Chapter 5:Curvature of a surface Flashcards
unit normal vectors?
for a regular parametrized surface σ : U → R^3
there are 2 unit normal vectors X_u and X_v
for each (u, v) ∈ U.
eg if intersects itself 4 normals at this point
Definition 5.1.1. The preferred unit normal vector
Definition 5.1.1. The preferred unit normal vector of the regular parametrization
σ : U → R^3 at the point (u, v) ∈ U is
n(u, v) = [σu(u, v) × σv(u, v)]/
[||σu(u, v) × σv(u, v)||]
This defines a map n: U → R^3
field of unit normals
surface in R^3 has a tangent plane containing vectors X_u and X_v. Normal to this plane is n
EXAMPLE:
normal vector of the sphere S^2 at (x, y, z) is
given by the radius vector, i.e. (x, y, z) itself, and it is also a unit vector. In
other words,
(x, y, z) = a unit normal vector of S^2
at the point (x, y, z).
UNIT
EXAMPLE:
normal vector of the sphere at (x, y, z)
Example 3.3.6 of S
2 \ {N, S}, the sphere without the North and South poles,
its the surface of rev of the open semicircle parametrized by γ:]-π/2,π/2[→ R^3
φ →(cosφ,0,sinφ)
the resulting para surface of rev is
σ: ]-π/2,π/2[ x R →R^3
σ(φ,θ) = (cosφcosθ,cosφsinθ,sinφ)
σ_φ=(-sinφcosθ,-sinφsinθ,cosφ)
σ_θ=(-cosφsinθ,cosφcosθ,0)
σ_φ x σ_θ=
(-cos^2(φ)cosθ,-sinθcos^2(φ),-cos^2(θ)sinφcosφ +sinφcosφsin^2(θ))
so n=-σ(φ,θ)
(into the sphere as neg)
The partial derivatives n_u and n_v
For σ : U → R^3 a parametrized surface the preferred unit normal vectors
form a vector valued function n: U → R^3
The partial derivatives n_u and n_v measure the variations of n in the u- and v-directions
n has constant
magnitude so
n_u and n_v are ORTHOGONAL to n
This means nu and nv are tangent vectors. Therefore we may write
−nu = W11σu + W21σv,
−nv = W12σu + W22σv
where W11, W12, W21, W22 are some functions of (u, v); the minus signs are
just a matter of convention.
(COLUMNS OF WEINGARTEN and neg, these show the normal vectors as we move in u or v direction )
*n_u and n_v lie in tangent plane (orthogonal to normal) so are linear combination of σ_u and σ_v, which form a basis for the tangent plane
Definition 5.2.1. The WEINGARTEN MATRIX
The Weingarten matrix or shape operator of σ is the matrix
W =
[W11 W12]
[W21 W22]
Definition 5.2.1. THE SECOND FUNDAMENTAL FORM
The second fundamental form of σ is the matrix II = [σuu · n σuv · n] [σuv · n σvv · n] =: [L M] [M N] , where this defines the functions L, M, N : U → R.
combos of -n_u and -n_v
SECOND partial derivatives
THM 5.2.2
Weingarten is expressible in terms of
The Weingarten is expressible in terms of the first and second
fundamental forms as follows:
W= [W_11 W_12 ] [W_21 W_22] = [E F ]^-1 [L M] [F G] [M N]
where the first fundamental form of a smooth regular surface is always invertible
(FFF is σu·σu, σvσv etc)
INVERSE
proof:
The Weingarten is expressible in terms of the first and second
fundamental forms
[E F ] [F G] W = [σu·σu σu·σv] [W_11 W_12 ] [σv·σu σv·σv] [W_21 W_22] = [σu σv ·W11+σu σv ·W21 ...] [... ...] =* [σuu · n σuv · n] [σuv · n σvv · n] = [L M] [M N] = ||
*by product rule
0=(σ_u· n)_u =σ_uu· n + σ_u· n_u
(=0 as σ_u orthogonal to n, which is normal to tangent plane of σ containing σ_u and σ_v)
Example:
Compute first and second fundamental forms and
thus the Weingarten matrix of the helicoid
σ : R^2 → R^3
σ(u, v) = (vcos u, vsinu, u),
**mistakes
σ_u=(-vsinu,vcosu,1) σ_v=(cosu,sinu,0) σ_uu=(-vcosu,-vsinu,0) σ_vv=(0,0,0) σ_uv=(-sinu,cosu,0)
σu·σu= (vsinu)^2 +(vcosu)^2+1 = 1+v^2
σu·σv= (-vsinucosu) +(vcosusinu) +0 =0
σv·σv=cos^u +sin^2u = 1
n= cross product σu x σv NORMALISED**** = _*(-sinu,cosu, -vsin^2(u)-vcos^2(u)) = _*(-sinu,cosu,-v) =1/√(1+v^2)(-sinu,cosu,-v)
σuu·n=1/√(1+v^2) =0
σvv·n=0
σuv·n=1/√(1+v^2) =1/√(1+v^2)
FFF^-1 = [1+v^2 0]-1 [ 0 ] =(1/[1+v^2])* [1 0 ] [0 1+v^2 ] MULTIPLY BY 1/DET***
W=
(1/[1+v^2])*
[1 0 ][ 0 1/√(1+v^2) ]
[0 1+v^2][ 1/√(1+v^2) 0 ]
=
[ 0 (1+v^2)^{-3/2} ]
[(1+v^2)^{-1/2} 0 ]
- n_u (u,v) = (1+v^2)^{-0.5}σ_v
- n_v (u,v) =(1+v^2)^{-3/2}σ_u
You can then deduce that as we move along the v-direction, i.e. along a horizontal line, −n keeps bending sideways towards the u-direction.
HELICOID -“spiral slide”
move in Direction σ_u then the normal changes a small amount perpendicular to the normal Vector in the tangent plane thus this is a linear combination of σ_u and σ_v
EXAMPLE:
Consider the parametrized surface of revolution (Example
3.3.4):
σ : R^2 → R^3
σ(t, θ) = (x(t) cos θ, x(t) sin θ, z(t)).
We assume that x(t) > 0 and that the generating curve is regular, so that σ is regular (Exercise 3.4.7.). The preferred unit normal field and the second fundamental form are
LEARN***
n(t,θ)=
[-z’(t)cosθ, -z’(t)sinθ, x’(t))]/[√(x’(t)^2 + z’(t)^2)]
(normalised vector)
II(t, θ) =
1/[√(x’(t)^2 + z’(t)^2)] *
[x’(t)z’‘(t) -x’‘(t)z’(t) 0 ]
[ 0 x(t)z’(t)]
EXAMPLE:
Consider the cylinder (Example 3.3.5):
σ : R^2 → R^3
σ(t, θ) = (cos θ, sin θ, t).
We have x(t) = 1 and z(t) = t, so that with the previous example
n(t,θ)=(-cosθ,sinθ,0)
II(t, θ) =
[0 0]
[0 1]
W=
[0 0]
[0 1]
Notice that the following unit normal vector points inwards and that
−nt = 0, −nθ = σθ
By the first equation, as we move along the t-direction, i.e. along a longitude line, −n remains the same. By the second equation, as we move along the θ-direction, i.e. along a latitude CIRCLE, −n keeps bending forward.
EXAMPLE:
Consider the parametrization of the sphere S^2
(Example 3.3.6)
σ :]−π/2,π/2[× R → R^3
σ(φ, θ) = (cos φ cos θ, cos φ sin θ, sin φ).
We have
x(φ) = cos φ and
z(φ) = sin φ, so that
n(φ, θ) =
(-cosφcos θ, -cosφsinθ, -sinφ)
II(θ,φ) =
[1 0 ]
[0 cos^2(φ)]
W(φ,θ) =
[1 0]
[0 1]
Note that the unit normal vector points inwards, and that
−n_φ = σ_φ,
−n_θ = σ_θ
By the first equation, as we move along the φ-direction, i.e. along a longitude,
−n keeps bending forward. By the second equation, as we move along the θ-direction, i.e. along a latitude, −n also keeps bending forward.
EXAMPLE: the parametrized catenoid σ : R^2 → R^3 σ(t, θ) = (cosh t cos θ, cosht sin θ, t). has Weingarten matrix
W_(t,θ) =
[−(cosh t)^−2 0 ]
[ 0 (cosh t)^−2]
(=
[(sinh^t +1)^0.5 0 ]^-1
[ 0 cosht]
*
[ cosht 0 ]
[0 cosht]
)
Definition 5.4.1.
The Weingarten map
Let σ : U → R^3 be a parametrized surface and (u0, v0) ∈ U.
The Weingarten map (curlyW)_(u0,v0) of σ at (u0, v0) is the map
(curlyW)(u0,v0) :
T(u0,v0)σ → T(u0,v0)σ,
γ˜(0) → −(d/dt)| _t=0 n(γ(t))
any tangent vector to a curve at X
map is independent of the basis chosen
3.4.6 if X∈T(u_0,v_0)
then there exists γ: ]−ε, ε[ → U st if we take γ˜ = σ ◦ γ then tangent vector is X γ˜^•(0)=X
diagram: tangent space contains γ~
THM 5.4.2
weingarten map and matrix relation
The Weingarten map curlyW(u0,v0)
is represented by the Weingarten matrix W(u0,v0)
in the basis {σu(u0, v0), σv(u0, v0)} of T(u0,v0)σ, in other words
curlyW(u0,v0)(aσu(u0, v0) + bσv(u0, v0)) =
W(u0,v0) *
[a]
[b]
EXAMPLE:
Choose an arbitrary point on the 2-dimensional sphere S^2
and compute the Weingarten map of the sphere at this point
for this curlyW of any point is identity mao
innter product
I( (a,b) , (c,d)) =
(c)
(a b) I (d)
=
[a]T [c]
[b] I [d]PROOF
THM 5.4.2
weingarten map and matrix relation
if γ(t)=(u(t),v(t)) and
(σ ◦ γ)(t)=γ~(t) then
γ~•(t)=u’(t)σ_u(γ(t)) +v’(t)σ_v(γ(t)) * by the chain rule.
Similarly (d/dt)(n(γ(t)) = u’(t)n_u(γ(t)) +v’(t)n_v(γ(t))**
we want γ st
γ~•(0) =aσ_u(u_0,v_0) + bσ_v(u_0,v_0)
taking γ(t)=(u_0+at, v_0 +by) for t∈]−ε, ε[ and small ε. By *:
γ~•(0)=aσ_u +bσ_v. Then def of map curlyW(aσ_u(u_0,v_0) +bσ_v(u_0,v_0))
=
curlyW(γ~•(0))
=
-d/dt (n(γ(t))|_t=0
=
-u'(0)n_u(u0,v0) - v'(0)n_v(u0,v0)
=-an_u -bn_v
=a(W_11σ_u + W_21σ_v) + b(W_12σ_u + W_22σ_v)
=
[aW_11 + bW_12 ]
[aW_21 + bW_22]
wrt basis {σ_u,σ_v}
=
W*
[a]
[b]
=
as required
**we know that as -n is a unit vector
d/dt (n(γ(t)) ∈T_γ(t) σ
and this up to a sign is curlyw_(u_0,v_0)X
PROP 5.4.4
a property of the weingarten map
curlyw
The Weingarten map curlyW is self-adjoint.
Namely, for any tangent vectors X, Y at (u0, v0),
we have
curlyW_(u0,v0) (X) · Y =
X · W_(u0,v0)(Y)
