Curves and Surfaces in Euclidean Space

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Lecture 7 A

Curves and Surfaces in Euclidean Space

p p p

CurvesinIR³ andIR^d,Frenetframes SurfacesinIR³,Gaussframes Firstandsecondfundamentalform Curvatureofsurfaces

c 2003–2008 Martin Welk, 2015 Martin Schmidt, 2019 Marcelo Cárdenas

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Curves inIR³

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Reminder about Curves and Curvature in 2D

p In 2D, a regular curve is characterised up to Euclidean transformations by the curvatureκ(as function of the curve parameter)

p At each curve point, there are tangent and normal unit vectors~tand~n, such that in arc-length parametrisation

cs=~t , css=κ~n

c c_ss

n

c_s t = x

Curve with tangent and normal vectors and first two derivatives at a point, from Lecture 2

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Curves inIR³

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Curvature Parameters of a Curve in 3D

p Consider regular curve in 3D Euclidean space,c:I→IR³, in arc-length parametrisation

p At each curve pointc(s), there are

a a unit tangent vector ~t(s), cs =~t(s)

a a unit normal vector ~n(s), css =κ(s)~n(s)

a a unitbinormalvector~b(s), ~b(s) =~t(s)×~n(s) (unique ifcis2-regular, i.e.css6= 0)

p In contrast to the 2D case,κ(s)can always be chosen nonnegative

p ~t(s),~n(s),~b(s)form an orthonormal system, theFrenet frameofc

n b

t c s( )

c

CurvecinIR³with Frenet frame atc(s)

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Curves inIR³ (3)

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Curvature Parameters of a Curve in 3D, cont.

p the derivative of~bis perpendicular to both~tand~b, i.e

~bs(s) =−τ(s)~n(s) with a functionτ(s), thetorsionofc

p Curvatureκ(s)and torsionτ(s)determine the curvec(s)up to rotations and translations

p Frenet-Serret equations(or Frenet equations):

d ds





~t

~n

~b



=





0 κ 0

−κ 0 τ

0 −τ 0









~t

~ n

~b





p The torsion can also be defined by

csss(s) =−κ(s)²~t(s) +κs(s)~n(s) +κ(s)τ(s)~b(s)

p The torsion vanishes identically if and only if the curve is contained in a plane

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Curves inIR^d

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Frenet Frames in Higher Dimensions

p Consider a curvecinIR^din arc-length parametrisation

p For each curve pointc(s), there is an orthonormal basis(~e1, . . . , ~ed)such that cs(s) =~e1, css(s)∈Span (~e1, ~e2),

c_s(k)(s)∈Span (~e1, . . . , ~ek), k≤d

p (~e1, . . . , ~ed)is theFrenet frameofc.

p Frenet equationsinddimensions:

d ds



















~e1

~e2

...

~ed



















=



















0 κ1 0 . . . 0 0

−κ1 0 κ2 0 0

0 −κ2 0 0 0

... . .. ...

0 0 0 0 κd−1

0 0 0 . . . −κd−1 0





































~e1

~e2

...

~ ed



















p Curvature functionsκi(s),i= 1, . . . , d−1(nonnegative fori≤d−2) determinecup to rotations and translations

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Surfaces in IR^d

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Surfaces

p Surface inIR^d: Differentiable functionσ:D→IR^d,D⊂IR² connected domain

p Graph (image) of a surfaceσ: Set of points inIR^dgiven by {σ(u, v)|(u, v)∈D}

Remark: Similarly as for curves, surfaces with identical graphs but different parametrisations are considered different.

Curves on a Surface

p σ:D→IR³ surface

p I⊂IRinterval, mappingI3p7→(u(p), v(p))∈D

p Byc:I→σ(D)⊂IR³, p7→σ(u(p), v(p)), a curve onσis given

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Surfaces in IR^d

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Related Definitions

p Regular surface: Surfaceσ: (u, v)7→IR^dis regular if the (Jacobi) matrix

Dσ:=





 σ_u¹ σ¹_v

... ... σ_u^d σ^d_v







has rank2everywhere inD.

p k-regularity can be defined analogously using higher order derivatives. We will always assume that surfaces are sufficiently many times differentiable.

p Tangent planeT_(u,v)σtoσatσ(u, v): image of Dσ(u, v) : T(u,v)IR²→Tσ(u,v)IR^d

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Surfaces in IR^d

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Reparametrisation

p Transforms a surface into another one with the same graph

p σ:D→IR^d surface

p D˜ ⊂IR² connected domain

p ϕ: ˜D→Ddifferentiable mapping with rank Dϕ= 2everywhere

p σ˜:=σ◦ϕ: ˜D→IR^dreparametrised surface

p orientation-preservingifdet Dϕ >0

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Gauss Frame for Surfaces in 3D

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Gauss Frame for Surfaces in 3D

p Restrict now to surfacesσ:D→IR³ in 3D Euclidean space

p Gauss frame(analog of Frenet frame for surfaces) consists of the three unit vectors

~t1(u) := σu

kσuk , ~t2(u) := σv

kσvk, ~n:= σu×σv

kσu×σvk

p First two vectors~t1 and~t2 of the frame lie in tangential direction,~n perpendicular to the surface

p In general,~t1 and~t2 are not orthogonal

p ~t1 and~t2depend on parametrisation

p Normal vector~ndoes not change under orientation-preserving reparametrisation, is reverted by orientation-changing reparametrisation

t₂ t₁ n

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First Fundamental Form for Surfaces in 3D

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First Fundamental Form

p Consider regular surfaceσ:D→IR³

p Use boldface lettersu,v, . . .for points and vectors inIR²

p Symmetric bilinear form

Iu(v,w) :=hDσ(u)v,Dσ(u)wi, v,w∈TuD

p p

iscalledfirstfundamentalformofσatu RegularityimpliesIu(w,w)6=0, for nonzero w

Incoordinates,IuisdescribedbyamatrixwhichwewillalsodenotebyIu

(u= (u, v)):

Iu=I(u,v)= E F

F G

,

E=hσu, σui, F =hσu, σvi, G=hσv, σvi

p MapDσ(u) : TuIR²→Tuσ⊂Tσ(u)IR³allows to transferIu also into a bilinear form onTuσ

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First Fundamental Form for Surfaces in 3D

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Transformation Properties of the First Fundamental Form

p The first fundamental form isinvariant under Euclidean transformations (including reflections) ofIR³:

Forψ:x7→Ax+b,A∈O(3,IR),b∈IR³, and˜σ:=ψ◦σone has

˜Iu(v,w) =Iu(v,w)

where˜Iu is first fundamental form of˜σ

p The first fundamental formtransforms under reparametrisationsas follows:

Letσ˜:=σ◦ϕ,ϕ: ˜D→D; then

˜Iu(v,w) =Iϕ(u)(Dϕ(v),Dϕ(w)) where˜Iu is first fundamental form of˜σ

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First Fundamental Form for Surfaces in 3D

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MeasurementsUsingtheFirstFundamentalForm

p Considersurfaceσ:D → IR³ andacurvec:[0,P] → D)onσ.Then

I(u(p),v(p))(cp(p), cp(p)) =c^T_pI(u(p),v(p))cp=E u²_p+ 2F upvp+G v_p² Lengthofc:

L[c] =

P

Z

0

kcp(p)kdp=

P

Z

0

q

E u²_p+ 2F upvp+G v_p² dp

p Anglebetween vectors in a point (here forσu, σv) cos <) (σu, σv) = hσu, σvi

kσuk kσvk= F

√EG

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First Fundamental Form for Surfaces in 3D

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Measurements Using the First Fundamental Form, cont.

p Areaof surfaceσ(D):

A[σ] = Z Z

D

q

detI(u,v)dudv= Z Z

D

pEG−F² dudv

p The bilinear form onT_(u,v)σdefined by the first fundamental form is a Riemannian metric on the surface (graph). It is obtained by restricting the Euclidean metric ofIR³ to the surface.

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Outlook and References

A

References

p G. Sapiro: Geometric Partial Differential Equations and Image Analysis.Cambridge University Press 2001

p W. Haack: Differential-Geometrie, Teil I.Wolfenbütteler Verlagsanstalt, Wolfenbüttel 1948 (in German)

14

c c _ss

n

c _s

t =

x

n

b

t c s ( )

c

t ₂

t ₁

n

n t

projection

κ n c