Generalized Curvatures Part 2 pptx

Chapter 2

Motivation: Curves

The length and the curvature of a smooth space curve, the area of a smooth surface

and its Gauss and mean curvatures, and the volume and the intrinsic (resp., extrinsic)

curvatures of a Riemannian submanifold are classical geometric invariants. If one

knows a parametrization of the curve (resp., the surface, resp., the submanifold),

these geometric invariants can be directly evaluated. If such parametrizations are

not given, one may approximate these invariants by approaching the curve (resp.,

the surface, resp., the submanifold), by suitable discrete objects, on which simple

evaluations of these invariants can be done. Our goal is to investigate a framework

in which a geometric theory of both smooth and discrete objects is simultaneously

possible. To motivate this work, we begin with two simple examples: the length and

curvature of a curve.

2.1 The Length of a Curve

This book deals essentially with curves, surfaces, and submanifolds of the Euclidean

space EN endowed with its classical scalar product < .,. >.

2.1.1 The Length of a Segment and a Polygon

If p and q are two points of EN, the length of the segment pq is the norm of the

vector −pq→, i.e., the real number

|pq| =
< −→pq,

−→pq >.

If P is a polygon, given by a (finite ordered) sequence of points v1,...,vn in EN, the

length l(P) of P is the sum of the lengths of its edges, i.e.,

l(P) =

n−1

∑

i=1

|vivi+1|.

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14 2 Motivation: Curves

2.1.2 The General Definition

Let us now give the classical definition of the length of a curve, using approxima￾tions to the curve by polygons. The length of a curve (without any assumption on

regularity) is usually defined as the supremum of the lengths of all polygons in￾scribed in it: let

c : I = [a,b] → EN

be a (parametrized) curve from a segment [a,b] ⊂ R into EN. If there is no possible

confusion, we identify the image Γ of c (i.e., the support of the curve c) with c

(Fig. 2.1).

Definition 1. Let S be the set of all finite subdivisions σ = (t0,t1,...,ti,...,tn) of

[a,b], with

a = t0 < t1 < ... < ti < ... < tn = b,

and denote by l(σ) the length of the polygon c(t0)c(t1)...c(ti)...c(tn). If

sup

σ∈S

l(σ)

is finite, one says that the curve c is rectifiable and its length l(c) (or l(Γ)) is this

supremum:

l(c) = sup

σ∈S

l(σ). (2.1)

It is well known that there exist continuous curves which are not rectifiable. The

most famous example is the Von Koch curve obtained as follows: start from an equi￾lateral triangle and consider each of its edges e. Take off the middle third e1 of

e and replace it with an equilateral triangle t1. Then, take off e1. The limit of this

process gives rise to the Von Koch curve, which is continuous but with infinite length

(Fig. 2.2).

Fig. 2.1 A smooth curve and its

approximation by a polygon line c (t4)

c (t3)

c (t2) c (t1)

c (t0)