclidean space (Rn, I I - I l E ) ; and Mn, the n-dimensional Minkowski space (Rn. II I l z r ) for n 5 3. Both of these spaces are examples of a finite dimensional normed vector

space. It will be necessary to understand this concept before proceeding further with

t his t hesis.

First of all. a vector is best thought of as an arrow sitting in some space, Xo

that extends from the origin of that space to some point in the space. A scalar is an

element a of a field <P (the field of interest here shall be the real numbers B). The

vector space arises wit h the definition of the following tmo operat ions:

1 . Addition

V Z ~ E X . I + f = f + . F .

V Z , & f € X . I + ( G + z ) = ( Z + ? j ) + = :

~ ! O E x such that I + O = Z , VZE XI and -

VZ E X 3! -Z such tha t Z + ( - I ) = 0.

VZ, y~ X and Va, ,d E Q I ive have:

A vector space is said to be fininite dimensional whenever it is spanned by a finite

basis. That is. each vector in the space can be uniquely specified bq- a finite Iinear

combination of basis vectors resulting in the n-tuple [ x l , xl, . . . , x,]. Thus the vector

5 is uniquely deterrnined by the coordinates xi, for i = 1. 2?. . . : n.

The next concept that will be explored from En and Mn is that of a norrn. The

notation that will be used here is II I l E for the Euclidean norm and 11 - IILbf for the

Minkowski norm. A vector space X is said t o be equipped with a n o m , or be a

normed vector space if VZ E X we can assign a non-negative number llZll (ivhich is

called the n o m of 2) such that the following properties hold:

1.2.1 The Minkowski Metric

In general. the properties that a vdid metric, d(J. satisfies are as follows. Given

elements Z? y. Z of a rnetric space X. ive have:

.A11 normed spaces are also metric spaces. That is, there esists a well-defined

distance function on the space. The metric function is usually denoted by d(5.j ' )

(read as the distance betiveen Z and 3. In normed spaces. the metric is exactly

1 1 - 1 . 1 shall use dE and dicr to represent the Euclidean and Minkowski rnetrics

respect ively The familiar Eiiclidean met ric is given by

Figure 1.2: The Minkowski geometry generated by the unit icircle' LI.

In order to obtain a Minkotvski metric we rnust consider t h e direction in which we

measure distance. Let C; be a closed convex set whose center is at O. As mentioned

earlier, the geometry of the space will be determined entirely by the choice of L.

To account for direction in determining the Minkowski distance between 2 and y? consider S and y', the vectors that extend to the boundary of U such that the Iine.

L i t joining 5' to f through the ongin is parallel t o t h e line, L2. joining the vectors 2

and Y (see example). First for the length of a single vector 2 we have

and t hus

This equation

squash or stretch

appears in a similar form in a paper by Petty [.Il. It allows us to

Euclidean spaces in a convenient way, the reasons for which will

become apparent in the next chapter.

Exarnple 1 Consider as / I the square in E2. With corners at ( 1 , 1). (-1.1). (1. -1). (-1. -

(see Jigure 1.2)). Let F = 31 and ij = [4 51. Thea &(Z, y3 = 'fi. &(Zr . y') =

and thus dLbf(T.y7 = 2.

This esample illustrates that the "distance" from any point on the boundary of

lJ, through O? to its diarnetrically opposite point has lengt h 2. Similarly di\r (O. 5) = 1

whenever i? E dCL

The above example is greatly simplified in that 5-y need not be pointing in direc-

tion that causes 5' and to be a t corners of li and it's length was also conveniently

chosen. However. it helps to illustrate the nature of this geornetry.

1.3 The Dual Space

This section is an introduction to the dual (or polar) space as it applies to the stiidy

of Minkowski clifferential geornetry. The motivation for incliiding this cliscussion is

that the isoperimetrix (see chapter 2) is a 90" rotation of the dual of the ball. In most

cases that followvt I will use the word " dual" to refer to some object in the -dual'

space and use the word "polary to refer to the action of forming the dual object.

To begin, we will need the following definitions.

Definition 3 (Linear functionals on Mn) A linear functional on Mn, f. is a

l inear mapping fmm Mn to R. T h a t is, f is a linear function operating on uectors in

Mn and refurning real nurnbers.

Definition

is the finite

4 (The dual space (Mn)- ) The dual space of Mn. denoted (Mn).'

dimensional normed uector space of a n h e u r functionals on Mn.

We shall also consider a convex set S as the intersection of al1 closed half spaces

H - which contain S. To facilitate this frarne of reference we define:

Definition 5 (Hyperplane) A hyperplane. H' in Mn is a translate of an ( n - 1)-

dimensional subspace which is given by a linear funetional f E Mn. That is for - x = (xI?. . . ,x,) E Mn and scalars CO, cl.. . . ,c, we have

H = (5 E Mn : f ( Z ) = CO + c i x i + . . . + ~ x , = O}. ( 1.5)

Definition 6 (The dual of a closed convex set) Let I< be a closed conrex set in

Mn. The dual 01 I<. denoted IC0, is giuen by

The duality arises as a result of the following important theorem

Theorem 1 I/ K is a closed convex set in Mn containing the origin then

Thos the dual of a dual set is the original set.

\Ve may definc a half space by al1 of t hc points in the spacc which lic on one sidc

of a hyperplane. or explicitly

H- = {S E Mn : f(Z) 5 O}. ( 1.8)

Note that for centrally symmetric convex sets: equation 1.6 is ecpivalent to re-

quiring that 1 f(I)l 5 1 (because of the symmetry of these sets). This being said. we

may nooiv describe the dual ball.

Definition 7 (The polar dual of the ball) Let B be the unit ball dejining some

Mn. The polar dual of B is given b y

Figure 1.3: The ball

The set Bo is also a closed convex set and as such we may define it to be the unit

ball of the dual space. The norm II - 11' on (Mn)=. defined by Bo. will be such that

and

Since our study rvill concentrate on M2 it would be useful to see an example of

the concepts in this chapter as they apply to two dimensions. Let the ball be given

as in figure 1.3. From ecluation 1.9 we must find al1 linea,r f u n c t i ~ n a l s ~ f. on M~ that

satisfy the given conditions. Now (M')- is also a finite dimensional normed vector

space with norm II 11'. Thus we may equivalently state 1.9 as

Figure 1.4: The vectors f of the dual ball

Figure 1.5: The dual ball

Figure 1.6: The polar of a hexahedron is an octahedron

and the boundary of the dual ball is given by

Geometrically, the procedure to construct Bo is as follows. Consider a point on

B given by a unit vector E and the tangent line H to B a t S. The vector f is that

one which is normal to H and with unit length with respect to the dual norm (see

figure 1.4).

Now with the ball as in 1.3 ive have that for al1 unit vectors on a part.icular line

segment (not including the corners), we get only one f vector (actually many copies

of the same vector). Then the intersection of al1 half planes in (M')~ which contain

these S points is the dual ball given in figure 1.5.

1.4 Concepts from Euclidean Differential Geom-

etry

There are many similarities between the study of differential geometry on Euclidean

spaces as compared to Minkowski spaces. This section will explore some of the im-

portant concepts from the Euclidean case that will be considered later. There are

two reasons for this section to appear here, firstly, as an introduction to what are the

considerations of differential geometry in a setting where our preconcevied notions of

length, area, angles, etc. a11 hold, and secondly as a frame of reference to contrast its

study in the Minkowski spaces M* and M?

1.4.1 The Inner and Cross Products

Both the inner and cross products are specialized Pmultiplication" of vectors that

each carry their own geometrical interpretations. The former is used to identify or-

t hogonality between vectors and the latter to measure the of t he parallelogram

(parallelepiped) generated by two (or more) vectors. It is important to note that the

inner product procluces a scalar and the cross product produces a vector orthogonal

to each of the product components with length equal to the -ares- described above

T h e inner product. also known as the dot product. of two n-dimcnsional nx tors

2, y' shall be denoted (5: 9. This product has two equivalent formulations

where 0 is the angle between S. f with respect to the plane t hat the two vectors span.

This inner product also h a the following properties:

1. If K y # O then (S'y3 = O 5 is orthogonal to f7

Consider the cross product, again otherwise referred to as vector product, of two

3-dimensional vectors, 5, E E ~ . This product is characterîzed by

Note that in general? the cross product requires n - 1 vectors in En. +

Thus. choosing z conveniently, we can explicitly express 3 x f as

where el = [ l , 0: O ] , eî = [O. 1, O], e~ = [O, 0.11 represent the standard basis vectors in

The properties of the cross product for Z. f, Z E E~ are:

3. 5 x y = O .F is a multiple of ij.

One other feature of I x must be considered? and that is direction. It follows

from the above definition that (Z, f . 5 x fi is a positive basis. That is, the matrix that

changes the basis (5? ',, 5 x $ into (e l$ e i7 e3 ) has a positive determinant (note that

the order of the vectors is important here).

I shall end this section with two relations that occur frequently when dealing with

inner and cross products.

Where A stands for the area of the parallelogram formed by I and ij'.

1.4.2 Curves

There are two divisions to the study of curves in differeotial geometry: global issues

and local issues. Consideration will b e given to both here. However, this section is

meant only to give a flavour for the topic in the farniliar settings E~ and E ~ . so I will

lirnit the discussion to global theory for curves in E2. and local theory for curves in

E ~ . In both cases, the curves will be considered to be parameterized by arc length s.

.4n extensive treatment may be found in [Il.

Curves in E ~ : Global Theory

There are several situations that arise rvhen dealing with curves in the plane. One of 4

the more interesting to consider is that of a simple closed curve. Let C(s) : [a. b] -.t

E~ be a twice differentiable curve in E' parameterized by arc length and note t h e

following defini t ions.

Definition 8 C is said to be closed whenever @)(cc) = @ ) ( b ) for n = O? 1.2. That

is C and its derivatives agree at a and b.

Definition 9 C iù: said to be simple whenever cl # c:, * C(q) # f ( c 2 ) Vcl. c2 E

(a , 6 ) . Thnt is the cwue does not intersect itself (see figures 1.7 and 1.8 respecliuely).

Two other important notions to consider here are orientation and the concept of

an interior. For simple closed curves in the plane, the interior shall be that part of

the plane that is bounded by the curve.

Figure 1.7: A simple closed curve

Figure 1.S: A non-simple closed curve

The convention chosen for the orientation of simple closed curves may be clescribed

as follows: if you can walk along the curve in the direction arising frorn its param-

eterization such that your left hand always points toaards the interior of t he curve

then the curve is positively oB'ented. Negat ive orientation is the same descript ion

with "right" replacing -left7.

Local Theory

This section is an introduction to the local theory of curves as studied in Euclidean

differential geometry for curves in E3 that have been pararneterized by arc iength S.

The concepts introduced here will also apply to E ~ . however some of them apply in

a degenerate way.

Let C ( s ) : (a? 6 ) E~ be a twice differentiable curve in E3 ( tha t is p ( s ) and

k = constant

C 7 = 0

+

Figure 1.9: The vectors 6, < and b for a circle.

C ' ( s ) both exist Vs E (a. 6) c W and that C ' ( s ) = i(s) is the unit tangent vector to

C at C(s)).

Definition 10 (Curvature) With C as above. we dejine k ( s ) = I I~ ' ' ( s ) [~E to be the

cuniature of C at &).

Geometrically we think of the curvature a t s as a rneasure of t h e rate of change

of the direction in ii-hich a unit tangent çcctor. q s ) , is pointing. Bcyond t here are

two other important unit vectors that arise, namely E, the unit normal vector in the

direction of p ( s ) (thus rnaking Z orthogonal to t) and b. t h e unit b i n o m a f vector

defined by b(s) = t ( s ) x 6 ( s ) pointing normal to the plane spanned bq- f l s ) and Z(s)

(see figure 1.9). This plane will be referred to as the oscdating plane. or the plane - that is tangent to the curve. It is important to note that in IC2. b has no meaning, in

any case it iç unnecessary since the osculating plane is aiways E? At this point. a further restriction is required on the curves t o be studied. Note

that C"(s) = B(s)i?(s), thus if C is a straight line, 6(s) and therefore the osculating

plane will not be defined. Those points s E (a. b) for which P ( s ) = O shall be referred

to as singular points of order 1.

Definition il (Torsion) With C as above and C r f ( s ) # O , iue define the torsion of

C at C(s) to be r(s) given in the relation V ( s ) = r ( s ) Z ( s ) .

This definition is arrived a t by ooting tha t C(s) = t (ç ) x R'(s). Since the product d

rule applies to cross products in the usual way we have that b f ( s ) = ?(s) x E ( s ) + q s ) x R f ( s ) = f i s ) x C r ( s ) . Ge~rnetrically~ this means that P(s ) is normal to both l (s)

and E(s ) and therefore parallel to Z ( s ) . T h e T function is simply the length of b'(s).

.A curve travelling in a plane in E~ hast by definition. r = O. The geometric

interpretation of this value is the measure OF the rate that the c u v e is pulling away

from the osculating plane.

h sirnilar analysis may be done regarding ? ( s ) and E f ( s ) . In fact these equations

are three of the most important ones to differential geornetry. They are collectively

known as the Frenet formulas:

Again here I shall try to explain geometrically what these yiiantities describe. The

first. ?. is a vector pointing in the direction of R and having a length equal to the

magnitude of t h e curvature at that particular point s on the curve. T h e second. Cf.

is a sum of the vector in the direction - h i t h "curvature length- and the vector in

the direction -b with length equal to the torsion. Finally. b' is. as described above.

in the direction of 5 but with length equal to the torsion.

r 2 ue n Theorem 2 (Fundamental Theorem of the Local Theory of Curves [II) C'

the differmtiable funetions k ( s ) > O and r ( s ) for s in some intentai 1. there erists

a regular parameterked cume C : I -+ E~ such that P ( s ) is the curvature? r ( s ) is

the torsion, and s is the arc length of C. Moreover, given nnother curve. El, that

satisfies the same abooe conditions as C . then B dgers from C by a rigid motion

(see Jigvre 1.10). That is B = F ( C ) + Zo for some orthogonal linear map F of E3.

Figure 1.10: Trvo helices differing by a rigid motion

The significaace of this theorem is that locally. the only information needed to

describe any curve (with the aforementioned properties up to a rigid motion). is

the curvature and the torsion of that curve. The proof of the fundamental theorem

involves a lengthy existence and uniqueness argument that is reminiscent of proofs of

this kind from various undergraduate courses such as differential equations.

1.4.3 Surfaces

A surface may be considered as a subset of E~ that has been deformed in some

fashion and embedded in E~ . This is a natural extension of one dimensional ctirves

in E~ as described previously. We shall require, as with curves, a certain degree of

smoothness to the surfaces we study here so that the concepts of calculus apply to

the entire surface. This section deals with how to make rigorous the idea of a surface

that is "smooth eoough' for our purposes (see figure 1.11). Working with these strict

conditions local propert ies will have global implications [l] .

Figure 1.11: An example of a regular surface in E3

Definition 12 (Regular surface) Civen a set S c E~ such that VZ E S . 3V. a

neighborhood of i i in E ~ . and a rnapping z: Cr - (\a' f l S ) of an open set I - C E~

onto (V n S ) c E~ and such thnt the following holdr

1. The rnapping ~r (nlso called the parameterixtion) is differentiable. That is

V ( u t a ) E V . the rnapping ~ ( u . u ) = ( x ( u . v). y ( u . o). z (u - o ) ) hns continuous

partial denùat ices of ail orders.

7. The mapping rr is a homeomorphisrn (i.e. rr-' exists and is continuous).

S. For al1 (ü E IL the differentinl dx,: E* - E~ is one-to-one.

then S is a regular surface.

Definition 13 (Differentiable function on a regular surface) A function f. de-

Jned in an open subset of a regular surface S , f : (V C S ) - IR. is said f o be

dgerentiable at a point ii E V ! i//or some parorneterkation ~r: ( I r c E') - S mith

â E *(II) c V , we have that f O rr: (Li c E ~ ) --t E is differentiable at A-'(;) .

the aboue holds Va E V then f is said to be differentiable in V .

Before proceeding further, it is necessary t o introduce the tangent plane. 7. h

regular surface has an infinite number of tangent vectors for each point à of the

surface (since they are one-dimensional vectors in E3). Perhaps suprisingl- al1 of

these vectors lie in a plane which is tangent to the surface at B. This particular plane

is denoted Tc. Note that the tangent plane is not unique for surfaces that are not

regular.

Theorem 3 (Tangent Plane) Let 6 E U c E'. and let S be the reguiar surface

given b y the rnapping T: Li - S . T h e n d z g ( E Z ) c E~ coineides with the set of al1

tangent vectors to S at ~ ( b ) .

Proof: Let 6 be a tangent vector to S at ~ ( b ) . Let ii: [(-el 6) c E ~ ] - [rr(U) c SI be the diffeomorphic reparameterization of 5' such that r ( b ) = %(O). Thus the curve.

A. defi ned by A = ?r-' O ii: (-c. e ) - U is differentiable. Now since d?rg(Ar(O)) = &. then E dq(E2) .

Conversely. if r2 is any vector in d q ( G ) for some G E E ~ . then W is the velocity

vector of the curve B: (-6, E) -t U given by &(a) = aG + 6 for a E (-cm e ) . Thus - t 2 = n'(O), where R = 7r O B and & must be a tangent vector. O

The choice of parameterization gives rise to a basis for %(SI that is given by m m i ; ) + ,, y T} (where ~ ( b ) = ü E S. (u . v ) E E') and shall be referred to as the basis

associnted to R. We shall denote

With this justification for the existence of the tangent plane we may proceed with

the types of measurements that are made on surfaces.

1.4.4 Measurements of Length, Angle and Area

The issues of this section are of primary importance to what follows in later chapters.

Ho~vever~ the tools developed for the study of these concepts in En will have to be

reworked to b e applicable in the more general setting of Mn.

1 shall begin with the introduction of two new terms that will make forma1 the

study of al1 or 'pieces" of some surface in E ~ . The first is a domain. A domain. D of

a surface S is an open connected subset of S such that its boundary is the image of

a circle by a differentiable homeomorphism whose differential is non-zero. except a t

a finite number of points. The closure of a domain, that is D U BD is what shall be

referred to as a region. -4 region is denoted by R.

The First Fundamental Form

Let S be a regular surface in E~ with E S. Then there is an induced inner product

on T@)l denoted by ( . ),. This induced inner product is equal to the usual Euclidean

inner product on E3. that is. for two tangent vectors within the tangent plane of S + - - 4

at âo t l : t 2 E 1;i(S) c E ~ . we have (c & = ( t , . t Z ) E . Corresponding to this induced Ta inner product is a quadradic f o m Q,- W given b - :

Definition 14 The qundradic form defined aboce in equation 1-24 is called the first

fundamentel f o m al the point ü on the regular surface S c E ~ .

We next express Qa in a conveoient basis. Consider the regular surface S rvith

à E 5 that is parameterized by ~ ( u , u ) with u : v as in ecpation 1.23. We have that a

tangent vector. FE G(.s). is tangent to a pararneterized c u r w C on the surface given

by C(p) = r ( u ( 3 ) . v(,d)). ,d E ( - E , e) such that â = (?(O) = 1i(u0. cO). Then.

Q.- = (C'(0). ?'(O)). = ( iT , U' f R, u'. 7ï. U' + T , U ' ) ~

= (T,, Z,),(U')* $ ?(Tu: z ~ ) ~ u ' u ' f (ru: ~ ~ ) à ( 2 . ' ) ~

To ease the notation we let.

Figure 1.12: A tubular surface in E3

and arrive at

Exarnple 2 (Cornpute the first fundamental form) C'o.n.sider the sirrfnre girvrn

by the parameterization ~ ( u . u ) = (cos(u) - ~ C O B ( O ) . s i r ~ . ( u ) - .7sir((u). c)' O 5 ZL < :37r? O 5 o 9 10 (see fig. 1.4.4). Taking the partial derivatives ruith respect to u and c

giues

ru = ( - s i n ( u ) , cos(u), O ) , r, = (2sin(o) . -2cos(v). 1).

Then. we must co.mpute E: F and G,

E(u, o) = s in2(u) + cos2(u) = 1:

F(u . u ) = ?(s in(u)s in(v) - cos(u)cos(u)) = -2 cos(u + TI), G(u. U ) = 4sin2(v) + . ~ c o $ ( v ) + l = 5 .

W h these coeficients, the jirst jundamental fonn is a s shown in equation 1.25.

Next we use this mathematical structure to measure length. angle and area.

Length

Let C be a parameterized curve on some regular surface SI C ( p ) : (O.po) - S We

know from calculus that the arc leagth L of S from p = O to p = po is given by:

c(c) = &? I ~ C ' ( P ) ~ ~ E C ~ P -

But, we now have that if C(p) = sr (u (p )? a ( p ) ) then

I I C f ( p ) l l ~ = ,/&O) = , / E ( ~ I ) ~ + 2Fufd + G ( u ~ ) ~

and t hus.

Angle

Let Cl: C2 be t1 arameterizecl curves on some reguiar surface SI C i ( p ) : (0: p ) - S

and & ( p ) : ( O < p ) S. The angle, 8: between the two curves ( that intersect at

p = po) is given by

and thus the angle 4 between the coordinate curves u = uo and u = co parameterized

by î r ( u _ uo) and R ( u ~ . L.) is

cos(4) = ( r u , r u ) - F -- I I ~ I E I I R . I I E J E '

Area

Let R c S be a bounded region of a regular surface S T h a t is. there exists a ball.

B(iSo: r O ) C E ~ . that contains R. Let *: I; c E~ -t S be a pararneterization such

that for Q c U we have n(Q) = R. Then the area, A? of R is given by

Chapter 2

The Isoperimetric Problem

The purpose of this section is to familiarize the reader with the isoperimetrix. 1.

Recall, as displayed in example 1, that with each new definition of a unit "circle".

cornes a new geometry on that space and as will be shown, a new isoperimetric

problem. In later chapters, t h e isoperimetrix 1 plays a central role in t h e study of

the spaces M' and M3.

Figure 2.1: Soap Bubbles

33

2.1 Introducing the Problem

The isoperirnetric problem in two dimensions may be stated as follows: C h e n an

area, what is the shape of the simple closed curve that contains this area such that

its length is a minimum'? The solution in E~ is the circle and in fact this result will

generalize to the n-sphere in En. Given this solution in E', we irnmediately have

that for any simple closed curve C c E~ with length L and enclosing area A? L is

a minimum when C is a circle. Thus, we obtain what is known as the Isoperimetric

Inequalit-y:

However, a satisfactory proof for the problem itself tlid not appear until 1S70 when

Karl Weierstrass formally displayed t hat a solution does indeed exist . Siiprisingly.

this was more than 2000 years after the problem \vas first posed by the early Greeks.

Since then a number of proofs have been given with varying degrees of generalization

and difficulty. 1 shall present one for the two-dimensional case due to E. Schmidt

(1939). This proof makes use of the well known theorern of Green.

Theorem 4 (Green's theorern) The area A enclosed 69 a simple closed cume

~ ( t ) = ( z ( t ) . y ( t ) ) thnt is d i f f ~ r ~ n t i n h l ~ ~lrno.sl ~ v e n j u h e r e . where 1 f [ a . b] is a n

ar-bilrary painmekr. is giuen by:

Theorem 5 (The Isoperimetric Inequality [II) Let C c EZ be a simple closed

cunte, diflerentiable almost eveywhere and zoith fength L and enclosing an area A.

Then equafion 2.1 holds for C . Eqvality is achieved if and only if C is a circle.

Proof: Let A l and A2 be any two parallel lines that bound C, translate them

towards each other until they first touch C and relabel them B1 and B2 respectively

(see figure 2.2). Let S be a circle of radius r that has B1 and B2 tangent to it ancl

Figure 2.2: Diagram for the Isoperimet ric Problern

which does not intersect C. Define a Cartesian coordinate system whose origin. 0.

lies at the center of S such t hat the y-axis is parallel to B 1.

Let C be pararneterized by arc length s E [O,L]. ~ ( s ) = (+). y(s)) such that C

intersects B2 and B1 at s = sl = O and s = s2 respectivel-.

Let S be given b - K ( s ) = (+): i j ( s ) ) for s E [O, LI. Xote tliat +) projects points

on C to the circle of radius r and so Qs E [O. L] we have that x ( s ) = Z(s). Thus the

distance between B1 and B2 is exactly 'Zr.

Xote that the top half of S is constructed from that part of C travelling from sl

to s2 and the bottom half of S is constructed from that part of C travelling from s?

to sl. Also, where C is not convex, that part of S will be traced out more than once

but this will have no net effect on the calculation of its area.

Using equation 2.2. we have

A = JO z y'ds,

where d is the area enclosed by S. This implies that

C A + rr' = 1 (zy' - yrf)ds 5

and making use of the following algebraic identity

(2 + J ~ ) ( ( x ' ) ~ + - (19' - yd)* = (z4' + J ~ ' ) * 2 O

we have that 2.4 is

Using that the arithmetic mean of two positive numbers is larger than or equal to

their geometric mean we have

and thus equation 2.1 follows.

If we now assume that equality holds in equation 2.1? then it must also hold in

equations 2.4, 2.5. 2.6, and 2.7. From equation 2.7. it follows t h a t d = nr2 and thus

L = 2irr. Note that r does not depend on the choice of direction of B2. Equality

between the equations 2.4 and 2.6 implies equality in 2.5.

1.e.

(xx' + y = O

which is to say.

The latter equation gives us that x =t ry' and since r does not depend on the

direction of B2, x and y may be interchanged to obtain = + r x f . Therefore.

x2 + IJ2 = r 2 ( ( x 1 ) 2 + ( y ' ) 2 ) = rZ

and C is a circle. O

2.2 Other Facets and General Comments

There are in fact ma- versions of the isoperimetric problem in the mathematical

literature. and 1 believe it will be useful to consider a few of them before proceed-

ing further. Essentially. each one is a constraint problem that is studied under the

framework of the calculus of uariations. The calculus of variations? in brief. is the

search for curves. surfaces. and their generalized counterparts in bigher dimensions.

for which a given function attains a maxinium or a minimum.

Dido's Problem

One of the more famous problems of this family is Dido 's Problern. The history of

this problem is as follows (see also [Il]). Dido was a Phoenician princess: circa 8-0

B.C., who Bed from her home to escape the tyranny of her brother. She and her

followers ended t heir flight on the north coast of Africa where the local natives agreed

t o g a n t her as much land as she could enclose with the hide of a bull. Dido then had

a bull's hide torn into thin strips and tied together forming a large loop with which to

enclose a piece of land along the coast. Her problem was to find the shape to enclose

with the strand of hide that would give her the most land. More formally stated.

the problem is to find the figure bounded by a curve mhich has the maximum area

for a given perimeter. The solution to this problem is a circle. Xotice the duality of

this problem to the one stated in the previous section: the former fixes the area and

attempts to minimize perimeter while the latter fixes the perirneter and attempts to

rnaximize area.

The Double Bubble

The study of some isoperimetric problems have been referred to as the mathematics of

soap bubbles (see figure 2.1) since a soap bubble forms a sphere and t hus demonstrates

the solution to the isoperirnetric problem in E3. An interesting variant has been given

in [12] which 1 h a l l describe below.

Figure 2.3: The Double Bubble

Given two equal volumes in E ~ . the surface with minimum surface area that

encloses t hese volumes has as its shape two pieces of two equal sized spheres with

each piece large enough that they meet dong a circle at an angle of 9. This shape

is the double bubble (see figure 2.3).

Aiong with a proof that this is indeed the shape that solves the given problem.

the authors of [12] have also developed an isoperimetric inequality for this case given

as follows.

For any surface enclosing two regions of equal volume^ V in E ~ . The surface area

d satisfies

A3 2 3437rV2.

Crystal Growth

Another cvay to consider the problem is to introduce an energy function, f. acting

on vecton normal to planes in E=. The muai physical interpretation of /(Z) in this

setting is the energy per unit area required to separate a crystal into two pieces along

a plane where 5 is normal to that plane. Thus for each crystal shape defined by the

intersection of several planes, you can determine the total energy required to forrn

that shape. Just as soap bubbles form a sphere to minimize the amount of energy

required to enclose a given volume, crystals grow in a fashion that minimizes the

amount of energy required for their formation. The procedure for finding the optima1

crystal shape that minimizes the energy function was developed in 1901 by G. LVulE

(see [13] for a full treatment of tliis subject) and the shapes produced by bis method

are known as iLVulff shapesY. The \Vulff shape is the isoperimetrix of crystallography.

For simplicity consider crystals in E2, or if you prefer, a tiling of the plane. The

problem then reduces to finding the collection of tiles that cover an area. Al leaving

no gaps and such that the boundary of A has minimal length.

A Connection With Plane ta ry Orbits

The path of the Earth around our Sun traces cu t an ellipse with the sun at one of the

foci. Kepler's Second Law of Planetary Motion states tha t equal areas are sivept out

in equal time periods. That is for the Earth to travel from a l to 61 dong its orbit

is the same amount of time as when it travels from a2 to 6-2 where the two sectors

Al and A2 have the saine area (see figure 2.4). The connection to the isoperimetric

problem has been displayed hy L. LVallen in [3]. Wallen has shown that if we travel

along the isoperimetrix ( t hat shape w hich solves the given isoperimet ric problem)

with a constant Minkowski speed (i.e. speed measured with a Minkowski metric).

then eyual areas will be swept out in ecpal times.

2.3 The Isoperimetrix

The purpose of this section is to formalize the concept of the isoperirnetris as it

pert ains to following discussions.

Figure 2.4: Kepler's Second Law

Definition 15 (The Isoperimetrix) Let P be the farnily of d l isopen'rnetric prob-

[ems. Then for P e P, the solution to P is the isoperimetrix Zp.

It should be ciear that for P on E2. we have that Zp is a circle. In general. 1

must be a conves body Otherwise its boundary could be pushed out to contain more

area or volume. Kote that the s i x of Z relates only to the particular statement of the

problem and its shnpe is a result of the space within which the problem was stated.

The spaces which are of particular interest to this work are M' and M~ so I shall

concentrate the study of Z to these cases.

2.3.1 The Isoperimetrix on M~

It should be noted that M~ is not a pre-defined finite-dimensional vector space? it's

particular definition depends on what the space X and norm II - II are defined to

be. In fact the familiar Euclidean space, E ~ , is a two-dimensional Minkowski space

characterized by IR2 equipped with the norm II - I l E . Consider the more unusual

Figure '2.5: The unit circle in the Manhattan norm

twvo-dimensional Minkowski space, (Et2, II - II). where II I I is defined such that if i? =

(x2? y*)-(xli gl) then 11Ü11 = 31x2-x1 If21 y*-yi 1. This norrn arises From what is known

as the Manhattan rnetric (with scaling factors) [ 3 ] . In this setting. the unit circle is a I diamond shape, as shown in figure 2.5, with vertices a t ($.O), (0. f ) . (-5.0)- (0. -?).

What cornes next is to solve the isoperimetric problem on this space. S~ippose

we clesire a closed curve C that has length L and encloses a masimum area. We

may assume, as previously discussed, that C is convex and suppose that C is not a

rectangle. Xow. enclose C with the smallest rectangle possible with two sides of the

rectangle parallel to each axis. as shown in figure 2.6. Let â be the point of intersection

of C and the bot tom of the rectangle which is closest to b (sec also figure 2.6) and let

F be the point of intersection of C and the right side of the rectangle which is closest

to b. If the point i? has coordinates (s, y) and the point Chas coordinates (s f t y') then

b is located at ( t, 9'). Notice that on t his space we have

Figure 2.6: The convex curve C

and also that

In fact. given this norm. the length of any vector can be measiired by siimming

the vertical and horizontal components. If a denotes the arc of C from to C then a

lies in the triangle whose corners are â,C?ë. This is due to both the construction of

the rectangle, which does not allow O to be below the line containing b- ii nor to the

right of the line containing c'- b. and to the convexity of C. mhich does not allow a

to be above the line containing C - Ü. Now. due to the variant of the squeeze t heorem

for convex curves, we have t hat

A similar argument may be used for each of the remaining arcs which lie inside the

rectangle, and thus a coovex curve which is not a rectangle has the same perimeter

as the rectangle which circumscribes it but with smaller area. Thus Z will be found

among the family of rectangles wit h some lengt h 1 and widt h ,W. Xotice t hat the sides

of Z are perpendicular to vertex vectors of this unit circle M.

We have fixed the length of the solution curve to be C. What remains is to

maximize A = fiu with the constraint 2(3f + 2w) = L. Notice that 241w = ( 3 f + y m ) 2 - (31 - 2w)2 and so = - -

4 (31 - 2 ~ ) ~ . Therefore A is maximized exactly

when 7 = 2 and the isoperimetrix, for this situation. is found.

The above example, wit h a small generalization on the norm. appears in the paper

by Wallen [:Il. From this point. Wallen moves on to find Z when the unit circle is

given by a Zn-sided polygon and then to the general case when the unit circle is given

by any closed convex curve in two dimensions.

The follocving are a iew important results concerning I that should be noted for

the t wo-dimensional case:

The isoperimetrix is a closed convex curve 161.

The isoperimetrix is symmetric about a central point [6].

The isoperimetrix is uniquely defined for each isoperimet ric problem in two

dimensions up to a translation [2].

The unit isoperimetrix, which 1 shall denote with Io- centered at the origin

is exactly a 90" counterclockwise rotation of the polar body of the unit circle

which has been scaled so that its lengt h is equal to twice its area [SI.

The particular choice of the definition of area does not affect the shape of Z, in

two dimensions but the scale is area dependent.

Figure 2.1: A scalar multiple of the isoperimetris

The final point in the above list arises in general for the samc rcason t hat thc sidcs

of the isoperimetrix in the Manhattan norm were perpendicular to vertex vectors of

the unit circle. A general discussion on U0 (refered to as the dual bail) may be found

in the book by Thompson [2].

Example 3 I f the bail B is giuen as i n figure 1.3 then Bo, rotated thro,ugh 90' gives

us, up to a scnh factor. the isoperimetriz shown in figure 2.7.

2.3.2 The Isoperimetrix on M3

The isoperimetrix in three-dimensional Minkowski space is a much more elusive object

than in two dimensions. T h e methods for the stuciy of 1 in Mn for n 2 2 were

pioneered by H. Busemann and require many of the concepts of graduate-level courses.

As such. the technical details required for the discussion of this subject are beyond

the scope of this work. Kowever, 1 shall endeavor t o explain the concepts involved

and refer the interested reader to chapter 5 of the recent compilation of Minkowski

geornetry by A. C. Thompson [-] for the specific details.

One of the most distinguishing differences between Z in M~ and Z in M~ is t hat in

M ~ , and in higher dimensions, the shape of an isoperimetrix depends on the definition

of area and is therefore not unique. There are two particular paths that lead from

the unit bal1 to the isoperimetrix that 1 shall discuss here- The first follorvs from the

Busernann definition of area and the second from t h e Holmes-Thompson definition.

In a Minkowski space with norm II - IlLbf, distance is a translation invariant quantit-

and so for vectors Z, Y translated by a we have

Thus it is reasonable to accept as an axiom that areas (or in general n-volumes)

also are translation invariant. Two of the consequences of our acceptance of this

axiom are

Given a (n-i)-dimensional hyperplane H which does not contain the origin and a

closed set S in HI the (n-1)-volume of S is equal to the (n-1)-volume of S' = TS

where T is the translation which translates H to the origin.

T h e (n-1)-volume rneasure of any fixed set in a hyperplane which passes tlirough

the origin is Haar measure.

The significance of a Haar measure is that given any two (n-1)-volume measures

axising from different geometries. Say X and p , then they will be scalar multiples of

each other.

- -

Figure 2.8: The plane H translated to H f

The follorving description will be centered on M3 but applies ecpally well to Mn

for n 2 2.

In order to measure the area of a setn in M3. i.e. a closed. compact 2-

dimensional region I ' embedded in some plane in M3. we must consider the following

Issues:

Let H be the plane (H for hyperplane in higher dimensions) that Ii lies in and

let H' be the translation of H to the origin O (see figure 2.8). The particular man-

ifestation of M ~ . that we assume has been predefined by some norrn. will inherit a

particular unit bal1 B.

Let IL be the area measure on H f corresponding to 24 = B n H f . Also let X be the

Euclidean area measure on H' corresponding to the circle C formed by H f intersected

with the Euclidean sphere. Note that X is not required to be Euclidean but because it

is so familiar and since we may choose any other Haar measure we like. it is convenient

for the purpose of calculation to do so. At this point we have the following:

:3. B 17 H f is a unit circle U

and because p and A are both Haâr measures we have

and t hus

.\(Kt) Notice that will be invariant with respect to the choice of area measure.

However. if we malce a subtle shift of our perspective and consider

then our choice of X becomes important.

The number - is the scaling factor For the area of Ii' (and t hus ii). This

representation for the scaling factor is also rather cumbersome so we do the folloiving:

Define H as the kernal of sorne lincar functionul udh unil ~ O I - I I L in llie du01 spuce h131

(see [IO]). That is H =

One desirable feature is

a to match up with the

{x : f (x) = O} and 11 f I l M 3 * = 1. NOIV we may define

to have the area of the unit circle in M3 to be defined to be

euclidean case so

Equation 2-17 is the Busemnnn definition of area in a kf inkowski space.

Figure 2.9: The support function

The Holmes-Thompson (H-T) definition follows along the same path as above up

to and including 2.16. The H-T definition makes use of a quantity known as the

uolume produet of the Blaschke-Santal6 inequality [SI. This product mil1 be defined as

the Euclidean measure of our set I< times the Euclidean measiire of its image in the

dual space. Ka. and will be denoted X(lijX(lic j. The Busemann definition chooses

to let p(U) = r but it has been shown to be very advantageous in m a n - sitiiations

[3] to let

PWI = wWw0 )

- L

and then we still have that - = o( f ) but it now takes the forrn

and in either case, O( f ) is exactly the support JÙnction of the isoperimetrix.

A support function. h. is defined as the function which takes as input unit vectors'

u. which radiate from the center of some convex body and return the distance from

the center of the body to the tangent hyperplane t ( u ) that is orthogonal to u (see

figure 3.9).

Chapter 3

Two-Dimensional D ifferent ial

Minkowski Geometry

3.1 Introduction

This chapter deals with the differential geometry of the space M' = (R'. II 1 1 ) . For

notational ease. 1 will remove the subscript &I from the norm. .As usual. II - II will

have al1 the properties of a norm and the distance between two points x. y E M' r d 1

be given by d(.t. y) = Ilx - 1/11.

In the thesis of Biberstein [6 ] . t h e Busemann definitioii ol area Ilas been eriipluyrd

as a starting point to describe angles and areas. Under the Busemann definition ive

have that the bal1 (unit circle in M ~ ) has an area of I; (by definition to agree with

the Euclidean case) and circumference that is bounded below by 6 (if the ball is an

affine regular hexagon) and above by 8 (if the ball is a parallelogram) depending on

the particular choice of ball. This last statement is a result of Golab's theorem [?].

Also under the Busemann definition, the isoperimetrix has an area that is equal to

the normalized volume product and a circuniference eqiial to twice that area (see

table 3.1).

One of the advantages, in terms of this paper, of the H-T definition of area is that

the isoperirnetrix is irnbued with more workable properties as given in table 3.2

Area A

Circumference C

Unit Circle U

&( := K

Isoperimet rix Io

Table :3.1: The Busemann unit circle and isoperimetrix

I

Area A

Isoperimet rix Io i Unit Circle U

Ai = .\(L() .\(UO) -

Circumference C

Table 3.2: The Holmes-Thompson unit circle and isoperirnet ris

6 5 CL( 5 S

The following is a discussion of both the work of Biberstein using the Busemann

definition and relating areas and angles to the ball, and a parallel discussion on how

the H-T definition relates t hese propert ies t o the isoperimet rix.

Let U be a closed conver curve centered at the origin O such that for an- point on

the boundary of U say x E 2 4 we have d ( x l 0) = 1. Then, as stated in the previous

chapter, the isoperimetrix I, is given by a multiple of RU0 where R is a 90° rotation

and U0 is the polar body. or dual. of U. The precise multiple depends on the definition

of area used. For a detailed description of the dual ball see (-1. Let ü. .ü E U be two different unit vectors on the unit circle. The angle between ü

and ü is twice the a-rea (in the sense of table 3.1) of tha t sector of U given by (x. y. O )

and denoted by o. Let 5. f E Z, be two different vectors on the isoperimetrix. The angle between 5

and y is both twice the area (in the sense of 3.2) of the sector (x. y. 0 ) and the arc

Iength of Z, from S to and denoted by 0 (see fig. 3.1).

Note that the above two points make use of different definitions of area (and

therefore different definitions of angle). This situation will continue in the work that

follows. The Busemann definition will be used for objects relating to the circle and the

Holmes-Thompson definition for objects relating to the isoperimetrix. The difference

should be clear €rom the context but I will continue to italicize the xord &arean to

remind the reader that care must be taken to keep the two defi nitions distinct.

Each definition of nrea used will give rise to a nem set of isoperimetrices. but

in ?-dimensions. an isoperimetrix which solves the isoperimetric problem under one

definition will be exact ly a scalar multiple of an isoperimetrix which solves the same

problem under another definition. The shape will be the same. Therefore the reader

is advised t hat Io and 1 are also definition dependent up to a clilation.

We must also define an axis in the plane as a frame of ceference to measure angles

against. Let this avis be the line which contains GO).

Definition 16 (Amplitude of unit vectors) The amplitude o / a unit uector =

G(d) is de f i ed to be twiee the a r a of the sector (&(O), G(c$)? O ) on U.

A t C lengch = chcta

Figure 3.1: The relationship of area to arc length of the isoperimetrix

Definition 17 (Amplitude of vectors on the isoperimetrix) The amplitude of O O

a uector t2 = t Z ( B ) is de jned to be the arc kngth of 1, from g(0) to & ( O ) (or equicn-

lentiy tmice Ihe aren of the sector (&(O). g(9) . 0 ) on 1,).

From now on, following t h e notation ahove. unit vertors and vectors on t h e

isoperimetrix shall be distinguished by the subscript labei (1.2) and t h e angle la- i.

bel (d. O). For example. = G ( Q ) will represent a vector on U and t 2 = fZ(8) will

represent a vector on Z,.

The unit circle ma,y be defined by al1 vectors Tl = & (4) radiating from O to 324

and represented by the function G(q5) where O 5 6 < 2ir

Similarly, t h e isoperimetrix may be defined by all vectors t2 = r z (0 ) radiating from

O to Z, and represented by the function g(6) where O 5 0 < ?a

The vector Z1 is defined by taking the derivative of < with respect to d for 6 E U and the vector Z2 is defined by taking the derivative of with respect to O in the

case F2 E Z,.

and t hus

Biberstein has shown. in the same way that 3-14 below is deriveci. that for the

unit circle, parameterized by area as above, the lollowing result holds.

For al1 vectors Zl[P) = Q i ) we have

det(fi(q5). ~ ~ ( 4 ) ) = 1.

Upon differentiating 3.5 with respect to o we get

det(f:(b). nl(6 ) ) + de t (G(*) . f i :(*)) = 0:

by the definition of the determinant and 3.3 we have

Thus 3.6 gives us

k t (G(4) : ?i'(d)) = d e t ( G ( 4 ) : ~ : ( q j ) ) = O (3.8)

and t hus we must have t hat fi; ( 4 ) (or equivalently C(q5)) lies dong the same line in

space as the vector Ti ( P ) .

We now wish t o show that similar results may be obtained for the isoperimetrix.

Let C = &) E c3 be a curve parameterized by an arbitra- parameter p. We have

that the magnitude of a tangent vector F= C'(p) to the curve is given by

where s represents arc Length. Thus for curves which are parameterized by arc length

we have that

That is. al1 of the velocity vectors of the curve have unit length. Because of this. it

is very convenient to use arc length to parameterize the boundary of the isoperirnetrix.

Let & = g(0) be some vector on 1.. We have from above that

has unit length. This is one of the key results of the chapter. [t gives us that the

derivative of vectors on Z, are vectors on U. Similarl . when ive start with vectors on

L1 and take the derivative. ive get vectors on 1,.

Consider the triangle T forrned by the vectors & O ) . r2(0 + 2.0) and g(0 + 1 0 ) - - t2 (8 j. Recali that the determinant measures the nren of the paralleIogram spanned

by two vectors and so

Following the usual limiting process for a derivative we have

and thus for al1 vectors Z2(8) = C ( B ) we have

Differentiating 3.14 with respect to 6 gives

det(i>,(O), i lz(@)) + det(&(Q 6;(0) ) = O

but. by definition we have

d e t ( i > , ( ~ ) . g 2 ( 6 ) ) = det(&(O), ?i2(e)) = o. Again. 3-15 yields

& ~ ( G ( O ) , c(0)) = d e t ( G ( e ) _ % ( O ) ) = 0. (3 .11)

and thus we rnust have that i?;(B) (or equivalently $ ( O ) ) lies along the same iine in

space as the vector & ( O ) .

Therefore. for both cases. the vector Z' is simply a scalar multiple of t which is

dependent on the parameter ( O or 4) . It will be convenient to represent 6' as in 3.18

and 3.19 in order to avoid confusion when discussing matters of curvature t hat are to

corne:

where 7[(6) > O is the proper scaling factor depending on o. Vo E [O. ' LT) .

Atso.

where -i2(8) > O is the proper scaling factor depending on 6. VO E [O.l?a).

-. Definition 18 (Norrnality) Consider two lines, A and B in M'. Let t' = t (6) be

a unit vector in the direction of A. The line A is said to be normal t o B i f B is in -C Ihe direction of the vector < = nl .

Definition 19 (Transversaiity) Consider two fines, A and B in M? Let & =

g(6) be a uector on the isoperimetrir i n the direction of A. The line A is said to be

transversal to B if B is in the direction of the vector = fi2.

There is also sorne notation that accompanies the ideas of transversality and

norrnality as follows. Consider a unit vector Cl. LVe Say ClL is the unit vector which

satisfies: Cl is normal to CF. For a vector Ü2 on the i~operirnetrix~ we Say Ü$ is the

unit vector which satisfies: üz is transversal to ÜZT. In the work of Busemann- who

was the pioneer. a second and confusing meaning was given to the terms -normal7

and 'transversal''. Therefore care must be taken, when reading papers on this topic.

how these definitions are stated. The following result will be useful to understand

how the confusion may arise.

Theorem 6 The vector U is normal to ü $and only i f .5 is transversal to Ü.

Proof Let Cl = ilI(@) E U and let ,Cl be the unit tangent vector to U a t Cl. Then

U1 is normal to Ül. Consider H. the tangent line containing Ül. There exists a linear

functional f E Li0 which supports U at Cl_ i.e. H = {x : f ( x ) = 1}. Now. the unit

(w.r.t UO) tangent vector to U0 a t f is some g and lies in the tangent line H'. By

duality, CI plays the same role in the dual space as f did in the ambient space and

therefore H' = { h : h(ûi ) = 1 ) . FVe know that U0 rotated 90' is an isoperimetris

Z. Xlso under this rotation. R1 f is in the direction of Cl, and g is in the direction

of Cl. Therefore since R f is transversal to Rg7 we have that FI is transversal to

Moreover each step is reversible. O

Definition 20 (Smoothness) '4 parameterized cun7e C is said to be smooth if it

has a continuous jrst deriuatiue at each point.

Definition 21 (Strict convexity) A concex set is said to be strictly convex i/

there are no line segments in its boundanj.

The following discussion requires that certain restrictions be placed on the unit

circle and the isoperimetrix. We shall restrict the class of unit circles and isoperi-

metrices to those that are smooth so that the curve traced out by a tangent vector

travelling about LI or 2, will also be differentiable everywhere. It is not necessary

t o stipulate the strict convexity of these objects because of the following two results

(which both hold in a more general setting than displayed here).

Theorem 7 Giuea a conuez set S with a srnooth boundary in some ~Cfiakowskï space

Mn , the polar body S0 E (Mn)' is stRctly convex.

Theorem 8 ((Cf)")" = U. (see afso 1.7)

Thus a smooth unit circle ensure a strictly convex isoperimetrix and a smooth isoperirnet ri

ensures a strict ly convex unit circle.

These condit ions give rise to the follorving properties:

For each vector <(ci) of U there is a unique line which fi(4) is normal to.

For each vector <(Q) of U there is a unique line which is normal to <(b) .

For each vector g ( 6 ) of Z. there is a unique line which is transversal to g(0).

0 For each vector r2 (6 ) of Io there is a unique line which g(6) is transversal to.

0 Let rl be a point on the line A in M ~ . Then for any vector which is normal to

A and with ends x1 and s2: the shortest distance from x h o A is exactly the

Length of that vector (see example 4).

a Let Cl. 21, and be unit vectors in M~ such that L;I is normal to Cl. Then we

have that for al1 such vectors 6

a Let S2? $2 and t2 be vectors on Z, in M' such that f2 is transversal to .F2. Then

we have tha t for al1 such vectors t:L

(see esample .5)

Figure 3.2: The minimum distance from a line to a point

Example 4 Let A be a fine in M* and z2 E M2 be some point not on -4. To jind

the shortest distance between A and r2 we trmslate U by 12 and dilate until A is

tangent to it. Let the point of tangency be 11 and denote the uector whose ends are

11 and x2 b y El. By this construction. El is normal to A and its lengih measures the

shorledt distance frur~r r 2 io A (set also fyarr 3.2).

Exarnple 5 Let .F2 ~z EZ, be Jxed and consider the area of the parnllelogram measured

69 Irlet(Zzl QI for al1 E Io. This area is a maximum when tue choose a 6 such thai

6 is transversal to &. Figure 3.9 displays three choices of & (namehj f21. f22 and

&) and the parallelograms spnnned by Z2 and &. The height. and therefore the nrea.

of the parallelogram is rnazimized when Ihe vector i2, ~ l ~ h i c h is trnnswrsal to .S2, is

used.

Note that example 4 may only be stated in terms of t he unit circle since this is

the object we use to measure distances. However, example 3 may be s ta ted either for

the unit circle or for the isoperimetrix.

Figure 3 . 3 : Maximizing the determinant on Z,

The relations displayed in 3.20 and 3-21 will also be given a special representation

of the form

Note that the functions aLc(Gl) only operates on unit vectors and so we shall

extend the definition by requiring that arr(cùt) = arc(Ûl) for al1 c E W\O.

Sirnilarly for ~ . ~ , ( 5 ~ ) we shall extend the definition by requiring that for & 6 1,: a&) = c q a ( L ) where (1 I I T o is tha t norm for which Io is the unit circle.

Ilt2 Ill0

The last new function to be introduced in this section will be the function 6.

This function will also have both a unit circle and an isoperimetrix version denoted

by blI and 6=,, respectively. First ive will discuss br,. This function takes a unit 4

vector t l = &($) and the unit vector that Tl is normal to? given by r: = e ( b ) .

and returns the value of t h e determinant of the two vectors (equivalently t h e area of

t h e parallelogram in t h e usual geornetnc interpretation of a determinant). This real

number will be non-negative as a result of the positive orientation of with respect

-1 - 6&(4)) = 644) = d e t ( t , . t l ) > 0.

Combining t he equations 3.20. 3.22 and 3.24 we obtain

bVe also note tha t is a unit vector in the direction of fi1 and as a result of 3-24

we have t hat

G(4) = &r(4)fll(o) (:326)

and thus? upon clifferentiation with respect to Q and using 3-18 and 3-26

We shall also denote the amplitude ?:(O) for sorne o by o ' = O l ( o ) and arrive

at the identity

T h e isoperimetrix version. hrO, has a major simplification d u e to [-] and t h e way

we have defined transversality. Recall tha t given a vector o n the isoperirnetrix. its

transversal is a unit vector in the direction of Z2 and consider t h e following.

Let P be the parallelogram formed by the span of I and y. If we again let A be

the Euclidean area rneasure then the Euclidean area of P is given by

.52

where sin(Z? tj) is the sine of the angle between S and f. Now there is also Minkowski

sine function which @es us a relationship

Definition 22 (Minkowski sine) Let 5

dimensional Minkou~ski space M'. Then

b~ sm(Z? y3 =

equivalent to 3-29? but first t h e definition.

and Y be two non-zero uectors in some 2-

the Minkowski sine srn(Z .3 is defined

where f is a linear functional in ( M ~ ) - such that f (2) = O.

This definition is defined so t hat if p is the Minkowski area measure Ive have t hat

Thus. if Q is the parallelogram spanned by & and GT we have

Xote tha t we may also choose f such tha t f (g) = 1 because of the following

argument. Let h be any linear functional for which h ( I ) = O for some vector S. Then

for any other vwtor ij # 5 that lies on t h e line -4 t l~finetl by h will he riich that

h ( f ) = O. Now we may translate il and have for al1 vectors t o n this translated line

that h ( t ) = c for some constant c E R. Finally we can let g = f h and have g(t ) = 1

(see figure 3.4).

Since & E I,, we have that f (6) = Il f Illo = 1 and so

p(Q) = 1 = brO(û), V6 E [O. 2 ~ ) . (:3.33)

Also. as we saw in 3.11, $ ( O ) = & ( O ) is a unit vectar. Therefore. corresponding

to 3.26, we have

h (XI = O

Y

Figure 3.4: The linear functional h

and so

3.2 CurveLength

R e d that for a class C' c u v e t C = Q p ) ? we define the -elernent of arc length' by

the relationship

and reparameterization by arc length of t h e curve gives us t hat 1 1 = 1.

Now for two points on the curve S = C(p l ) and y' = where pl < p,: t h e

length L of t h e arc of C betiveen 5 and i j is given by

- 1 Thus for an arc c of the unit circle we utilize 3.26 by letting Il?, ( 4 ) 11 = 9 - and t hus

giving us that the length of the unit circle is

Similarl- the length of the isoperimetrix is given by

2 7 1

J16r.0 = lZr I ~ B = 2a (13.40)

and this result for the isoperimetrix agrees with the result mhich arises from the

Holmes-Thompson definition (see table 3.2)

3.3 Curvature in the Minkowski Plane with re-

spect to U

Consider the class C3 curve C = C ( s ) parameter id by arc length S. Let Tl(o) = -n;

be a unit tangent vector of C and let Gl ( b ) = < (4). We say the amplitude of fi is the

same as its amplitude would be if it were radiating from the origin to the boundary

of U.

The Frenet representation of the curve C with respect to U shall be denoted

(Ct <, Cl), to ernphasize that at each point of the curve ive must be aware of the

vectors Tl and ill (see figure 3.5).

W e define the quantity given by

The no-1 veccor

The unit tangenc vector

Figure 3.5: The Frenet representation of the curve C with respect to the unit circle

to be the itfinkowski curvatvre of the curve wi th respect to U. and using t h e chain

rule we immediately obtain the following relations:

and

Thus the Minkowski representation of the Frenet formulas with respect to U are

and

We shall also need to define a quantity, which does not arise in the Euclidean

discussion of differential geometry but is necessary in this setting. that we shall cal1

the anti-curvature. where

Whether the curve is given by an arbitrary parameter p or parameterized by

arc length s, the curvature and anti-curvature may be given by explicit representa-

tions. First we will consider kl for C = &I). We begin by combining equations

det(G(4) . f i l (@)) = 1 (3 .5) and % = klRl (3.45) to get

det (g: 2) = k1

and by the chain rule we have

and so

Next we apply the chain rule again to 3.49

and so kl becomes

Finally? we apply t he norm to 3.49

and obtain

d C d 2 e

ALI = <let (z- dp2) (:3.-51)

II $11" -.

CVe may. in à much longer process. derive a formula for kl when C = C(p). The - +

process is to begin with det(G(d)? ~ ~ ( 4 ) = 1 and 9 = -kltl . Upon repeated

application of the chain ride and the appropriate substitutions we obtain

The process for representing LI and kl for C = C ( s ) again follows by observing d e tha t if p = S. then 11 zll = 1 and we get

I I - , = det (5; - - 2) 7

and

Clearly it is possible to have a curve with kl = O at some points, in which case

equation 3.60 is not defined. To avoid this problem we add the stipulation that

wheoever kl = O t hen il = 0.

However cumbersome equations 3.57 and 3-58 appear to be, they yield sorne in-

teresting and useful results when applied to the unit circle. For this special case. let

K I and il represeot the curvature and anti-curvature of U respectively and denote

al1 derivatives with respect to p with a prime rather thao in the notation of Liebniz.

Then for any & = G ( Q ) on U we have that

and t hus

6; = 71~1. (z3.6'3)

Xow, using 3.58 for 6 = G ( $ ) on U and again denoting derivatives with respect

to p with a prime we have

which may be manipulated using 3.3. 3.18 and 3.26 into

3.4 Curvature in the Minkowski Plane with re-

spect to 1,

Consider again the class c3 curve C = C(S) parameterized by arc length S. Let

The tangenc vtctor

Figure 3.6: The Frenet representation of the curve C: with respect to the isoperimetrix

be a tangent vector of C which lies on Z, and let % ( O ) = $ ( O ) . The above definition

gives us t hat

The Frenef r e p r ~ s ~ n t a t i o n of the curve C with respect to Io shall be denoted

(Cl 6, to emphasize, as before? that at each point of the curve we must be amare

of the vectors 6 and Z2 (see figure 3.6).

We define the quantity given by

to be the ~Clinkowski cuniature of the curve with respect to 1,: and again using t h e

chain rule obtain:

Thus the Minkowski representation of the Frenet formulas wit h respect to Z, are

dt; - = k2Z2. cls

The anti-cuntature with respect to Z,. k2. is given by

We also have explicit representations of k2 and depending on whether the curve

C is given by an arbitra- parameter p or parameterized by arc length S. 1 will again

show the derivation for the arbitrarily pararneterized ctirve case. C = C ( p ) . \Ve have

that det(Z2: 6 ) = 1 and using 3.71

By the chain rule we have

and so

Next we apply the chain rule again to 9-75, substitute the result into 3.76 and replacing

al1 deterrninants which contain 2 in both places with a zero we have that k2 becornes

Finally, we apply the isoperirnetric norm to 3-75

rearranging we have

and t hus

When p = S . 3.83 becomes

k2 = det (::y - - 5) The formulas for k2 are similarly obtained.

Xoiv we will investigate the curvature of Io using 3.84

and so

as we would expect from 3-73.

Studies of the Isoperimetrix under the Buse-

mann Definition

There is an alternate- but eqtiivalent. method of defining an i s o p e r i m e t r i ~ ~ used by

Biberstein (61, t hat does not require explicit mention of t h e dual space. Recall that

one feature of the unit circle tha t we desired was that the curve traced out bu a

tangent vector travelling d o n g U be a smooth closed curve. In fact this curve is the

isoperimet rix [:3].

The representation of a n isoperimetrix 1 given by Biberstein [6] is

T = z-pn,(+) (337)

where à is the point in the center of I. ,8 # O is the scaling factor mith respect to Io

and (0) are the normal vectors on the unit circle. Thus 1, is given -

Note that Biberstein refers to Z as an anti-circle and reserves the word isope rimetriz

for that special solution to the isoperimetric problem which has its area equal to half

of its arc length and centered a t t h e origin. However, each 1 is a solution to some

isoperimetric problem and so 1 shall also refer to them as isoperirnetrices and reserve

Io for that special case.

The following theorem is due to Busemann

Theorem 9 Let ill be a unit uector and C be a scalar 60th depending on a parameter 4 svch that % and 5 ezist. Also l e t Ü be sonze jhed point. Then the eume C = a +CÜI

is an arc of sorne isoperimetrix centered at ü if and onlg if is no~rmal to Cl(*).

Proof: The forward 'if" direction of the proof follows directly from the definitions

of an isoperimetrix? of normality and theorem 6. In the other direction? we have that

since û1(q5) is an arbitra- unit vector we may choose t o relabel it as -?(4) so that

C = Z - C ~ . Now, making use of 3-27 upon differentiation with respect to O we have

is normal to iTl(4), w have tliat since % is only in the and assiirning that 7

direction of &

Cg + C'du = O. dir

The above equation is true when

where ,3 is a constant. Substituting 3.91 into :3.89 ancl using 3.26 we have

and thus C is an isoperimetrix.

3.5.1 Arcs, Curvature, Length and Area of an Isoperimetrix

Consider an isoperimetrix given by 1 = a- D)Ci(d). Differentiation with respect to q

and using 3-18 gives

and so

T herefore 3 -93 gives

and so

Thus for a unit circle which has non-zero positive curvature. we have that the

isoperimetrices also have kl > 0.

Note tha t for ,8 = 1 t h e geornetric interpretation of ~ ~ ~ ( 4 ) is t ha t it is the curvature

of So a t the point where t h e vector whose amplitude is Q is tangent to Io. As a direct

consequence. the anti-curvature of a smooth curve C a t some point P E C is equal to

the curvature of C at p divided by the curvatiire of Io at the point PE Io such that

the line tangent t o Io at f is parallel to the line tangent to C a t P Combining t h e definition of anti-curvature (J.47) with 3.97 we have that the anti-

ciirvature of an isoperimetrix relative t o the unit circle is

and t hus:

Theorem 10 The anlicuruature of an isoperimetriz is a constant and is qua1 to the

reciprocal of ils scaling factor (wth respect to Zo).

Let c represent arc length on Io and consider 3.96 with 3 = 1

Then for an arbitrary curve parameterized by arc length C = C ( s ) we have that

- k, 1 dd 1 dQd< d< k l = - = -- = --- = - 71 71 ds 11 d< ds ds'

(3.100)

Geometrically. the anti-curvature of a n arbitrary curve is the ratio of the element

of arc length of the isoperirnetrix to the element of arc Length of the curve.

Next we will consider the length of an isoperimetrix LT which we may obtain

directly frorn 3-96 by integration.

and t hus

The area of a n isoperimetrix Al is also easily obtained and is given h -

-1 cornparison of 3.101 and 3.103 immediate!~ gires us t ha t

To complete this section. 1 shall present the following two theorerns and a corollary.

Theorem 11 (Existance and Uniqueness 1) Let kI = kl ( s ) be an arbitranj con-

tinuous funclion. s~ an arbitrary number in dorn(lil). Co an arbitrary point. and &, an a rb i t raq u n i t uector. There ezists a unique cume C = C ( s ) such that the follooing

hold

2. the unit tangent cector (g) = Co, and s=so

3. kl ( s ) is the eumature

Proof Let represent the

of C at the point &).

amplitude of ilo. We have

Letting Ül(s) = <(d(s)) , we have

4 4

Ù i ( s a ) = Z I ( ( ~ ( S ~ ) ) = LI(&) = 41

and so C is given by

C' = Co + /' ül(s)ds so

and C is t he desired curve. a

Theorern 12 (Existence and Uniqueness I I ) Let il = bl(s) be an urbitrary con- -.

tinuous junction. s o an nrbitrary nurnber in dom(&). Co an arbitraiy point. and Cl

an arbitrary unit tiector. There erists a unique cume C = c(s) such that the folloiuing

ho id

2. the unit tangent vector (g) = Cl, and s=so

3. k l ( s ) is the anti-cuntahrr ofC at the point C ( s ) .

Thas L l ( s ) uniquefy determines C (up to a rigid motion) .

Proof Let < represent arc length of I, and let be the point on Io such that the

unit tangent vector to lo at is equal to ill. W e have

Then Zl(s) is the unit tangent vector of Io a t ~ ( s ) and so ül(so) = &. The curve is

then given by

and C is the desired curve. fl

As an immediate consequence, ive have

CoroIIary 1 A curue with constant anti-cumature is an isoperimetrir.

3.6 Studies of the Unit Circle under the Holmes-

Thompson Definition

Cinder the Holmes-Thompson definition much of the work of the preceeding section

becomes unnecessary in terms of describing the isoperimetrix. However. this gain is

balanced by a loss in terms of describing the unit circle as summarized in table 3.2.

In the previous section we used the nonnal uectors nt of the unit circle to define

an isoperimetris and in particular Io = -ni (6) . This gives us that

II6111, = 1- (:3.110)

Similady. we have tha t for the normal vectors on the isoperimetrix. 3.1 1 grives us

and so we may. given a smooth Zoo define circles C in terms of n o n a l ceciors of Io.

This gives thêt the circle. centered at b with radius r ) is

C = b- & ( O ) ?

and the unit circle will be given by

Theorem 13 Let Ü2 be such that IlÜ211ro = 1 and let $ be a scalar both depending

on a parameter 0 such that and $$ ezist. rllso let b be some jixed point. Then - d e ( 0 ) the c u m e C = b + tbÜ2 is an arc of some circle centered at b if and only if 7 is

iransversal to E2 ( O ) .

Proof: T h e forward direction of this proof also follows direct13 but this tirne frorn

the definitions of a circle. of transversality and theorem 6. In the other direction. ive

have that since iL(0) is an arbitrary vector of unit length with respect to Z, we may -<(el choose to relabel it as - -

I1~2llz, - cr(Z2)G2 = C2a so that

and so

dz72(0) and using the hypothesis that 7 is transversal to il2. ive have t hat is only in

the direction of and thus

i I - OC' - a'p = - ( ~ l i * ) ~ = o.

The above equation is triie when

a+ = q

where 7 is a constant. Substituting 3.118 into 3.114 we have

and thus C is a circle.

As a result of equation 5.85 we obtain the following result.

Theorem 14 The curvature of an isoperirnetrix is a constant and is equnl to the

reciprocal of ifs scaling factor (uith respect to Z,).

3.6.1 Arcs, Curvature, Length and Area of a Circle

Consider the circle given by C = 6 - 7G2(0). Differentiation with respect to B and

using 3-19 gives

This extra factor of Ilgll in equation 3.120 is a problern when determining the

lengt h of a circle. but from Golab's theorem, we are already given that the length

of the circle is not fixed. LVe may however. still obtain the following existence and

uniqueness results corresponding t o those of the previous section.

Theorem 15 (Existance and Uniqueness III) Let k2 = k212(s) be an arbitrary

continuous function. so an arbitrary number in dorn(k2), Co an n r b i h n j point. and

iio an arbitraq unit ceetor with respect to Z.. There ezists n unique cwve C = C(s)

such that the following hofd

4 -. 1. C ( s o ) = CO?

. k2(8) is the curvatnre of C at the point C(s).

As a n immediate consequence. we have

Corollary 2 A curue with constant cvrvature is an isoperirnetrix.

Theorem 16 (Existence and Uniqueness IV) Let i2 = k2(s) be an arbitrary

continuous f u ~ c t i o n ~ so an arbitrary number in Co an nrbitrary point, and

ü2 an arbitrary unit vector luith respect to Io. There erists a unique cume C = Ç ( s )

such that the following hold

($1 2. khe unit tangent uector = Ü2, and

3. k2(s) is the anti-cumature of C at the point e ( s ) .

Then G2(s) uniquely de tennines C ( u p to a rigid motion) .

The proofs of these two theorerns follow in the same way as their counterparts

above.

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