Curves and surfaces - sorbonne-universite · 2017-05-09 · Curves and surfaces: introduction...

Curves and surfaces

Chantal Oberson Ausoni

4.8.2014

ICS Summer school Roscoff - Visualization at the interfaces 28.7-8.8, 2014 1

Curves and surfaces: Outline1. CURVES AND SURFACES: INTRODUCTION

• Intuitive definitions• Curve and surface representation• Surface approximation and meshing

2. DIFFERENTIAL GEOMETRY

• Curves� Tangent to a curve� Arc-length, unit-speed parametrization� Curvature of a 2D-curve� Curvature of a 3D-curve

• Surfaces� Regular and “explicit” surface patches� The tangent plane� The first fundamental form� The second fundamental form� Curvatures� Theorema egregium� Gauss-Bonnet


Curves and surfaces: introduction

Curves: intuitive definitionWhat is a curve? Wikipedia: “In mathematics, a curve (also called a curved line in oldertexts) is, generally speaking, an object similar to a line but which is not required to bestraight.”

Most of the time, the curves are described by means of their Cartesian equation: forexample y � 2x = 1. So we can think of a 2D-curve as a set of points:C = {(x, y) 2 R2|F (x, y) = c}, for a real c and a function F : R2 �! R.For a 3D-curve, we need a pair of equations,

(F1(x, y, z) = c1F2(x, y, z) = c2

For example,(

x = 0y = 0

describes the line following the Oz axis.

We call such curves level curves.



Parametrized curvesThere is another very useful way to think about curves. A curve can be viewed as the pathtraced out by a moving point. The curve is then described by a mapping � of a parametert: �(t) is the position of the point at time t.

DEFINITION 1. A smooth parametrized curve is given by a smooth mapping � : I =]a, b[�! Rd.The image of � is called the associated geometric curve. Here, smooth meansthat � has continuous derivatives of all orders: �0, �00, . . . exist and are continuous.

EXAMPLE 1. The straight line through p in the direction v can be written �(t) = p+ tv andthus is a parameterized curve whenever v 6= 0.EXAMPLE 2. The circle about the origin of radius r can be written �(t) = (r cost, r sint).EXAMPLE 3. The helix can be written �(t) = (r cost, r sint, a t) for constants r and a.



Surfaces: intuitive definitionWhat is a surface? A set of points in R3 such that every point p on the surface has aneighborhood (perhaps very small) continuously deformable into a little flat open disk. Inother words, it looks like a piece of R2 in the vicinity of any given point.As an example, the surface of the Earth appears to be nearly flat to an observer on it.



Preliminaries: some topological conceptsDEFINITION 2. A set U ⇢ Rn is called open, if every point p 2 U is center of a openball also contained in U . An open ball in Rn of centre p and radius r is the set of pointsB(p, r) = {x 2 Rn | kx� pk < r}.

DEFINITION 3. A map f : X ⇢ Rn �! Y ⇢ Rm is said to be continuous if for every openset V ✓ Y , the inverse imagef�1(V ) = {x 2 X | f(x) 2 V } is an open subset of X.

It corresponds to the usual condition:

f is continuous in a 2 X if, for all ✏ > 0, there exists a � > 0 such that:kx� ak < � ) kf(x)� f(a)k < ✏.

DEFINITION 4. A map f : X ⇢ Rn �! Y ⇢ Rm, continuous, and having a continuousinverse, is called homeormophism. If such an homeomorphism exists between X and Y ,the are said homeomorphic.



Surface patches, surfaces, atlasDEFINITION 5. A surface patch is a smooth homeomorphism f : U ! f(U) ⇢ R3 definedon an open subset U of R2. Here smooth means that f has continuous partial derivativesof all orders.

The points (u, v) of U are called the parameters of f .

DEFINITION 6. A subset S ⇢ R3 is a surface if, for every point p 2 S, there is an open setW ⇢ R3, containing p, with S \W a surface patch. A collection of such surfaces patches,with images covering all S, is called an atlas of S.



Examples of surfacesEXAMPLE 4. The map f : R2 �!: R3, with f(u, v) = p+ u a+ v b and is a surface patchdescribing the plane through p in the directions a, b.EXAMPLE 5. The image of f : [0,2⇡[⇥R, defined by f(u, v) = (cos u, sin u, v), is acircular cylinder. Since : [0,2⇡[ is not open, we have to consider at least two surfacepatches to have an atlas of this surface. For example, f1 and f2, defined like f but on]0,2⇡[⇥R and ]� ⇡, ⇡[⇥R.EXAMPLE 6. For the sphere, the parametrizations f1 and f2 (for example) constitute an atlas:f1 :]0,2⇡[⇥]� ⇡/2,⇡/2[ is defined by f1(', ✓) = (cos ✓ cos', cos ✓ sin', sin ✓),f2 :]0,2⇡[⇥]�⇡/2,⇡/2[ is defined by f2(', ✓) = (�cos ✓ cos',�sin ✓,�cos ✓ sin').Both exclude the half of a big circle.

Illustrations by Pressley, Elementary Differential Geometry



Surfaces: implicit expressionLike for curves, it is sometimes possible to define a surface by an equation F (x, y, z) = c.For example, the unit sphere of example 3 is given by the equation x2+ y2+ z2 = 1 andthe cylinder of example 2 given by x2 + y2 = 1. For curves like for surfaces, it will becalled the implicit expression of a surface.

For a surface, if the equation is linear in variables x, y, z, it represents a plane (see the example 1); if it is of

second degree in the variables x, y, z, it represents quadrics. In general, the locus of zeros of a set

polynomials is known as an algebraic variety, and is studied in the branch of algebraic geometry.

More precisely, given a function F : R3 �! R, the preimage F�1(a) of a value a is asurface, under the condition that a is a regular value:

@F

@x(m),

@F

@y(m),

@F

@z(m)

!

6= (0,0,0),

i.e., the derivative is non-zero, for all m in F�1(a) (implicit function theorem).

Note that the unit sphere x2 + y2 + z2 = 1 is a surface, whereas the circular conedefined by x2 + y2 = z2 is not a surface.



Surface representation: summaryThere are mainly three ways to represent surfaces: parametric, implicit and explicitmethods.

1. In parametric representation the coordinates of a point (x, y, z) of the surface patchare expressed as smooth functions of the parameters u, v in a open rectangle:

x = x(u, v), y = y(u, v), z = z(u, v), u1 < u < u2, v1 < v < v2 ,

2. An implicit surface is defined as the locus of points whose coordinates (x, y, z) satisfyan equation of the form

F (x, y, z) = 0 .

3. If the implicit equation above can be solved for one of the variables, say z, as a functionof the other two x, y, we obtain an explicit surface:

z = h(x, y) .

In other words, the surface is the graph a function h : R2 �! R.

Given an implicit equation, it is always possible at least locally around a point m, when @F@z

(m) 6= 0

(implicit function theorem).



From implicit to explicit:PROPOSITION 1. Let F : R3 ! R be a class C1 function and ⌃ be the surface de-fined by F�1(a). Let p = (x, y, z) be a point in ⌃ and such that the affine map-ping (dF )(p) : R3 ! R is surjective. Then, there exists locally, around p, an explicitparametrization through a graph f : (x, y) 7! (x, y, h(x, y)).

More precisely, there exists an open set U ⇢ R2, containing p, and a class C1 function h

defined on U such that

(x, y) 2 U and F (x, y, z) = a , z = h(x, y) .

This result is a variant of the implicit function theorem.



Curve and surface representationGeometry Parametric Implicit Explicit

plane curves⇢

x = x(t)y = y(t) F�1(0) y = f(x)

t1 < t < t2 F : R2 �! R

space curves

8<

:

x = x(t)y = y(t)z = z(t)

F�1(0) \G�1(0) y = Y (x) \ z = Z(x)

t1 < t < t2 F,G : R3 �! R

surfaces

8<

:

x = x(u, x)y = y(u, v)z = z(u, v)

F�1(0) z = F (x, y)

u1 < u < u2, v1 < v < v2 F : R3 �! R

Table 1: Representation of curves and surfaces.



ExampleEXAMPLE 7. Consider the unit circle curve for example.

• The parametrization �(t) = (cos t, sin t), t 2 R describes this curve.

• It is also possible to define the curve as F�1(1), with F (x, y) = x2 + y2 (implicitmethod).

• It is not possible to define the circle as y = f(x), since every x 2]� 1,1[ correspond

to two points y =q1� x2 and y = �

q1� x2.

Given a parametrization f(u, v) = (0,1,0) + u (1,0,1) + v (1,0,0) of a plane, thecomposition f � � parametrizes a curve on this plane, which is an affine transform of acircle, more precisely an ellipse.



Surface approximation (meshing)

The parametric or implicit representation of a surface is convenient for defining the surfaceas a mathematical object, for modeling and manipulation, but it is not very convenient forhandling the surface by computer.

Position of the problem:

• For drawing or displaying a surface or for engineering computations (FE analysis), adiscrete representation as a union of polygons is easier to handle.

• Meshing is the process of computing a representation consisting of pieces of simplesurface patches. In the easiest case, the result will be a triangulated polygonal surface.

• Sometimes, a surface is already given as a mesh, but one wants to construct a different,”better” mesh. This is known as remeshing, refinement or decimation problems.

• Meshing is also related to surface reconstruction.



Surface approximation (meshing)DEFINITION 7. An isotopy between two surfaces ⌃,⌃0 ⇢ R3 is a continuous mapping� : ⌃⇥ [0,1] ! R3 which, for any fixed t ✓ [0,1], is a homeomorphism �(·, t) from ⌃onto its image and which continuously deforms ⌃ into the mesh ⌃0 : �(⌃,1) = ⌃0.In addition, the approximation error ✏ is the largest distance by which a point is moved bythis homeomorphism:

kx� �(x,1)k ✏ , for all x 2 ⌃ .

The two surfaces are homeomorphic to each other and their Hausdorff distance is atmost ✏.

This definition brings topological correctness and geometrical closeness to the originalsurface. In addition, other criteria may be introduced:

• Normals: the normals of the mesh should not deviate too much from the normals ofthe surface;

• Smoothness: adjacent faces should not form a sharp angle;

• Density: impose an upper bound on the size of the mesh triangles.

• Shape: control element skewness (quality).


Differential geometry primer

CurvesOur goal is to examine the parametric representation of curves and to define a measure ofthe extent to which a curve is not contained in a straight line: the curvature. We will alsodefine a measure of the extent to which a curve is not contained in a plane: the torsion.

DEFINITION 8. Two parametrized curves f : I ! R3 and g : J ! R3 represent the samecurve if there exists a diffeomorphism � : J ! I such that g = f � �, called a parameterchange.

• The mapping � preserves the orientation if �0(t) > 0.

• The variable t is called the parameter of the curve;

• A point t0 2 I is regular if f 0(t0) 6= 0 and singular in the other cases. A paremetrizedcurve is regular if the mapping f is an immersion, i.e., if f 0(t) 6= 0 for every t 2 I.

• At each regular point f(t0), the vector f 0(t0) is tangent to the curve. The affine linepassing by f(t0) and directed by f 0(t0) is called the tangent to the parametrized curveat the point f(t0).



The tangent to a curve

• This definition can be interpreted in a geometric fashion, by thinking of a particlemoving along the trajectory (the path) f , its position at time t corresponding to f(t).The velocity of the particle at time t is then given by

f 0(t) =df

dt(t) = lim

�t!0

f(t+�t)� f(t)

�t.

Figure 1: Tangent at a point to the curve ↵ and velocity vector [O’Neill].



Position of a planar curve f with respect to its tangent.Using a Taylor formula, we write

f(t) = f(t0) + (t� t0)f0(t0) +

1

2(t� t0)

2f 00(t0) + o((t� t0)2) .

• If f 0(t0) and f 00(t0) are two independent vectors, the curve is located on only oneside of its tangent, in the vicinity of t0; the curve is contained in the half space definedby the tangent and containing f 00(t0).

• On the other hand, if f 0(t0) and f 00(t0) are colinear and if, for example, f 000(t0) isindependent of f 0(t0), we can pursue the development

f(t) = f(t0)+(t�t0)f0(t0)+

1

2(t�t0)

2f 00(t0)+1

6(t�t0)

3f 000(t0)+o((t�t0)3) ,

such that the curve intersects its tangent, and presents an inflection point.



Metric properties of curves.We know that the direction of the tangent at a given point to a parametrized curve isgeometric, the tangent vector however, depends on the given parametrization. It is thisinteresting to find a parametrization that gives a unit tangent vector at every point.

Length of an arc of a curveLet f : I �! R3 a parametrized curve. If we want to calculate the length of a part of thiscurve, we can divite it into segments, each of which corresponding to a small increment �tin t, calculate the length of each linear segment and add up the results. Letting �t tend tozero should give the exact length⇤.

Note that the length of a small segment between f(t) and f(t+ �t), kf(t+ �t)� f(t)k,is, when �t is small, nearly equal to kf 0(t)k · �t.DEFINITION 9. The arc-length of the curve f starting at f(t0) is the diffeomorphism ' givenby

t 7! '(t) =Z t

t0kf 0(u)k du . (1)

Its derivate is given by d'dt (t) = d

dt

R tt0kf 0(u)k du = kf 0(t)k.

⇤Curves for which this limit exists are called rectifiable.



Arc-lengthThe name arc-length for ' is also used for the corresponding parameter s and is justified bythe fact that

s2 � s1 = '(t2)� '(t1) =Z t2

t1kf 0(t)k dt ,

which corresponds to the definition of the length of the arc between f(t1) and f(t2).

The function ' allows to define a change of parametrization using the arc length. Themapping g = f � '�1 is another parametrization of the same curve that verifies theproperty (proved using the chain rule)

kg0(s)k = 1 , 8 s 2 J .

Such parametrizations are called unit-speed parametrizations.

In summary,

• The length of the arc of curve between f(t1) and f(t2) is independent of the parametriza-tion of this arc.

• For every regular parametrized curve, there is a unit-speed parametrization, it is ob-tained through any given parametrization and the arc-lengh diffeomorphism associatedto it.



Curvature

Let f : I ! R3 be a unit-speed parametrization of a curve.

DEFINITION 10. The scalar function , defined on I by

(s) = kf 00(s)k

is called the curvature function of f and k(s) is null if and only if f(s) is an inflectionpoint.

• The vector function t(s) = f 0(s) is called the unit tangent vector function to f .

• At every point where (s) 6= 0, the vector f 00(s)/(s) is:

� a unit vector

� orthogonal to t(s). (The relation kf 0(s)k2 = 1 implies by derivation that hf 0(s), f 00(s)i = 0.)

We can define the normal vector n(s) as the unique unit vector such that {t(s), n(s)}is an orthonormal direct basis of R2.



Curvature of a 2D-curve, Frénet formulasDEFINITION 11. The scalar number k(s) such that

t0(s) = k(s)n(s) . (2)

is called the algebraic curvature of the curve at the point corresponding to the parameter s.

Figure 2: The sign of the algebraic curvature [Pressley].

We get also the formula

n0(s) = �k(s)t(s) (3)

both relations (2) and (3) are called the Frénet formulas. They express the derivatives oft(s) and n(s) in the basis {t(s), n(s)}.ICS Summer school Roscoff - Visualization at the interfaces 28.7-8.8, 2014 22


Osculating circleLet f be a unit-speed parametrization of our curve. We can draw in every point a circle

• of radius ⇢(s) = 1/(s) (the radius of curvature)

• of centre C(s) = f(s) + ⇢(s) sign(k(s))n(s) (the centre of curvature).

This circle is said osculating circle to the curve f , it approaches the curve at best at f(s).

It means that if �s is a unit-speed parametrization of this circle

�s(s) = f(s), �0s(s) = f 0(s) and �00s (s) = f 00(s).

In other words, f and �s agree up to derivatives of order 2.

Figure 3: Osculating circle.



Curvature of a 3D-curveDEFINITION 12. The plane defined by t(s) and n(s) is called the osculating plane to thecurve f at s. We define the binormal vector at s:

b(s) = t(s)⇥ n(s).

The frame (f(s); t(s), n(s), b(s)) is a direct orthonormal frame of R3. The three orthog-onal unit vectors (t(s), n(s), b(s)) form the Serret-Frénet trihedron at s .

Figure 4: Local Frénet trihedron in dimension three and in dimension two [doCarmo].



Curvature and torsion of a 3D-curveWe can show that the vector b0(s) is colinear to n(s), and thus

b0(s) = ⌧(s)n(s)

where the scalar function ⌧(s) is called the torsion of the curve f at s.

We have the Frénet formulas

t0(s) = k(s)n(s) (4)

n0(s) = �k(s) t(s)� ⌧(s) b(s) (5)

b0(s) = ⌧(s)n(s) (6)

The derivatives t0(s) = k(s)n(s) and b0(s) = ⌧(s)n(s) expressed in the basis(t(s), n(s), b(s)) give the curvature k(s) and the torsion ⌧(s), delivering information onthe local behavior of the curve f in the vicinity of s: ⌧(s) measures the curve’s attempt totwist out of the plane in which it finds itself momentarily trapped.



Regular surface patchTwo surface patches (f, U) and (g, V ) are equivalent if there exists a smoothdiffeomorphism ' : V ! U such that g = f � '.DEFINITION 13. A surface patch defined by a parametrization f : U ! R3 is called regularpatch if the differential of f at each point of U has rank 2 (i.e., it is the rank of the Jacobianmatrix of f ).

The Jacobian matrix of the parametrization f is the matrix function

(u, v) 7! Jf(u, v) :=

0

@@f1

@u(u, v) @f1

@v(u, v)

@f2

@u(u, v) @f2

@v(u, v)

@f3

@u(u, v) @f3

@v(u, v)

1

A .

DEFINITION 14. A subset S ⇢ R3 is a regular surface if, for every point p 2 S, there is anopen set W ⇢ R3, containing p, with S \W a regular surface patch.



Explicit surface patchesAmong all possible representations of a surface patch, we can consider an explicitCartesian parametrization

f : (x, y) 7! (x, y, h(x, y)).

The surface patch is the graph of the function h. The rank of the Jacobian is 2, implying theregularity of the surface patch. It has the following ramarkable property: the subspacedf(m)(R2) spanned by @f(m)/@x and @f(m)/@y never contains the z-axis.

Figure 5: Graph of a function of two variables.


Differential geometry primerThe tangent plane

Let f be a regular surface patch. Since the Jacobian matrix of f in m has rank 2, the twovectors @f(m)/@u and @f(m)/@v generate a 2-dimensional subspace df(m)(R2). Thisplane is independent of the chosen parametrization. We define the tangent plane to ⌃ atthe point p = f(m) by

Tp⌃ = df(m)(R2) .

This is a affine vector plane of R3 containing p. It contains the tangent vectors to all thecurves in the surface patch.

Example: Consider the saddle z = y2 � x2 and thepoint p = (1,2,3) on this saddle. The saddle canbe parameterized by f(u, v) = (u, v, v2 � u2) andp = f(1,2).The basis tangent vectors are @f

@u = (1,0,�2u), @f@v =(0,1,2v). At the point p these vectors are (1,0,�2)and (0,1,4).A tangent vector has the form X = (X1, X2,�2X1+4X2) and corresponds to the local coordinate vectorX = (X1, X2) in the basis given by (1,0,�2) and(0,1,4).ICS Summer school Roscoff - Visualization at the interfaces 28.7-8.8, 2014 28


Relative position of tangent plane and surfaceLet ⌃ be a surface, p 2 ⌃, f an explicit parametrization through a graph aroundp = f(m). If we write

p =@f

@x(m) q =

@f

@y(m)

we observe that the equation of the tangent plane at p is z = px+ qy, and if

r =@2f

@x2(m) s =

@2f

@x@y(m) t =

@2f

@y2(m)

then, with a Taylor expansion, in the neighborhood of m we have

h(x, y) = px+ qy +1

2(rx2 + 2sxy + ty2) + o(x2 + y2) ,

in which the quadratic term represents the gap between a point (x, y, h(x, y)) of ⌃ andthe corresponding point (x, y, px+ qy) in the tangent plane.



Digression on quadratic forms and symmetric bilinear formsA symmetric bilinear form B on R2 is a bilinear mapping B : R2 ⇥ R2 �! R, linear inboth variables, and symmetric: B(X,Y ) = B(Y,X). So it is like the giving of asymmetric matrix A such that

B((x1, x2), (y1, y2)) =⇣x1 x2

⌘A

y1y2

!

.

A quadratic form over R2 is the giving of a function q : R2 �! R with

q(x, y) =⇣x y

⌘A

xy

!

,

where A is a symmetric 2⇥ 2-matrix.

• if q(x, y) > 0, for all (x, y) 6= (0,0) 2 R2, we say that q is positive definite.

• if q(x, y) < 0, for all (x, y) 6= (0,0) 2 R2, we say that q is negative definite.

• q is positive or negative definite iff det(A) > 0.



Particular points on the surfaceHence, the position of the surface with respect to its tangent plane is described by thequadratic form

Q(x, y) = rx2 + 2sxy + ty2.

The sign of Q(x, y) = h(x, y)� px� py tells on wich side of the tangent plane lies thesurface �.

1. if Q is definite (positive or negative), that is to say if rt � s2 > 0, the surface lies onone side, the point p is said to be elliptic;

2. if Q is nondegenerate and its sign changes, that is to say if rt � s2 < 0, the point p issaid to be hyperbolic;

3. if Q is degenerate but nonzero, that is to say if rt � s2 = 0 (r, s, t not all zero), thepoint p is said to be parabolic;

4. if Q is zero (with r, s, t zeroes), the point is said to be planar.



Examples of elliptic, hyperbolic, parabolic, planar points

Figure 6: Position of a surface with respect to its tangent plane (elliptic, hyperbolic, parabolic and planarpoints).

Figure 7: Example of a torus.



Length of a curve on a surfaceLike before, let f be a regular surface patch, p = f(m):

• a vector w of the tangent plane Tp⌃, p, can be written as

w = a@f

@u(m) + b

@f

@v(m)

and its Euclidean norm is

kwk2 = a2��@f

@u(m)

��2+ 2ab

⌧@f

@u(m),

@f

@v(m)

�+ b2

��@f

@v(m)

��2.

• if t 7! c(t) = (u(t), v(t)) is a curve traced in the domain of parameters, the lengthof the corresponding curve 7! f(u(t), v(t)) traced on the surface is

L(f � c) =Z t2

t1l(t) dt

where l(t) =

ru0(t)2

��@f@u(c(t))��2+ 2u0(t)v0(t)

D@f@u(c(t)), @f

@v(c(t))

E+ v0(t)2

��@f@v(c(t))��2.



The first fundamental formThis equation giving the length of a curve on a surface can be simplified by the definition,in every point of the surface, of a symmetric bilinear form on the tangent space, called firstfundamental form. It is induced canonically from the Euclidean scalar product product ofR3.

Like before, let f : U �! R3 be a regular surface patch, p = f(m): Let us define on U ,

• E(u, v) =��@f@u(u, v)

��2

• F (u, v) =D@f@u(u, v),

@f@v(u, v)

E,

• G(u, v) =��@f@v(u, v)

��2

The first fundamental form is the giving in every (u, v) 2 U of the symmetric bilinear form

Qu,v(X,Y ) =⇣X1 X2

⌘· E(u, v) F (u, v)F (u, v) G(u, v)

!

· Y1Y2

!

.



Properties of the first fundamental form1. The functions E, F and G depend on the parametrization f . The quadratic form does

not depend on the parametrization, since if two parametrizations are related through adiffeomorphism ', the corresponding matrices are related through:

E0(u, v) F 0(u, v)F 0(u, v) G0(u, v)

!

= JT'

E(u, v) F (u, v)F (u, v) G(u, v)

!

J',

meaning that they are similar.

2. The area of a domain ⌦ traced on a surface patch f : U ! R3 is given by the integral

A(⌦) =ZZ

⌦

��@f

@u^

@f

@v

�� dudv =ZZ

⌦

qEG� F2dudv ,

and is thus independent of the parametrization.



Example: a surface of revolutionEXAMPLE 8. A surface of revolution

Surface of revolution([Pressley]).

A surface of revolution is the surface ob-tained by rotating a plane curve, calledthe profile curve, around a straight linein the plane, here Oz. If (f, g) isa unit-speed parametrization of the pro-file curve, k(f 0(s))2 + (g0(s))2k = 1. Theparametrization of the surface can be written

r(u, v) = (f(u)cosv, f(u)sinv, g(u)).

Then,

@r

@u= (f 0(u)cosv, f 0(u)sinv, g0(u))

@r

@v= (�f(u)sinv, f(u)cosv,0)

and the first fundamental form is given by the matrix 1 00 f(u)2

!

.



Second fundamental formLike before, let f be a regular surface patch. We introduce the unit normal vector n(p) asthe unit vector of direction

@f

@u⇥

@f

@v.

If the surface lies in a plane, this normal vector is constant. Consequently, curvature isrelated to changes in n; we can study these changes by differentiation.DEFINITION 15. For every vectors X,Y of the tangent plane Tp⌃ and for every point p of⌃, we define the symmetric bilinear form

IIp(X,Y ) = �hTpn(X), Y i ,

where Tpn is the linear tangent mapping to n. It is called the second fundamental form ofthe surface ⌃ at point p and can be written with a symmetric matrix:

IIp(X,Y ) =⇣X1 X2

⌘ L MM N

! Y1Y2

!


Differential geometry primerComputation and example of the surface of revolution

The coefficients L,M,N can be computed using the equations

L = fuu · nM = fuv · nN = fvv · n

EXAMPLE 9. Surface of revolution

@2r

@2u= (f 00(u)cosv, f 00(u)sinv, g00(u))

@2r

@u@v= (�f 0(u)sinv, f 0(u)cosv,0)

@2r

@v@v= (�f(u)cosv,�f(u)sinv,0)

so that the second fundamental form is given by the matrix f 0(u) g00(u)� f 00(u) g0(u)

0 f(u) g0(u)

!

.

For the unit sphere S2, generated by the profile curve f(✓) = cos✓, g(✓) = sin✓, we

have: sin2✓ + cos2✓

0 cos2✓

!

=

1 00 cos2✓

!



Curvature of curves on surfacesConsider a unit vector X 2 Tp⌃ and let ⇡X be the plane spanned by X and n(p). Theintersection of ⇡X with ⌃ is a (planar) curve traced on ⌃ and X is tangent to it. Inparticular, we can define its algebraic curvature KX . The mapping X 7! KX associateswith every vector X 2 Tp⌃ a scalar value, which is given by IIp(X,X).THEOREM 1 (Euler). If all the KX are not equal, there exists a unique direction, representedby a vector X1 (resp.X2), for which KX is minimal (resp. maximal).DEFINITION 16. The directions of X1 and X2 are called the principal curvature directionsat point p. The curvatures KX1

and KX2are the principal curvatures.

Figure 8: Principal directions and curvatures.



Gaussian and mean curvaturesDEFINITION 17. The Gauss curvature at p is the determinant (resp. the trace) of �Tpn andcorresponds to the product (resp. to the sum) of the principal curvatures.DEFINITION 18. The mean curvature at p is half the trace of �Tpn and corresponds to themean of the principal curvatures.DEFINITION 19. Minimal surfaces are surfaces for which the mean curvature is zero every-where.COROLLARY 1. The point p is elliptic if and only if K(p) > 0, and is hyperbolic if and onlyif K(p) < 0.

Figure 9: Characterization of a point p of ⌃ using the sign of Gauss curvature K(p), [Audin].



Theorema egregiumIf ⌃1 and ⌃2 are surfaces patches, a smooth map f : ⌃1 �! ⌃2 is called a localisometry if it takes any curve in ⌃1 to a curve of the same length in ⌃2. If a local isometryf : ⌃1 �! ⌃2 exists, we say that ⌃1 and ⌃2 are locally isometric.THEOREM 2 (Theorema egregium of Gauss). The Gauss curvature of a surface is invariantunder local isometry.

For example, a piece of paper (K = 0) cannot be bent into a sphere (K = 1/R2)without crumpling.

The theorem is sometimes expressed by saying that the Gaussian curvature is an intrinsicproperty of a surface, for it implies that the Gaussian curvature could be measured by abug living in the surface.LEMMA 1. The Gauss curvature can be expressed in coordinates by the expression

K(u, v) =LN �M2

EG� F2 .

EXAMPLE 10. Example of the surface of revolution:

K(u, v) =�f 00(u)

f(u)



Gauss-Bonnet theoremThe Gauss-Bonnet theorem relates geometrical properties (curvature) with topological ones(Euler number). A simplified version states:THEOREM 3 (Gauss-Bonnet theorem). Let R be a simply-connected region on a surface,whose boundary consists of a finite number of geodesic curves meeting at corners. Then

X

corners✓i +

Z

RKdA = 2⇡.

EXAMPLE 11. Triangle on a sphere

Take the 3D-sphere of radius R. Let us consider the triangle onthe figure as our region. The sum of angles is 3 ⇡

2. The integral

of the Gaussian curvature is 1R2 ·Area(triangle) = 1

R2 · 4⇡R2

8 .Both contributions hence sum to 2⇡.



Gauss-Bonnet theorem: a more general versionTHEOREM 4 (Gauss-Bonnet theorem). Suppose S is a compact two-dimensional Riemannianmanifold with boundary @S. Then

Z

MK dA+

Z

@Mkg ds = 2⇡ �(S)

• K is the Gaussian curvature of S

• kg is the geodesic curvature of @S (can be understood as the curvature of this curve inthe surface)

• �(S) is the Euler characteristic of S: The Euler characteristic of a closed orientablesurface can be calculated from the number of "handles" g as � = 2� 2g.


Curves and surfaces - sorbonne-universite · 2017-05-09 · Curves and surfaces: introduction...

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