Chapter 2 Global theory of curves Flashcards
smooth
A vector funct is smooth on I if it extends to an infinitely differentiable vector function defined on open interval J containing I
e.g I=(a,b) inff diff on x
I=[a,b]-smooth funct on I extends to smooth funct on (a-e,b+e) e>0
PARAMETRISED CURVE
A PARAMETRISED CURVE is a SMOOTH vector function
REGULAR
if γ’(t) =/ 0 for all t in I
VELOCITY
γ’
SPEED
|γ’|
e.g of parametrised curves
simple circular curve
when is it regular
parametrised plane curve
γ:R to R^n
γ(t) = (rcos 2πt, rsin 2πt) r>0
regular
γ’(t) =(-2πr sin2πt, 2πr cos 2πt)
|γ’(t)| =2πr>0 never 0
e.g of parametrised curve
is it regular
γ:R to R^3
γ(t) = (t^3,t^2)
γ’(t) =(3t^2,2t)
=(0,0) for t=0
not regular
ACCELERATION
γ’’
LENGTH
L(γ) = Integral_[a,b] |γ’(t)|.dt
e.g of parametrised curves
straight line
when is it regular
give another repara
γ:R to R^n
γ(t) =a+tv a,v in R^n
regular when γ’(t) =v iff v=\0
γ~(𝜏)= a+ (tan𝜏)v
𝜏 in J=(-pi/2,pi/2) is a reparameterization of γ
φ(𝜏) = tan𝜏 regular
but
𝜏^3 wouldnt be regular
-𝜏 would be traversed in opposite direction
circular helix
is it regular?
γ(t) = (rcost,rsint,ht) r>0
regular
RPC:
for any h in R
γ’(t)=(-rsint,rcost,t)
if h=0 but periodic so never equal to 0
DEF
RE-PARAMETRISATION
of a parametrised curve
γ:I to R^n
is another parametrised curve
γ~ : J to R^n s.t
γ~= γ∘φ
where
φ:J to I is a SMOOTH SURJECTIVE funct s.t φ’(t)>0 for any t in J (strictly increasing)
RETAINS IMAGE, DIRECTION TRAVERSED, #TIMES IF CLOSED
γ~= γ∘φ
where
φ:J to I is a SMOOTH SURJECTIVE funct s.t φ’(t)>0 for any t in J (strictly increasing)J to I is a SMOOTH SURJECTIVE funct s.t φ’(t)>0 for any t in J (strictly increasing)
φ
is called the parameter transform
allows us to evaluate original PC using different input φ(t) for t in J
and is a BIJECTION
as injective
γ~= γ∘φ
used to reparametrise
consider
γ(t) = (a + t(b-a)) t in [0,1]
thus we can always define a PC on [0,1] from [a,b]
using the reparametrisation
φ’(t)>0
φ(t) = a +t(b-a)
φ(0)=a
φ(1)=b
can be used for smooth surjective funct
PROPN 2.1
which re-parametrisations of regular closed curves are RPC
EVERY re-para of an RPC is an RPC
proof:
by chain rule
γ~’(𝜏)= (γ∘φ)’(𝜏)
= φ’(𝜏) γ’(φ(𝜏) =/0
we know
φ’(𝜏)>0 for all 𝜏 in J
γ’(t) =/0 for all t in I
thus PC γ~ is
REGULAR
length of arc
Let γ : I → R
n be a PC. For a closed interval [t_0, t_1] ⊂ I the quantity
L(γ| [t_0, t_1]) =
integral_[t_0,t_1]
|γ’(t)| dt
is called the length of the arc
γ| [t_0, t_1].
arc length for repara
prop 2.2
The length of an arc does not change under a re-parametrisation
proof: prop 2.2
The length of an arc does not change under a re-parametrisation
Let ˜γ : J → R^n be a re-parametrisation of a PC γ : I → R^n, that is ˜γ = γ ◦ϕ, where ϕ : J → I is a function that satisfies the hypotheses of Definition II.3. Suppose that ϕ maps [τ_0, τ_1] → [t_0, t_1] bijectively. Then by the change of variables formula, we obtain
L( ˜γ| [τ0, τ1]) =
∫{τ_0,τ_1] |γ˜’(τ )| dτ =
∫{τ_0,τ_1] |γ’(ϕ(τ ))| ϕ’(τ )dτ =
∫_{t_0,_1] |γ’(t)| dt
= L(γ| [t_0, t_1]),
ϕ’(τ ) > 0 t_0 = ϕ(τ_0), t_1 = ϕ(τ_1).
change of vars integral fromula
∫{a,b] f(x).dx = ∫{c,d} f(g(t__g’(t).dt
g([c,d]) to [a,b]
example: considering PCS
γ_ℓ:R to R^2
γ_ℓ (t)=(cos2πℓ, sin2πℓ)
γ_1(t)=(cos2π, sin2π)
are they reparametrisations?
γ_1 =γ_ℓ ◦ϕ
ϕ(t)= ℓt
t to ℓt
but
NO: check ARC LENGTH
|γ_ℓ ‘(t)| = 2πℓ
L(γ_ℓ|[0,1]) = ∫_{0,1} 2πℓ .dt
=2πℓ
|γ_1 ‘(t)| = 2π
L(γ_1|[0,1]) = ∫_{0,1} 2π .dt
=2π
not the same
note their image looks the same!
UNIT SPEED CURVE
A PC γ : I → R^n is called a unit speed curve (USC) if |γ’(t)| = 1 for any t ∈ I.
MAKES IT EASIER TO FIND LENGTH
PROPN 2.3 USC exist?
Let γ : I → R^n be an RPC. Then there exists a re-parametrisation γ˜ of γ that has unit speed.
PROOF:
Let γ : I → R^n be an RPC. Then there exists a re-parametrisation γ˜ of γ that has unit speed.
Proof. Pick a point t0 ∈ I and consider the arc-length function
s(t) = integral_[t_0,t] |γ’(τ )| dτ.
Clearly, s : I → R is a smooth function, and s’(t) = |γ’(t)| > 0. Denote by J the image of s; it
is an interval in R . Then s : I → J is bijective, and there is an
inverse function
ϕ = s−1: J → I.
By the inverse function theorem, ϕ is smooth,
ϕ’(τ ) =
(1/s’(ϕ(τ ))) > 0.
ϕ can be used as a parameter transformation
γ˜’(τ ) = γ’(ϕ(τ ))ϕ’(τ ) chain rule
= γ’(ϕ(τ ))/s’(ϕ(τ ))
= γ(ϕ(τ ))/ |γ’(ϕ(τ ))| ,
thus USC
|γ˜’(τ )| = 1.
DEFN
normal component of acceleration
γₙ’‘(t)
found by
[γ’‘(t)- [γ’‘(t) *γ’(t)][|γ’(t)|²] γ’(t) ]
CURVATURE VECTOR OF
γ
k(t)
(1/|γ’(t)|²)
[γ’‘(t)- [γ’‘(t) *γ’(t)][|γ’(t)|²] γ’(t) ]
=γₙ’‘(t) /|γ’(t)|²
RECALL IN EXAM
diagrams for acceleration, velocity vectors
For a curve compute acceleration and velocity
taking projection of acceleration on velocity
v =γ’(t)/|γ’(t)|
taking orthogonal component and γ’’ prokected onto notmal
ie γ’’ projected onto γ’
true or false
the curvature vector orthogonal to velocity?
true always
k(t) γ’(t)
=
(1/|γ’(t)|²)
[γ’‘(t)γ’(t)
- [γ’‘(t) γ’(t)][|γ’(t)|²] γ’(t)γ’(t) ]
=0??
true or false reparametrisation keeps arc length
true, it will have same properties, if we find a phi s,t it doesnt clearly this isnt a repara
PROP 2.4
curvature vector
how does is change under re-parametrisation?
The curvature vector is unchanged under a re-parametrisation, that is if
γ˜ : J → R^n is a re-parametrisation of an RPC γ : I → R^n, then
˜k(t) = k(ϕ(t)) for any t ∈ J,
where γ˜ = γ ◦ ϕ,
˜k is the curvature vector of γ˜, and k is the curvature vector of γ
if USC
dot prod
γ’ * γ’=
γ’’ * γ’
=1
=0
as if USC velocity orthogonal to acceleration
if our curve isnt USC?
we can always reparameterise
into a USC
preserving image
traversed in same direction
same #times
regularity
length
and curvature preserved
EXAM EXAMPLE
given a curve check its regular:
Given 2 parametrised curves determine whether related by parametrisation or not
γ_1(t)= a+tv line in R^n
γ_2(t) = r(cos 2pit, sin2pit)
γ_3(t) = a +t^3v
circle
take line and wrap around circle
1 &3:
1 regular 3 not regular as γ’_3(0)=0 so not
1 and 2: straight line has curvature k(t)=0 so cant be repara of a circel as regions where curvature never equals 0
Given 2 parametrised curves determine whether related by parametrisation or not
γ_2(t) = r(cos 2pit, sin2pit)
defined for [0,1]
γ_4(t) = r(cos 4pit, sin4pit)
both regular
velocity >0
γ’_4(t) = 2γ’_2(t)
length preserved?
|γ’_2| = 2pir
|γ’_4| = 4pir
length arc different so not repara
integrating over [0,1] gives double
CLOSED CURVES
DEFORMATIONS OF A CIRCLE
e.g
point end at same place
SPHERE IN R^3
2 parameters
2D object in R^3,R^4,..
def T-Periodic
A vector-function f : R → R^n is called T-periodic (or periodic with period T > 0) if f(t + T) = f(t) for any t ∈ R.
e.g
periodicity of
f(t) = sin(2πt)
g(x) = sin (x )
h(t) = t^2
f(t) = sin(2πt) 1-periodic smooth function,
g(x) = sin x 2π-periodic smooth function.
On the other hand, any unbounded continuous function h : R → R
(e.g. h(t) = t) can not be T-periodic with T > 0
continuous funct on closed interval period?
are bounded, may be periodic
PROPN 2.5 when do we have a periodic extension
A smooth function γ : [a, b] → R^n has a smooth (b − a)-periodic extension
IFF
all derivatives at the ends coincide
γ⁽ᵐ⁾(a) = γ⁽ᵐ⁾(b)
for any non-negative integer m.
L5 PROOF
A smooth function γ : [a, b] → R^n has a smooth (b − a)-periodic extension
IFF
all derivatives at the ends coincide
γ⁽ᵐ⁾(a) = γ⁽ᵐ⁾(b)
for any non-negative integer m.
LONG SUMMARISE??
uses that ends coincide then periodic can be shown as adding f(a+T) =f(b)?
(T₁, T₂)-equivariant
A real-valued function ϕ : R → R is called (T₁, T₂)-equivariant for given positive real numbers T₁& T₂ , if
ϕ(t + T₂) = ϕ(t) + T₁ for any t ∈ R.
ie
evaluated at t + T₂
same as evaluating at t and adding T₁
diagram 2T_2 for 2T_1 increments
example
ϕ(t) = 2πt
(T₁, T₂)-equivariant
( 2π, 1)-equivariant
for any t in R
ϕ(t + 1) = 2π(t + 1)
= 2πt + 2π = ϕ(t) + 2π.
how are T periodic functs and (t_1,t_2) equivariant functs linked
*any T-periodic function
is (0, T)-equivariant function.
- derivative of a (T_1, T_2)-equivariant function is a T_2-periodic function.
*equivariant functions often occur as integrals of periodic functions
L5:
Let f : R → R be a T2-periodic function. Can you show that the function ϕ, defined below, is
(T1, T2)-periodic:
ϕ(t) = ∫{0,t} f(s)ds,
where
T1 =∫{0,T_2} f(s)ds
ϕ(t +T_2) =
∫_{0,t+T_2} f(s)ds
using change of vars
𝜏=s-T_2
d𝜏/ds =1
𝜏_1= s_1-T_2=0
𝜏_2 =s-T_2 -t = ?
= ∫{0,T_2} f(s)ds + ∫{T_2,t+T_2} f(s)ds
= T_1 + ∫_{0,t+T_2} f(s)ds
= T_1 +ϕ(t)
PROPN 2.6 equivariant function composed with periodic function gives
Let ϕ : R → R be a (T1, T2)-equivariant function. Then for any T1-periodic vector-function f : R → R^n the composition
f ◦ ϕ is a T_2-periodic vector function.
proof:
For any t ∈ R
f ◦ ϕ(t + T2) = f(ϕ(t + T2)) = f(ϕ(t) + T1) = f(ϕ(t)) = f ◦ ϕ(t),
where in the second relation we used equivariance of ϕ, and in the third – the periodicity of f.
PROP 2.7
when to we have a (T_1,T_2) equivariant extension
Let ϕ : [c, d] → [a, b] be a surjective smooth function. Then it has a smooth (T1, T2)-equivariant extension ϕ¯ : R → R with
T_1 = (b − a) and
T_2 = (d − c)
IFF
ϕ(c) = a,
ϕ(d) = b, and
ϕ⁽ᵐ⁾(c) = ϕ⁽ᵐ⁾(d)
for any integer m > 0.
Proof. (For MATH5113M only.) The proof follows an argument similar to the one in the proof of
Proposition II.5.
CLOSED PARAMETRISED CURVE
A vector-function γ : [a, b] → R
n is called a closed parameterised curve (CPC) if it has a (b − a)-periodic extension ¯γ : R → R^n that is a smooth map.
“no distinguished start /end”
when is a closed parametrised curve regular
A closed parameterised curve γ
is called regular (RCPC), if its periodic extension ¯γ is regular
simple circular curve
an example of a closed regular curve
γ : [0, 1] → R^2
γ(t) = (cos(2πt),sin(2πt)).
example:
is this a closed parametrised curve
Consider a PC γ : [0, 1] → R
2 given by γ(t) = (t^2 − t, sin 2πt)
γ(0) = γ(1) but not a CPC
If γ would have a smooth periodic extension, then γ’(0) = γ’(1).
but
γ’(t) = (2t − 1, 2π cos 2πt)
⇒
γ’(0) = (−1, 2π) /= (1, 2π) = γ’(1)
γ_0(t)=(cos(2pit), sin(4pit)) t in [0,1] is this a RCPC
REGULAR CLOSED PC?
has extension
γ¬(t) = (cos(2pit, sin4pit) t in R
γ_0(t)=2pi(-sin(2pit), 2cos(4pit)) periodic function in [0,o.5] will vanish when both sin cos =0 but =/0
is this a closed PC?
γ:[0,1] to R^2
γ_0(t)= (t^2-t, sin2pi*t)
no because if it was a CPC it would have a smooth periodic extension and
γ’(0)=γ’(1) at end points but these arent equal
γ’(0)= (-1, 2pi) =/ (1, 2,pi) = γ’(1)
FOR A SMOOTH PERIODIC EXTENSION TO EXIST
MUST HAVE DERIVS PERIODIC
AND
COINCIDE AT a and b end points
when is a CPC a reparameterization
A CPC ˜γ : [c, d] → R^n is called the re-parametrisation of a CPC
γ : [a, b] → R^n if there exists a surjective map
ϕ : [c, d] → [a, b] that has a (T_1, T_2)-equivariant extension
¯ϕ : R → R with
T1 = b − a and
T2 = d − c s.t
¯ϕ is smooth, ¯ϕ’(t) > 0 for any t ∈ R, and
γ¯˜ = ¯γ ◦ ϕ¯, where
γ¯˜ and ¯γ are periodic extensions of ˜γ and γ respectively.
where does a CPC repara come from
We know any PC γ:[a,b] to R^n can be reparametrised to a PC defined on [0,1]
γ~(t)= γ(a+t(b-a)) t in [0,1]
if γ is a CPC then PC γ~ defined is also a CPC
can use the above repara parameter as it is a ((b-a),1) equivariant extension
with deriv >0
SIMPLE
A closed parametrised curve (CPC) γ : [a, b] → R^n is called simple if the restricted
map γ| [a, b) is injective
no self intersects
e.g
is this simple?
CPC γ` : [0, 1] → R^2
γ_ℓ(t) = (cos 2πℓt ,sin 2πℓt), where ℓ is
an integer
is simple if and only if ℓ = 1 or = −1.
Geometrically γ_ℓ wraps around γ_1 ` ℓ times; if ℓ > 0 it is traversed in the same direction, while
if ℓ < 0 – in the opposite.
0/w self intersects and loops
true or false
being SIMPLE depends on the parametrisation
false
SIMPLE
doensnt dep on repara
as for each point there is a unique t in [a,b] meaning unique tau in [alpha,beta]
HOMOTOPY OF CLOSED CURVES
space of all paths joining points
DEF
REGULAR HOMOTOPY
A regular homotopy from a closed curve α : [0, 1] → R^ n to a closed curve β : [0, 1] → R ^n is a continuous map F : [0, 1] × [0, 1] → R^ n that satisfies the following properties:
(i) α(t) = F(0, t), β(t) = F(1, t) for any t ∈ [0, 1], and for any fixed τ ∈ [0, 1] the map[
t in [0, 1]
t → F(τ, t) ∈ R^n
is an RCPC.
(ii) For any integer k > 0 the derivatives (∂k/∂tkF(τ, t)) are continuous in τ
i idea of deformation
ii ensures curvature change continuously under deformation
regular homotopy key points
*MAP will be a FAMILY OF CLOSED CURVES
each value of tai
smooth derivatives
deformations change
alpha(t) F(0,t)
beta(t) F(1,t)
F(tau,t) some other closed curve
derivative property ensures curve geometry affected, acceleration and velocity affect curvature
in exam will need to know
definitions of curvature
regular homotopy
stating these
given two curves alpha and beta, asked to prove RH by checking conditions and give formula for F
EXAM EXAMPLE
Γ _1(t)=(cos 2πt,sin 2πt)
t in [0.5,3/2]
Γ _1(t)_overline = overlineγ_1(t)
periodic extension same as
(cos 2pit, sin 2pit)
so
φ_overline : R to R
φ_overline (t) = t -0.5
φ(t) =t -0.5 for [0.5,3/2]
φ_overline (t+1) = φ_overline (t) +1
LHS: t +1 -0.5 = t -0.5+1 = RHS
γ:[a,b] to R^n is a CPC
overlineγ :R to R^n a (b-a) periodic extension
φ:[0,1] to [a,b]
φ(t)=a+t)b-a)
φ’(t)=b-a>0
φ(t+1) = a+(t+1)(b-a) = overline_φ(t) + (b-a)
and thusφ overline is (b-a,1) equivariant and this we have
γ~ using this as a repara parameter is a repara of γ
note arc lengths are the same for γ and overline_γ for a closed curve
HOMOTOPY CAN CHANGE GEOMETRY OF THE CURVE
EXAM EXAMPLE
check that the simple circle
α(t) = (cos 2πt,sin 2πt)
is regularly homotopic to the ellipse β(t) = (2 cos 2πt,sin 2πt), where t ∈ [0, 1].
2) kth derivatives wrt t of F(𝜏,t)
F(τ, t) = ((1 + τ ) cos 2πt,sin 2πt), where τ, t ∈ [0, 1],
CONDITIONS
1) F(0,t)= α(t) & F(1,t)= β(t)
fix 𝜏 in [0,1]
Show F(𝜏,t) is an RCPC
F_𝜏: t to F(𝜏,t):
check if periodic extension is regular
OVERLINE_F_𝜏: R to R^2
F‾_𝜏(t) = ((1+𝜏)cos2pit, sin 2pit)
is a smooth 1-perioduc funct
deriv ert t=/0 checked >= 4pi^2(s^2+c^2) = 4pi^2 >0
((1+𝜏) ∂ᵏ/∂tᵏ cos2pit, ∂ᵏ/∂tᵏ sin2pit)
1+𝜏 depends on tau continuously (polynomial of first order)
deforming by homotopy methods
we can deform for example by horizontal lines or rays from the origin
check velocity o this extension has no 0’s ie never equal to 0
a periodic is easier to check for intervals
For example consider
Γ _1(s)=(cos 2πs,sin 2πs)
t in [0.5,3/2]
same image as [
Γ _2(t)=(-cos 2πt,-sin 2πt)
t in [0,1]
homotopy?
same image as
Γ _2(t)=(-cos 2πt,-sin 2πt)
t in [0,1]
by repara change of vars t=s-0.5
F(tau,t)=
[cos pi𝜏, -sin pi𝜏]
[sin pi𝜏 cos pi𝜏] *
[cos 2pit]
[sin 2pit]
dep continuously on tai
rank 1
rotation fraction tau/pi
if tau=1 angle is pi
[-1 0]
[0 -1]
closed curves for integer vales of ℓ
gamma_ℓ
are these regularly homotopic
no
recall in exam regularly homotopic property PROP 2.8
The relation of being regularly homotopic is an equivalence relation on the set of closed regular curves
(1) reflexivity α ∼ α, RH to itself
(2) symmetry α ∼ β =⇒ β ∼ α,
(3) transitivity α ∼ β and β ∼ γ =⇒ α ∼ γ.
proof
The relation of being regularly homotopic is an equivalence relation on the set of closed regular curves
(1) reflexivity α ∼ α, RH to itself
(2) symmetry α ∼ β =⇒ β ∼ α,
(3) transitivity α ∼ β and β ∼ γ =⇒ α ∼ γ.
(1) For any RCPC α : [0, 1] → R^n we define F(τ, t) = α(t). satisfied.
(2) Let α and β : [0, 1] → R
n be two RCPC for exists a regular homotopy from α to β, F : [0, 1] × [0, 1] → R^n ,F(0, t) = α(t), F(1, t) = β(t). Then the map
G(τ, t) := F(1 − τ, t), where τ, t ∈ [0, 1]
defines a regular homotopy from β to α, that is G(0, t) = β(t) and G(1, t) = α(t). The conditions
in Definition II.18 for G follow from similar conditions for F. For example, for any fixed τ the map
t 7−→ G(τ, t) is an RCPC, since G(τ, t) = F(1 − τ, t) and t 7−→ F(1 − τ, t) is an RCPC. We also see
that for any k > 0 the derivative
∂^k/∂tk G(τ, t) = ∂ k/ ∂tk F(1 − τ, t)
is continuous in τ , as a composition of continuous maps.
(3) Suppose that α is regularly homotopic to β via a regular homotopy F, and β is regularly homotopic to γ via G. Consider the map H : [0, 1] × [0, 1] → R^n
, H(τ, t) =
{F(2τ, t), τ ∈ [0, 1/2];
{G(2τ − 1, t), τ ∈ [1/2, 1].
H satisfies condition a regular
homotopy from α to γ
PROP 2.9
Every RCPC is regularly homotopic to
Every RCPC is regularly homotopic to an RCPC of unit length.
proof: uses map
F:[0,1] x [0,1’ to R^m
F(tau,t) = (1-tau + tau/L) gamma (t)
and verifies conditions
Proposition II.10. Let γ˜ : [0, 1] → R^n be a re-parametrisation of an RCPC γ : [0, 1] → R^n. Then
Proposition II.10. Let γ˜ : [0, 1] → R^
n be a re-parametrisation of an RCPC γ : [0, 1] → R^n. Then γ
and γ˜ are regularly homotopic.
PROOF
Proposition II.10. Let γ˜ : [0, 1] → R^
n be a re-parametrisation of an RCPC γ : [0, 1] → R^n. Then γ
and γ˜ are regularly homotopic.
Since ˜γ is a re-parametrisation of γ there exists a smooth function ¯ϕ : R → R such that:
* ϕ¯(0) = 0, ¯ϕ(1) = 1, and ¯ϕ’(t) > 0 for any t ∈ R;
* ϕ¯(t + 1) = ¯ϕ(t) + 1 for any t ∈ R;
* γ¯˜ = ¯γ ◦ ϕ¯, where γ¯˜ and ¯γ are 1-periodic extensions of ˜γ and γ respectively.
We define a map F¯ : [0, 1] × R → R
n by setting
F¯(τ, t) = ¯γ(τϕ¯(t) + (1 − τ )t).
We claim that for any τ ∈ [0, 1] the map F¯_τ : t 7→ F¯(τ, t) is smooth, 1-periodic, and has non-vanishing
derivative. Indeed, it is smooth as a composition of smooth functions. To verify 1-periodicity, we
write
F¯_τ (t + 1) = ¯γ(τϕ¯(t + 1) + (1 − τ )(t + 1)) = ¯γ(τϕ¯(t) + τ + (1 − τ )t + (1 − τ )) =
γ¯(τϕ¯(t) + (1 − τ )t + 1) = ¯γ(τϕ¯(t) + (1 − τ )t) = F¯τ (t).
Finally, computing the derivative we obtain
F¯’τ (t) = ∂/∂tF¯(τ, t) = ¯γ’(τϕ¯(t) + (1 − τ )t)(τϕ¯’(t) + (1 − τ )).
Since ¯γ’ /= 0, we obtain that for any τ ∈ [0, 1] the derivative F¯’ τ(t) 6= 0. These properties show that
for any τ ∈ [0, 1] the restriction of F¯
τ to [0, 1] defines an RCPC.
Now we define the homotopy F between γ and ˜γ by setting
F(τ, t) := γ(τϕ(t) + (1 − τ )t).
Clearly, F(0, t) = γ(t) and F(1, t) = γ(ϕ(t)) = ˜γ(t). We verify that the conditions of Definition II.18
hold. For any fixed τ ∈ [0, 1] the RPC F¯_τ is an 1-periodic extension of t → F(τ, t), and hence,
the latter is an RCPC. To verify the second condition of Definition II.18 we should show that the
derivative ∂^kF¯(τ, t)/∂t^k
is continuous in τ for any t ∈ R. The latter is a consequence of the fact that
F¯(τ, t) is defined as a composition of the maps whose all derivatives are continuous in τ .
to save time:
notes from now on only thms and statements
examples in written notes!
:(
Corollary II.11. Every RCPC γ is regularly homotopic to
Every RCPC γ is regularly homotopic to an RCPC of constant speed L, where L is the length of γ
Corollary II.12. Every RCPC γ is regularly homotopic to
Corollary II.12. Every RCPC γ is regularly homotopic to an RCPC of unit speed.
Proof. The statement is direct consequence of Proposition II.9 and Corollary II.11
UNIT TANGENT VECTOR
v(t) = γ’(t)/ |γ’(t)|
unit normal vector
uses components of unit tangent vector
n(t) := (−v2(t), v1(t)).
like rotation 90 degrees
unit normal and unit tangent vector
vectors v(t) and n(t) are orthogonal and form a positively oriented basis of R
signed curvature
k(t) be a curvature vector to γ. As we know, it is orthogonal to v(t), and since γ is a plane curve, we conclude that
k(t) = κ(t)n(t)
for some real-valued function κ(t). The function κ(t) is called
the signed curvature of γ
signed curvature and curvature
t |κ(t)| = |k(t)|.
Geometrically the signed curvature function κ(t)
measures the rate of change of a tangent line direction: when κ > 0 the curve is bending towards the
unit normal, while when κ < 0 it is bending away from the normal.
diagram of smother version of a kink
___B__
| |
—-A-| |——–
signed curvature positive A
curve bends towards unit normal
points up ish
signed curvature negative B
curve bends away from unit normal
points upish
left for readin: THM 2.13 Fundamental thm of plane curves
Given a smooth real-valued function
κ : [a, b] → R, a point t0 ∈ [a, b], and vectors γ0, v0 ∈ R^2
such that |v_0| = 1, there exists a unique unit speed parametrised curve γ : [a, b] → R^2 whose signed curvature equals κ(t) and γ(t0) = γ0,
γ’(t_0) = v_0.
Lemma
θ function
Let γ : [a, b] → R^2
be a regular parametrised curve. Then there exists a smooth function
θ : [a, b] → R such that the unit tangent vector v(t) satisfies the relation
v(t) = (cos θ(t),sin θ(t)).
If θ1 and θ2 are two such functions, then they differ only by an integer multiple of 2π, that is
θ1(t) = θ2(t) + 2πm, where m ∈ Z is a constant. In particular, the quantity
θ(b) − θ(a) uniquely determined by PC
angle parametrisation for unit tangent vector
v(t)= (Cos(θ(t), sinθ(t))
ROTATION INDEX for an RCPC
times unit tangent vector v(t) winds around the unit circle S^1
r(γ) :=
(1/2π)(θ(1) − θ(0)) ∈ Z
is called the rotation index of γ.
anticlockwise counting positively
angle is from 0 anticlockwise
e.g draw plane RCPC given
γ1(t) = (cos(2πt),sin(2πt)) and
γ2(t) = (cos(2πt),sin(4πt)), where t ∈ [0, 1].
Compute (analytically
or visually) how many times the unit tangent vector winds around the unit circle S^1
for each curve.
circle anticlockwise
curvature 1
derivative used in formula for rotation index =1
v(t)=(1/2pi)(deriv)
used to find theta functs
figure 8
curvature =0 never makes full turn
unit tangent vector:
γ’(t)=(-2pisin(2pit), 4picos(4pit))
v(t)= (-sin(2pit, 2cos(4pi*t)
=(cos(θ(t), sin(θ(t))
cos(2pit + (pi/2))
= cos(pi/2)cos(2pit) - sin(pi/2)sin(2pit)
=-sin2pit
…..difficult
analytically starts clockiwise turns anti but never makes full #turns so 0
θ for γ _ℓ(t)
rotation index = ℓ
θ(t) = (pi/2) + 2piℓ*t
PROP 2.15 rotation index geoemtric formula
Let γ : [0, 1] → R^2 be a plane RCPC. Then its rotation index satisfies the relation
r(γ) = (1/2π)
integral_{0,1} κ(t)|γ(t)| dt,
where κ is the signed curvature of γ
proof: we know smooth theta exists s.t we can write the unit tangent vector.
our first deriv will be multiple of this( modulus of first deriv) then subbin into formula
PROP 2.16 regularly homotopic does rotation index change
Regularly homotopic RCPCs have the same rotation index
proof: defining a regular homotopy the rotation index defined as a formula for any curve in the homotopy takes integer values so must be constant by IVT
important thme
WHITNEY GRAUSTEIN
Two regular closed plane curves are regularly homotopic if
and only if they have the same rotation index
thm 2.18 Hopf
Let γ be a simple regular closed plane curve. Then its rotation index r(γ) equals either 1 or −1.
corollary 2.19 for simple regular closed plane curves regularly homotopic to
Any simple regular closed plane curve is regularly homotopic either to the standard circle
[0, 1]
t → (cos(2πt),sin(2πt))
or to the reversed standard circle
[0, 1]
t→ (cos(2πt), − sin(2πt)).
Corollary II.20. For any simple plane RCPC
integral_[0,1] |κ(t)| |γ’(t)| dt > 2π
Corollary II.20. For any simple plane RCPC γ : [0, 1] → R^2 the following relation holds
integral_[0,1] |κ(t)| |γ’(t)| dt > 2π;
the equality occurs
IFF
κ(t) does not change sign
(that is, κ(t) > 0 everywhere or κ(t)<=0 everywhere).
proof in notes
if rotation index differs
curves arent RH to each other
circular curve cant be deformed to curve of length 2pi st
signed curvature (t) >1
changing curvature changes length everywhere
1=r(gamma)=r(gamma~) if deformed
same length
contradiction as r(gamma~) = fromula using signed curvatiure > 1
if we have a closed curve with RI ℓ then its homotopic to
curve with RI ℓ
so given a curve we cab either
find the integral formula fro RI
or guess and prove by constructng a regular homotopy
applying whitney graustein thm
TOPOLOGICAL DEGREE METHOD
for RI counts #pre-images with signs
1) fix direction curve traversed
tangent traverses anticlockwise for t in [0,1]
2) find points with same curvature
signs will show curvature
drawn a sketch
chosen all points with same tangent/direction upwards
curvature at 90 degrees anticlockwise to this drawn fixed unit bector
signed curvature
positive if points into a curve
negative if points out of a curve
added
in exam we can do this or cumpute by constructing RH
cos 2pi*t, -sin2pit)
complicated drawing is actually regularly homotopic to circle traversed once clockwise
L5:
when does a periodic extension exist?
when we have closed curves
any closed curve can be parametrised to USC
L5:
Bounded domain Ω
subset in R^2
open subset bounded by a SIMPLE REGULAR CLOSED PC
image of surrounds it as a boundary
an example
γ[0,1] to R^2 simple
restriction part injective with no self intersection until 0,1
restrict to [0,1) so we can use
common qs L5
among all domains in R^3 of a given area A find boundary with least length
among all simple regular closed curves of given length L find a curve that encloses a domain of greatest area
find a region of greatest area bounded by a straight line aand a curve of a fixed lengthwhose ends lie on the line (answer disk part of circle)
ISOPERIMETRIC INEQUALITY
Let Ω ⊂ R^2
be a domain bounded by a simple
regular closed parametrised curve γ : [0, 1] → R^2
. Let A be the area of Ω, and L be the
length of γ. Then the inequality
4πA <= L^2 holds, and the equality is achieved if and only if
Ω is a disk.
Lemma II.2 (Green’s formula).
Let P(x, y) and Q(x, y) be two smooth real-valued functions
defined on a domain Ω. Then the following relation holds:
∫∫_Ω [∂P/∂x + ∂Q/∂y]dxdy =
∫_γ (P dy − Qdx),
where the right hand-side above is the anti-clockwise integral along the boundary curve
γ = ∂Ω.
example
Let Ω ⊂ R^2 be a domain bounded by a simple regular closed unit speed curve γ : [0, L] → R^2
; that is |γ’(t)| = 1 for any t ∈ [0, 1], and hence, L equals to the length of γ
L(γ) =
∫[0,L] |γ’(t)| dt =
∫[0,L] 1dt = L
Let n(t) be a unit normal vector to γ such that
det(γ’(t), n(t)) = 1 for any t ∈ [0, L]. In
other words, if γ’(t) has coordinates
(x’(t), y’(t)), then
n(t) = (−y’(t), x’(t)).
We claim that
the area A of the domain Ω satisfies the following relation:
2A = −∫[0.L] (γ(t) · n(t))dt.
Indeed, choosing P(x, y) = x and Q(x, y) = y, by Green’s formula we obtain
2A =
2∫∫Ωdxdy =
∫∫_Ω (∂P/∂x + ∂Q/∂y) dxdy =
∫_γ (xdy − ydx) =
∫[0,L] (xy’ − yx’)(t)dt =
− ∫[0,L] (γ(t) · n(t))dt.
WIRTINGERS INEQUALITY
Let f : R → R be a smooth L-periodic function such
that ∫_[0,L] f(t)dt = 0.
Then the following inequality holds
∫[0,L] f^2(t)dt <=
(L^2/4π^2)
∫[0,L] (f’(t))^dt;
the equality occurs if and only if f(t) = a cos(2πt/L) + b sin(2πt/L) for some a, b ∈ R.
REMEMBER THE NAME?
integral over period =0
tranlation left
curvature
total curvature
For a given RCPC γ : [0, 1] → R
n the quantity
µ(γ) =
integral_{0,1} |k(t)| |γ’(t)| dt,
where k(t) is the curvature vector, is called the total curvature of γ
propn 2.4 for total curvature
Proposition II.4. Let γ˜ : [0, 1] → R
n be a re-parametrisation of an RCPC γ : [0, 1] → R^n.
Then the total curvatures of γ˜ and γ are equal, µ(˜γ) = µ(γ).
Theorem II.5 (Fenchel-Borsuk).
Let γ : [0, 1] → Rn be a simple regular closed parametrised
curve. Then its total curvature µ(γ) is at least 2π, and it equals 2π if and only if the curve lies in an affine 2-plane and its signed curvature does not change sign.
The statement above is a special property of closed curves. Simple examples show that the total curvature is not a topological invariant, that is it changes under the deformations of a curve.
isotopy
An isotopy of R^n is a continuous map Φ : [0, 1] × R^n → R^n such that for any fixed τ ∈ [0, 1] the map R^n x→ Φ(τ, x) ∈ R^ n is a homeomorphism, that is bijective,
continuous, and the inverse map is also continuous
ambient isotopic l5
Definition II.3. Two simple RCPCs α : [0, 1] → Rn and β : [0, 1] → R
n are called ambient
isotopic if there exists an isotopy of Rn such that:
* Φ(0, x) = x for any x ∈ Rn;
* Φ(1, α(t)) = β(t) for any t ∈ [0, 1].
knotted
A simple RCPC is called unknotted if it is ambient isotopic to a plane
circle. Otherwise, it is called knotted.
Theorem II.6 (F´ary-Milnor).
Let γ be a knotted simple RCPC. Then its total curvature
is at least 4π, that is µ(γ) > 4π.
note l5
Tangent indicatrix. Let γ : [0, 1] → R
n be an RCPC, and v(t) = γ’(t)/ |γ’(t)| be its
unit tangent vector. In the sequel we use the notation S^{n−1}
for a unit (n − 1)-dimensional
sphere centred at the origin in the Euclidean space R^n
S^{n−1} = {(x1, . . . , xn) ∈ R^n
: x^2_1 + . . . + x^2_n = 1}.
tangent indicatrix
For a given RCPC γ : [0, 1] → R
n the parametrised curve
Γ : [0, 1] → S^{n−1} ⊂ R^n
Γ(t) = v(t) = γ’(t)/|γ’(t)|
is called the tangent indicatrix of γ
Proposition II.7. The total curvature and tangent indicatrix
Proposition II.7. The total curvature of an RCPC γ : [0, 1] → R^n is equal to the length of the tangent indicatrix Γ of γ, that is L(Γ) = µ(γ)
great circle
A great circle, or equator, in a unit sphere S
n−1
is an RCPC γ : [0, 1] → R
n
of the form
γ(t) = cos(2πt)e1 + sin(2πt)e2,
where e1 and e2 are orthonormal vectors
Fact II.1. For any two different points p and q in a sphere S
n−1
there exists a great circle
γ that passes through p and q. Moreover, if p and q are not anti-podal, p /= −q, then such
a great circle is unique.
Suppose that given points p and q ∈ S
n−1
lie in a great circle γ. Then they divide it
into two arcs γ1 and γ2, and we define the spherical distance between p and q by setting
distS(p, q) = min{L(γ1), L(γ2)}.
Note that distS(p, q) 6 π for any points p and q, and the equality occurs if and only if p
and q are anti-podal, that is p = −q. Besides, distS(p, q) = 0 if and only if p = q. Note
also that distS(p, q) = π/2 if and only if the points p and q, viewed as vectors in R
n, are
orthogonal.
Fact II.2. Let γ : [0, 1] → S
n−1 ⊂ R
n be a PC such that γ(0) = p and γ(1) = q. Then the
spherical distance is not greater that the length of γ, that is distS(p, q) 6 L(γ). Moreover,
the equality is achieved if and only if γ is an arc of a great circl
Exercise II.7. Check that the distance function distS(p, q) satisfies the triangle inequality
distS(p, q) 6 distS(p, z) + distS(z, q) for any points p, q, and z ∈ S
n−1
