strain energy of bézier surfaces - diva portal

Strain Energy of Bézier Surfaces

Department of Mathematics, Linköping University

Erik Bråmå

LiTH-MAT-Ex-2017/16-SE

Credits: 16 hp

Level: G2

Supervisor: Milagros Izquierdo,Department of Mathematics, Linköping University

Examiner: Göran Bergqvist,Department of Mathematics, Linköping University

Linköping: March 2018

Abstract

Bézier curves and surfaces are used to great success in computer-aided designand nite element modelling, among other things, due to their tendency of beingmathematically convenient to use. This thesis explores the dierent propertiesthat make Bézier surfaces the strong tool that it is. This requires a closerlook at Bernstein polynomials and the de Castiljau algorithm. To illustratesome of these properties, the strain energy of a Bézier surface is calculated.This demands an understanding of what a surface is, which is why this thesisalso covers some elementary theory regarding parametrized curves and surfacegeometry, including the rst and second fundamental forms.

Keywords:

Surface geometry, Bézier curves, Bézier surfaces.

URL for electronic version:

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-145645

Bråmå, 2018. iii

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-145645

Acknowledgements

I would like to express my gratitude to my supervisor, Professor Milagros

Izquierdo, for her guidance and support. Since I had no real experience inthe eld of dierential geometry before working with this thesis, her help hasmeant a great deal to me, espeacially regarding the direction of the thesis andthe choice of litterature. I also want to thank her for her availability through-out the entirety of this work. Furthermore, I would like to thank my examiner,Professor Göran Bergqvist, for his help, espeacially with the administrative as-pect of this thesis. Lastly, I want to thank my opponent, Mari Ahlquist, for herhelpful and insightful critisism concerning the layout of the thesis.

Bråmå, 2018. v

Nomenclature

S surface

γ parametrized curve

N standard unit normal

n normal of a curve

S2 the unit sphere

T tangent plane

fxipartial derivative of a function with respect to xi

G the Gauss map

W the Weingarten map

〈 , 〉 symmetric bilinear form

I the rst fundamental form

II the second fundamental form

B Bézier curve

X Bézier surface

Bni Bernstein polynomial

b control point

× vector product

· dot product

‖ , ‖ length of a vector

Bråmå, 2018. vii

Contents

1 Introduction 1

2 Surface Geometry 3

2.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Parametrization of a Surface . . . . . . . . . . . . . . . . . . . . 62.3 The First Fundamental Form . . . . . . . . . . . . . . . . . . . . 102.4 The Second Fundamental Form . . . . . . . . . . . . . . . . . . . 13

3 Bézier Surfaces 21

3.1 Bézier Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Bézier Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Results and Future Work 31

4.1 Strain Energy of a Bézier Surface . . . . . . . . . . . . . . . . . . 314.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Bibliography 37

A Fundamental Concepts 39

B Matlab 41

Bråmå, 2018. ix

List of Figures

4.1 Strain energy surface of Bézier surface with parameter valuesa = 0.1 and w = 200 . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Strain energy surface of Bézier surface with parameter valuesa = 0.2 and w = 20 . . . . . . . . . . . . . . . . . . . . . . . . . . 32



Bråmå, 2018. xi

List of Tables

4.1 Strain energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Bråmå, 2018. xiii

Chapter 1

Introduction

Bézier curves and surfaces are named after the French engineer Pierre Bézier,who is one of the founders of the eld of geometric modelling. He worked atRenault where he used them to design car bodies. There, he developed theCAD/CAM system known as UNISURF. A good account of Bézier's work canbe found in Pierre Bézier: An engineer and a mathematician of Pierre-JeanLaurent and Paul Sablonnière ([Laur-Sabl]).

Bézier surfaces have a wide range of applications in the eld of CAD andcomputer modelling because they possess certain algorithmic properties whichenables ecient methods for rendering and analyzing shapes. To explain why,this thesis covers the fundmamentals of dierential geometry, including the the-ory of surface geometry. We follow the material in Elementary Dierential Ge-

ometry by Andrew Pressley and Dierential Geometry of Curves and Surfaces

by Manfredo P. do Carmo.In surface geometry we regard every part, or patch, of a surface as being

equivalent to an open subset of a plane. One way to look at it is that thesurface is the shape the plane takes as it is being deformed by bending andstretching. We say that they are homeomorphic and that the map betweenthem is a homeomorphism. This map is also called the parametrization of thepatch. By studying the behavior of curves going through a point of a surface,we can dene local metric properties of the surface at this point. This is doneby dening a vector space, which we call the tangent plane, which is spannedby tangent vectors of the curves going through this point. The rst and second

fundamental forms are symmetric bilinear forms dened on the tangent plane.Together, they determine the metric properties of the surface. The inverse

function theorem allows the neighborhood of every point of a surface to beparametrized as a Monge patch. This enables the strain energy of the surface

Bråmå, 2018. 1

2 Chapter 1. Introduction

at this point to be expressed in terms of the fundamental forms. Since they areindependant of the choice of parametrization, the strain energy can be calculatedfor any given parametrization.

A Bézier curve is dened by its control polygon and the Bernstein basis poly-

nomials. The control polygon is a set of points and the Bézier curve is containedwithin the convex hull of these points. Thus, given a control polygon, there isexactly one corresponding Bézier curve. The Bernstein basis polynomials inducecertain practical properties, such as the de Castiljau algorithm which enablesevery point of the Bézier curve to be recursively dened by the points of its con-trol polygon. Not only that, but the derivatives are recursively dened as well.A Bézier surface, which can be seen as the product of two Bézier curves, alsohas these properties. By extension, this means that the fundamental forms of aBézier surface are given by the corresponding control polygon. This simpliesthe process of calculating the strain energy, among other things, considerablyand is one of the reasons why Bézier surfaces are so prevalent in technical appli-cations such as CAD computer-aided design. Chapter 3, of which the majorityis based on the material presented in Applied Geometry for Computer Graphics

and CAD by Duncan Marsh, is dedicated to Bézier curves and surfaces. A briefaccount can also be found in Bonneau's dissertation.

This thesis covers the calculation of the strain energy of a Bézier surfaceand uses the results found in the dissertation in Variational Design of Rational

Bézier Curves and Surfaces by Georges-Pierre Bonneau. Strain energy is acentral concept of the design and production of many technical implementations,such as cars and airplanes. In chapter 4 a comparison of the strain energy of fourdierent Bézier surfaces is presented, mainly for the purpose of giving the readera glimpse of the vast possibilities this mathematical tool has to oer. A shortintroduction to future work and possible continuations of this thesis concludesthis chapter and can be further studied in the dissertation of Bonneau.

Chapter 2

Surface Geometry

Before introducing the concept of a Bézier surface we must rst have a rmunderstanding of surface geometry. Loosely speaking, a surface is a subset ofR3 where the local environment, which we call a patch, of every point of thesurface appears to be a part of a plane. Indeed, we will regard every such patchas being essentially equivalent to a part of a plane, allowing the surface to locallybe parametrized by an open subset of R2. The surface will then be the unionof all these patches, in a way resembling a patchwork quilt. A helpful analogywould perhaps be the one where the open subsets of R2 are referred to as chartsof the surface. This collection of charts that together make up the whole surfacewould then be called the atlas of that surface, much like an atlas of the surfaceof the Earth, containing charts of dierent countries and continents.

By studying the properties of curves going through a point on a surface, wecan study the local properties of that surface in a small neighborhood of thatpoint. With every point of the surface belonging to at least one patch, curvesgoing through this point can be expressed in terms of the parametrization of theassociated patch. This will enable the dening of two important properties ofthe surface, namely the rst and second fundamental forms, which are essentialto the process of calculating the strain energy of a surface.

In section 2.1 we will start by looking at some basic properties of parametrizedcurves after which section 2.2 covers the parametrization of a surface. This willbe followed by two sections about the rst and second fundamental forms. In sec-tions 2.1 through 2.4 we follow the material presented [Pressley] and [Do Carmo].An account of the notion of strain energy at the end of section 2.4 can be foundin [Bonneau].

Bråmå, 2018. 3

4 Chapter 2. Surface Geometry

2.1 Curves

In this section we cover only what is required in order for the reader to be able tofollow the material in section 2.2. For a more comprehensive account of curves,see the introductory chapters of [Do Carmo] or [Pressley].

Denition 2.1.1 (Parametrized curve). A parametrized curve γ is a map γ :(α, β)→ Rn, for some open interval (α, β). We write γ(t) = (γ1(t), γ2(t), . . . , γn(t)),t ∈ (α, β), and dene the following properties:

• The tangent vector of γ at the point γ(t) is given by

γ(t) = (γ1(t), γ2(t), . . . , γn(t)).

• γ(t) is a regular point of γ if γ(t) 6= 0. We say that γ is a regular curve

if all its points are regular.

• The arc-length of γ starting at the point γ(t0) is the function s(t) givenby

s(t) =

∫ t

t0

‖γ(u)‖ du.

• The speed of γ at the point γ(t) is ‖γ(t)‖, and γ is said to be a unit-speedcurve if ‖γ(t)‖ = 1 for all t ∈ (α, β).

• The curvature at a point p of a regular curve is the magnitude of thedeviation of γ from the tangent line, i.e. the line through p parallel to thetangent vector. Denoted κ, the curvature of γ is given by

κ =‖γ × γ‖‖γ‖3

(2.1)

or, in the case of a unit-speed curve, by κ = ‖γ‖.

Example 2.1.2 (Regular and unit-speed parametrizations). The upper halfof the unit circle in the xy-plane in R3, with the x-axis excluded, is givenby y =

√1− x2, where x ∈ (−1, 1) and z = 0. Let γ1 and γ2 be dierent

parametrizations of this curve, such thatγ1(θ) = (cos θ, sin θ, 0), θ ∈ (0, π)

γ2(t) = (t,√

1− t2, 0), t ∈ (−1, 1).

The respective tangent vector is then given byγ1(θ) = (− sin θ, cos θ, 0), θ ∈ (0, π)

γ2(t) =(

1, −t√1−t2 , 0

), t ∈ (−1, 1)

,

2.1. Curves 5

where ‖γ1‖ = 1 ∀ θ ∈ (0, π) and ‖γ2‖ = 1√1−t2 . We conclude that both

parametrizations are regular and that γ1 is unit-speed whereas γ2 is not. Sincethe curvature is a property of the curve itself, one might suspect that it isindependent of the choice of parametrization so long as it is regular. Indeed,

γ1(θ) = (− cos θ,− sin θ, 0), θ ∈ (0, π)

γ2(t) =(

0, −1(1−t2)3/2 , 0

), t ∈ (−1, 1)

,

and by equation (2.1) we have that

κ1 = ‖γ‖ = 1

κ2 =‖γ2 × γ2‖‖γ2‖3

=‖(0, 0, 1

(1−t2)3/2 )‖‖(1, −t√

1−t2 , 0)‖3= 1,

and thus κ1 = κ2.

The dot notation is frequently used in physics to indicate the derivative withrespect to time, and has a natural interpretation. If we imagine a curve beingthe trace of a particle moving through space, then the parametrization would tellus the speed, among other things, of the particle at every point in time. Theexpression "unit-speed" can then be interpreted as the speed of the particlebeing constantly 1. It turns out there is a way to always nd a unit-speedparametrization of a regular curve.

Proposition 2.1.3. A parametrized curve γ has a unit-speed reparametrizationif and only if it is regular. If γ has a unit-speed reparametrization γ, then it isessentially parametrized by its arc-length s, i.e. γ(s(t)) = γ(t).

Thus, proposition 2.1.3 enables the assumption that every regular parametrizedcurve is unit-speed. Now, consider a point p and a curve γ going through thispoint. Since we have assumed that γ is unit-speed, we know that

γ · γ = 1. (2.2)

Dierentiating (2.2) with respect to t yields

γ · γ + γ · γ = 0 ⇐⇒ γ · γ = 0.

Hence, γ and γ are orthogonal unit vectors. If we let t = γ, n =1

κγ and

b = t × n, where κ = ‖γ‖, then t,n,b form an orthonormal basis of R3

called the Frenet trihedron. Here, t is the tangent vector, n is the principal


normal vector and b is the binormal vector of γ at p. The planes spanned byt,b, n,b and t,n are called the rectifying, normal and osculating plane,respectively. Furthermore, the equations

t = κn

b = τn

n = −τb− κt

are called the Frenet formulas, where κ is called the curvature and τ is calledthe torsion. Thus, κ and τ measure the deviation of γ from its rectifying andosculating plane, respectively, at the point p. In other words, a curve with nocurvature is a straight line and a curve which is completely contained within aplane has no torsion. Together, the curvature and torsion completely determinesthe local properties of a curve up to an isometry of R3.

2.2 Parametrization of a Surface

In the introduction of this chapter we used the phrase "essentially equivalent"when referring to the relation between the surface patch and the associated opensubset of R2, but we shall need to be more specic. The following denitionuses the terms in the introductory analogy and can be found in [Pressley].

Denition 2.2.1 (Surface). A subset S of R3 is a surface if, for every pointp ∈ S, there is an open set U ⊆ R2 and an open set W in R3 containing p suchthat U is homeomorphic to V = W ∩ S. If σ : U → V is a homeomorphism,we say that (U,σ) is a chart of S such that σ(u0, v0) = p, (u, v) ∈ U, and σ isthe parametrization of V.

For the remainder of this thesis we will assume that every σ is a smooth

map, see A.0.1. A denition of homeomorphism can be found in A.0.2.

Remark 2.2.2. In most cases we will only write σ when actually referring toa chart (U,σ) of a surface S. It is then implied that σ is a chart of S with thedomain U such that σ parametrizes a patch V of S.

As mentioned earlier, one way to dene the local properties of a surface S isto study the behaviour of curves in S passing through a point p ∈ S. Let γ besuch a curve. We will dene a tangent vector of S at p to be the tangent vectorof γ at p. Of course, there is an innite amount of curves passing through p,which leads us to dening the tangent plane of S at p to be the set of all tangentvectors of S at p.

2.2. Parametrization of a Surface 7

Denition 2.2.3 (Tangent vector and tangent plane). A tangent vector v toa surface S at a point p ∈ S is the tangent vector at p of a curve in S passingthrough p. The tangent plane TpS of S at p is the vector space consisting ofall tangent vectors to S at p. Let σ : U→ V be a parametrization of an openset V on a surface S containing a point p ∈ S, and let (u, v) be coordinates inU. The tangent plane of S at p is the vector subspace of R3 spanned by thevectors σu and σv (the derivatives are evaluated at the point (u0, v0) ∈ U suchthat σ(u0, v0) = p).

If σu,σv is a basis of the tangent plane TpS and γ is a curve through p,then γ(t) = σ(u(t), v(t)) and, by the use of the chain rule, the tangent vectorγ can then be expressed in terms of the basis σu,σv as

γ(t) = u(t)σu + v(t)σv. (2.3)

This result will be used in section 2.3 where we dene the rst fundamentalform of a surface.

Denition 2.2.3 claries the need for the charts of S to be smooth. However,we need to make sure that the charts t together properly to cover the wholeof S correctly. In other words, we would like S to be a smooth surface. Thisrequires the charts to be regular and compatible.

Denition 2.2.4 (Regular parametrization). A parametrization σ : U→ R3 iscalled regular if it is smooth and the vectors σu and σv are linearly independentat all points (u, v) ∈ U. The standard unit normal of a surface S at a point pis then dened to be

Np =σu × σv‖σu × σv‖

,

where Np 6= 0 if σ is regular.

Now, let σ and σ be regular charts of S such that σ : U→ V and σ : U→ Vand let Ω = V ∩ V be an open set. Thus, Ω is parametrized by both σ andσ. We say that the charts are compatible if the transition function Φ : U→ U,where Φ = σ−1 σ, is a smooth bijective map and its inverse, Φ−1 : U → U,is smooth. In this case, Φ could also be called the reparametrization of Ω. Tosummarize:

Denition 2.2.5 (Altas of a smooth surface). A surface S is smooth if itscharts are regular and compatible. A collection of such charts is then called anatlas of S. Furthermore, we say that the maximal atlas of S is the union of allatlases of S.


Example 2.2.6 (Catenoid). The resulting surface of revolution, S, when re-volving the catenary curve x = cosh z around the z-axis is called a catenoid andis parametrized by

σ(u, v) = (coshu cos v, coshu sin v, u), u ∈ R, v ∈ [0, 2π].

However, the fact that σ(u, 0) = σ(u, 2π) means that σ is not a bijective mapand hence not a homeomorphism. In order for σ to be a chart of S, we have torestrict v to, for example, the open interval (0, 2π). By adding the chart

σ(u, v) = (cosh u cos v, cosh u sin v, u), u ∈ R, v ∈ (−π, π)

we nd that the union of the two charts cover the whole of S. Some calculationsreveal that both σ and σ are regular. The reparametrization u = u and v = v−πis a linear map and is therefore smooth and bijective with a smooth inverse,which means that the charts are compatible. We can conclude that σ and σtogether make up an atlas of S. By changing the intervals of v and v in asuitable way, one can nd innitly many atlases of S.

There is one last restriction we shall have to impose on a surface which hasto do with its orientation. The standard unit normal of the tangent plane atp ∈ Ω ⊆ S should be independent of the choice of parametrization. However,if σ and σ are compatible charts of Ω and Φ is the transition map Φ : σ → σ,then (

σuσv

)=

(∂u∂u

∂v∂u

∂u∂v

∂v∂v

)(σuσv

),

where J(Φ) =

(∂u∂u

∂v∂u

∂u∂v

∂v∂v

)is the Jacobian matrix of the transition map Φ

and both σu,σv and σu, σv are bases of the tangent plane TpS. If p and pare the same point in Ω, expressed in terms of their associated parametrizations,then the standard unit normal at p is given by

Nσ =σu × σv‖σu × σv‖

=det(J(Φ))

‖det(J(Φ))‖σu × σv‖σu × σv‖

= ±Np,

where the sign depends on Φ.

Denition 2.2.7 (Orientable surface). We say that a surface S is orientableif there exists an atlas A such that, if Φ is a transition map between any twodierent compatible charts in A, then det(J(Φ)) > 0 where Φ is dened.

A casual way of putting it is that we need to be able to tell the "inside" and"outside" of a surface apart. This means that by walking on the outside of anoriented surface one can never end up on the inside.

2.2. Parametrization of a Surface 9

Example 2.2.8 (Möbius strip). The Möbius strip is an example of a surfacethat is not orientable.

Remark 2.2.9. Henceforth we will assume that all surfaces are smooth andorientable. This simplication will not lead to any major restrictions in laterchapters.

By studying the change of the standard unit normal one can study thecurvature of a surface. We will introduce two new concepts which will be usedto dene the second fundamental form of a surface.

Let Np be the standard unit normal of the tangent plane TpS. Since |Np| =1 we can dene a map that takes each point p ∈ S to a point on the unit sphere,S2.

Denition 2.2.10 (Gauss map). The map that takes points on a surface S toa point on the unit sphere S2 is called the Gauss map and is given by

G : S → S2

p 7→ Np

A more detailed motivation regarding the following denition can be foundin [Pressley].

Denition 2.2.11 (Derivative of a map). Let f : S → S be the smooth mapthat takes points p ∈ S to points f(p) ∈ S. The derivative Dpf of f at p isthen the map that takes tangent vectors of S at p to tangent vectors of S atf(p), in other words Dpf : TpS → Tf(p)S.

The derivative of G is then the linear map from TpS to TG(p)S2. Here, G(p)is determined by its tangent plane TG(p)S2, i.e. the plane perpendicular to G(p)going through the origin. But, since G(p) = Np and Np is determined by thetangent plane TpS, it follows that TG(p)S2 = TpS and thuslyDpG : TpS → TpS.Denition 2.2.12 (Weingarten map). The map

W : TpS → TpSW = −DpG

is called the Weingarten map.

Thus, W describes the way the standard unit normal changes when movingacross a surface. The greater the curvature of the surface, the faster the changeof the standard unit normal. Note that the Weingarten map is self-adjoint, seeA.0.5. This characteristic will be used in section 2.4.

Remark 2.2.13. The minus sign inW = −DpG is only a convention to simplifyfuture calculations.


2.3 The First Fundamental Form

When measuring the distance between two points in R3 one simply draws aline between the two points and then measures the lenght of the line. Thedistance between two points on a surface is in general not the length of a straightline joining the two points, however, hence we will need to dene the metricproperties of a regular smooth surface. To do this, we must dene a symmetric

bilinear form of a surface, see A.0.4.

Example 2.3.1. The dot product on Rn is perhaps the most well-known ex-ample of a symmetric bilinear form.

Let S be a surface and let σ : U→ V be a chart of V ⊆ S with (u, v) ∈ U ⊆R2. In section 2.2 we saw that the tangent vector of a curve γ moving througha point p ∈ S can be written as γ(t) = u(t)σu + v(t)σv in the basis σu,σvof the tangent plane TpS, see equation 2.3. Let w1,w2 ∈ TpS be two tangentvectors expressed in terms of the basis σu,σv as w1 = µ1σu + υ1σv andw2 = µ2σu + υ2σv, respectively, where µ1, µ2, υ1, υ2 ∈ R. We can then denean inner product of TpS to be the dot product restricted to tangent vectors, i.e.

〈w1,w2〉 = w1 ·w2 (2.4)

= µ1µ2(σu · σu) + (µ1υ2 + µ2υ1)(σu · σv) + υ1υ2(σv · σv). (2.5)

This bilinear form is called the rst fundamental form, I, of S and can be writtenin matrix form as

I(w1,w2) =(µ1 υ1

)( σu · σu σu · σvσu · σv σv · σv

)(µ2

υ2

). (2.6)

We also make the following denotations:

E = σu · σu, F = σu · σv and G = σv · σv. (2.7)

Denition 2.3.2 (First fundamental form). Let TpS be the tangent plane ofS at the point p and let w1,w2 ∈ TpS be tangent vectors to S at p. Thesymmetric bilinear form I, dened as

I : TpS × TpS → R(w1,w2) 7→ 〈w1,w2〉,

such that I(w1,w2) = w1 ·w2, is called the rst fundamental form of S at p.Given a parametrization σ, the matrix of the rst fundamental form is then

dened as FI =

(E FF G

), with E,F and G as in (2.7).

2.3. The First Fundamental Form 11

It is important to note that the rst fundamental form is, in general, dierentfor each point of the surface. The fact that I(w1,w2) = I(w2,w1) varies thestatement of the rst fundamental form being symmetric.

Example 2.3.3 (Plane). Let Π be the plane in R3 parametrized by σ(u, v) =(u, v, k), where (u, v) ∈ R2 and k ∈ R. Since σu = (1, 0, 0) and σv = (0, 1, 0),

the rst fundamental form of Π is FI =

(1 00 1

). If we let w1 = aσu + bσv

and w2 = cσu + dσv be two tangent vectors in the tangent plane of a point inΠ, then we have that

I(w1,w2) =(a b

)( 1 00 1

)(cd

)= ac+ bd,

as expected.

Example 2.3.4 (Torus). A torus with axis of revolution the z-axis in R3 isparametrized by

σ(θ, ϕ) = ((R+ r cos θ) cosϕ, (R+ r cos θ) sinϕ, r sin θ), 0 < r < R,

where r is the radius of the tube and R is the distance from the z-axis to thecenter of the tube. We will nd the matrix of the rst fundamental form of thetorus.

σθ = (−r sin θ cosϕ,−r sin θ sinϕ, r cos θ)

σϕ = (−(R+ r cos θ) sinϕ, (R+ r cos θ) cosϕ, 0)

‖σθ‖2 = r2 sin2 θ cos2 θ + r2 sin2 θ sin2 ϕ+ r2 cos2 θ = r2

‖σϕ‖2 = (R+ r cos θ)2 sin2 ϕ+ (R+ r cos θ)2 cos2 ϕ = (R+ r cos θ)2

σθ · σϕ = r sin θ(R+ r cos θ) sinϕ cosϕ− r sin θ(R+ r cos θ) sinϕ cosϕ = 0

∴ FI =

(r2 00 (R+ r cos θ)2

)

It is worth noting that the rst fundamental form is independent of ϕ, which isto be expected since the torus is symmetric with respect to the z-axis.

Example 2.3.5 (Catenoid). We will nd the rst fundamental form of the


catenoid dened in example 2.2.6.

σu = (sinhu cos v, sinhu sin v, 1)

σv = (− coshu sin v, coshu cos v, 0)

E = ‖σu‖2 = sinh2 u+ 1 = cosh2 u

G = ‖σv‖2 = cosh2 u

F = σu · σv = 0

∴ FI =

(cosh2 u 0

0 cosh2 u

)As in the case of the torus, the catenoid is also symmetric with respect to thez-axis and thus the rst fundamental form is independent of v.

Proposition 2.3.6 (Length of a Curve). The length of γ(t) is∫‖γ(t)‖ dt =

∫ √I(γ(t)) dt =

∫ √Eu(t)2 + 2Fu(t)v(t) +Gv(t)2 dt, (2.8)

where σ(u(t), v(t)) is the parametrization of S and E,F and G are the coe-cients of the rst fundamental form dened in (2.7).

Example 2.3.7 (Meridian of a Torus). The meridian of a surface of revolutionis the intersection of the surface and a half-plane, with the axis of revolution asits boundary. Let γ be the meridian of a torus where ϕ = ϕ0. In this case, themeridian is a circle with radius r, so the length of γ should be 2πr, see example2.3.4. We calculate the length of γ using the results found in the propositionabove. The meridian can be expressed in terms of the parametrization of thetorus as

γ(t) = σ(θ(t), ϕ0),

where θ(t) = 2πt, t ∈ [0, 1], and ϕ = ϕ0 is a constant. In example 2.3.4 wefound the rst fundamental form of a torus to be

FI =

(r2 00 (R+ r cos θ(t))2

).

The length of γ is then∫ 1

0

√I(γ(t)) dt =

∫ 1

0

√r2θ(t)2 dt =

∫ 1

0

2πr dt = 2πr.

Continuing, we will use the rst fundamental form of a surface to calculatethe area of a part of that surface. Let S,U,V, and σ be the same as in the

2.4. The Second Fundamental Form 13

introductory discussion of this section. The area of a small rectangle ∆A in Uwith sides ∆u,∆v corresponds approximatly to the area of the parallelogram∆A spanned by ∆uσu,∆vσv on V. That is,

∆A = ∆u∆v and ∆A = σ(∆A) ≈ ‖∆uσu ×∆vσv‖ = ∆u∆v‖σu × σv‖.

As (∆u,∆v)→ (0, 0) we get dA = dudv and dA = ‖σu×σv‖dudv, which leadsus to the following proposition.

Proposition 2.3.8 (Area of a Surface Patch). Let V ⊆ S and let σ : U→ Vbe a parametrization of V, where U ⊆ R2. The area of V is then given by

A(V) =

∫∫U

‖σu × σv‖ dudv =

∫∫U

√EG− F 2 dudv, (2.9)

where E,F and G are the coecients of the rst fundamental form.

Remark 2.3.9. Using Lagrange's identity and equation (2.7) yields

‖σu × σv‖ = (‖σu‖2‖σv‖2 − (σu · σv)2)1/2 =√EG− F 2.

The ability to determine the area of part of a surface is of particular intrestas it is required in order to calculate the strain energy of a surface.

2.4 The Second Fundamental Form

We have seen that the rst fundamental form can be used to measure distancesand areas on a surface. In order to determine the curvature of a surface wewill dene another symmetric bilinear form of a surface, namely the second

fundamental form.Recall that the Weingarten map describes the change of the standard unit

normal at a point on a surface as one moves away from the tangent plane. Fromdenition 2.2.12 we know that W is a self-adjoint map, i.e. W : TpS → TpS.As a consequence of this, W is adjoint to the rst fundamental form I.

Let w1,w2 ∈ TpS, with σu,σv being a basis of the tangent plane TpSand let, in this case, N be the standard unit normal of S at p. That is, w1 =µ1σu + υ1σv and w2 = µ2σu + υ2σv. The bilinear form

II(w1,w2) = I(W(w1),w2)

= (µ1W(σu) + υ1W(σv)) · (µ2σu + υ2σv)

= (µ1(−Nu) + υ1(−Nv)) · (µ2σu + υ2σv)

= µ1µ2((−Nu) · σu) + µ1υ2((−Nu) · σv) + µ2υ1((−Nv) · σu)

+ υ1υ2((−Nv) · σv)= µ1µ2(N · σuu) + (µ1υ2 + µ2υ1)(N · σuv) + υ1υ2(N · σvv),


is called the second fundamental form of the surface S at the point p. In the laststep used the fact that dierentiating N · σu = 0 and N · σv = 0 with respectto u and v, respectively, yields

σuu ·N = −σu ·Nu

σuv ·N = −σu ·Nv

σuv ·N = −σv ·Nu

σvv ·N = −σv ·Nv

The second fundamental form can be written in matrix form as

II(w1,w2) =(µ1 υ1

)( σuu ·N σuv ·Nσuv ·N σvv ·N

)(µ2

υ2

).

We also make the following denotations:

L = σuu ·N, M = σuv ·N and N = σvv ·N. (2.10)

Denition 2.4.1 (Second fundamental form). Let TpS be the tangent planeof S at the point p and let w1,w2 ∈ TpS be tangent vectors to S at p. LetW be the Weingarten map and let N be the standard unit normal at p. Thesymmetric bilinear form II, dened as

II : TpS × TpS → R(w1,w2) 7→ 〈W(w1),w2〉,

such that II(w1,w1) = I(W(w1),w2), is called the second fundamental form ofS at p. Given a parametrization σ, the matrix of the second fundamental form

is then dened as FII =

(L MM N

), with L,M and N dened as in (2.10).

Some verications will reveal that II(w1,w2) = II(w2,w1), indicating thatthe second fundamental form is indeed symmetric.

Example 2.4.2 (Torus, continued). Continuing example 2.3.4, we calculate the


second fundamental form of the Torus.

σθθ = (−r cos θ cosϕ,−r cos θ sinϕ,−r sin θ)

σϕϕ = (−(R+ r cos θ) cosϕ,−(R+ r cos θ) sinϕ, 0)

σθϕ = (r sin θ sinϕ,−r sin θ cosϕ, 0)

σϕθ = σθϕ

σθ × σϕ = −r(R+ r cos θ)(cos θ cosϕ, cos θ sinϕ, sin θ)

N =σθ × σϕ‖σθ × σϕ‖

= −(cos θ cosϕ, cos θ sinϕ, sin θ)

N · σθθ = r(cos2 θ cos2 ϕ+ cos2 θ sin2 ϕ+ sin2 θ) = b

N · σϕϕ = cos θ(R+ r cos θ) cos2 ϕ+ cos θ(R+ r cos θ) sin2 ϕ

= cos θ(R+ r cos θ)

N · σθϕ = r sin θ cos θ sinϕ cosϕ− r sin θ cos θ sinϕ cosϕ = 0

∴ FII =

(r 00 cos θ(R+ r cos θ)

).

Example 2.4.3 (Catenoid, continued). Calculating the second fundamentalform of the Catenoid yields some interesting results.

σuu = (coshu cos v, coshu sin v, 0)

σvv = (− coshu cos v,− coshu sin v, 0)

σuv = (− sinhu sin v, sinhu cos v, 0)

σu × σv = coshu(− cos v,− sin v, sinhu)

N =σu × σv‖σu × σv‖

=1

coshu(− cos v,− sin v, sinhu)

N · σuu = −1

N · σvv = 1

N · σuv = 0

∴ FII =

(−1 00 1

).

The catenoid is a minimal surface. This kind of surfaces has many interestingproperties, but is outside the scope of this thesis. More information regardingminimal surfaces can be found in [Do Carmo] and [Pressley].

Proposition 2.4.4. Let FI and FII be the matrices of the rst- and secondfundamental forms, respectively, and let W be the matrix of the Weingartenmap. Then,

W = F−1I FII. (2.11)


Proof. We rst note that, since I is a symmetric bilinear form, det(FI) > 0 andthus the inverse F−1I exists. Suppose that σu,σv is a basis of the tangentplane at a point on a surface and let N be the standard unit normal at thispoint. Then W(σu) = −Nu and W(σv) = −Nv, where Nu and Nv can bewritten as

−Nu = aσu + bσv

−Nv = cσu + dσv, a, b, c, d ∈ R. (2.12)

The matrix of the Weingarten map in the basis σu,σv is thenW =

(a cb d

).

Applying the rst fundamental form now yields the following equations:I(−Nu,σu) = −Nu · σu = a(σu · σu) + b(σu · σv)I(−Nv,σu) = −Nv · σu = c(σu · σu) + d(σu · σv)I(−Nu,σv) = −Nu · σv = a(σu · σv) + b(σv · σv)I(−Nv,σv) = −Nv · σv = c(σu · σv) + d(σv · σv)

Which, written in matrix form with the use of the coecients of the rst- andsecond fundamental forms, is equivalent to(

L MM N

)=

(E FF G

)(a cb d

).

Matrix multiplication of F−1I from the left gives us the expression in (2.11).

Now, since W is self-adjoint and assuming that S is a smooth oriented sur-face, σ can be chosen in such a way that there exists a basis t1, t2 of thetangent plane such that t1 and t2 are orthogonal eigenvectors of W, see A.0.5.

Proposition 2.4.5. For each point p of a surface S there exists a basis t1, t2of TpS such that t1 and t2 are the eigenvectors of the matrix of the Weingartenmap, i.e.

W =

(κ1 00 κ2

), κ1, κ2 ∈ R. (2.13)

The eigenvectors t1 and t2 of the Weingarten map are called the principal vectorsand the corresponding eigenvalues κ1, κ2 are called the principal curvatures ofthe surface S.

Let γ be unit-speed curve through a point p on S. Then γ is a tangent vectorto S at p. Thus γ, N and N×γ are mutually perpendicular unit vectors. We candeduce the following relationship between the curvature of curves on a surfaceand the second fundamental form.


Denition 2.4.6 (Normal- and geodesic curvature). Let γ be a unit-speedcurve through a point p on a surface S and let N be the standard unit normalat that point. Then γ = κnN + κg(N × γ, where κn is the normal curvature

and κg is the geodesic curvature of γ. That is,

κn = γ ·Nκg = γ · (N× γ)

κ = κ2n + κ2g,

(2.14)

where κ = ‖γ‖ is the curvature of γ and κn = II(γ, γ). The principal curvaturesκ1 and κ2 are the maximum and minimum values of the normal curvatures ofcurves through p.

Denition 2.4.7 (Mean- and Gaussian curvature). We dene the mean- andGaussian curvature of a surface as

H =1

2trace(W) and K = det(W), (2.15)

respectively.

The mean curvature is a measure of the curvature of a surface at a certainpoint. By combining 2.4.5 and 2.4.7 the mean- and Gaussian curvature can beexpressed in terms of the principal curvatures as

H =1

2(κ1 + κ2) and K = κ1κ2. (2.16)

This together with 2.4.4 implies that the mean and Gaussian curvature canbe expressed in terms of the coecients of the rst and second fundamentalforms as

H =LG− 2MF +NE

2(EG− F 2)and K =

LN −M2

EG− F 2. (2.17)

Example 2.4.8 (Monge patch). Let p be a point on a surface S. Then thereis a patch V ⊆ S containing p such that σ : U → V is given by σ(u, v) =(u, v, f(u, v)), where f : U → R is a dierentiable function. We calculate the


rst and second fundamental forms of this surface patch, called a Monge patch.

σu = (1, 0, fu), σv = (0, 1, fv)

E = ‖σu‖2 = 1 + f2u , G = ‖σv‖2 = 1 + f2v , F = σu · σv = fufv

∴ FI =

(1 + f2u fufvfufv 1 + f2v

)σuu = (0, 0, fuu), σvv = (0, 0, fvv), σuv = (0, 0, fuv)

σu × σv =1√

1 + f2u + f2v(−fu,−fv, 1)

N =1√

1 + f2u + f2v(−fu,−fv, 1)

L = N · σuu =fuu√

1 + f2u + f2v, N = N · σvv =

fvv√1 + f2u + f2v

,

M = N · σuv =fuv√

1 + f2u + f2v

∴ FII =1√

1 + f2u + f2v

(fuu fuvfuv fvv

)

We also make some preparatory work by calculating the mean and Gaussiancurvature introduced in propostion 2.4.7.

H =fuu(1 + f2v )− 2fuv(fufv) + fvv(1 + f2u)

2(1 + f2u + f2v )3/2(2.18)

K =fuufvv − f2uv

(1 + f2u + f2v )3/2. (2.19)

In fact, by the inverse function theorem stated in A.0.3, every surface canlocally be seen as a Monge patch. This turns out to be quite useful. If we, for amoment, regard U in example 2.4.8 as being a thin elastic plate, then V is theshape U takes as the deformation f is being applied to it. This deformationstrains the plate and generates stresses in the material. The energy required toproduce this deformation is called the strain energy. The coordinate system ofR3 can be chosen in such a way that p = σ(0, 0) = (0, 0, f(0, 0)) is the origin andf(0, 0) = fu(0, 0) = fv(0, 0) = 0. According to the theory of elasticity, the strainenergy is then proportional to the expression fuu(0, 0)2+2fuv(0, 0)2+fvv(0, 0)2.


We relate this expression to the rst and second fundamental form by

κ21 + κ22 = (κ1 + κ2)2 − 2κ1κ2 = 4H2 − 2K

=(fuu(1 + f2v )− 2fuv(fufv) + fvv(1 + f2u))2

(1 + f2u + f2v )3− 2

fuufvv − f2uv(1 + f2u + f2v )3/2

,

which at the point p is

κ21 + κ22(0, 0) = (fuu(0, 0) + fvv(0, 0))2 − 2(fuu(0, 0)fvv(0, 0)− fuv(0, 0)2)

= fuu(0, 0)2 + 2fuv(0, 0)2 + fvv(0, 0)2.

Since the shape of a surface is the same regardless of choice of parametrization,the principal curvatures at each point of the surface is independent of the choiceof parametrization. This means that the strain energy of a surface can becalculated for any given parametrization.

Proposition 2.4.9 (Strain Energy of a Surface). Let V be a patch of a surfaceS and let σ : U→ V be the parametrization of V. The strain energy W of Vis then given by

W =

∫∫U

(κ21 + κ22)√EG− F 2 dudv, (2.20)

where κ1, κ2 are the principal curvatures and E,G and F are the coecients ofthe rst fundamental form.

We remind the reader of proposition 2.3.8. In 2.4.9 above, (κ21+κ22)√EG− F 2

is the strain energy of the area element dA =√EG− F 2 dudv.

Remark 2.4.10. The theory of elasticity is not within the scope of this thesisbut a brief account can be found in [Bonneau].

Chapter 3

Bézier Surfaces

We now focus on a specic kind of surfaces known as Bézier surfaces. These sur-faces have certain interesting properties which makes them easy to manipulateaccording to one's preferences. The convex hull property contains the surfacewithin a set of points in R3. The recursive property substantially reduces theamount of calculations required to nd the rst and second fundamental formswhich, in turn, makes calculating the strain energy a rather straightforwardprocedure. In this chapter we enumerate and study these properties. In section4.2 the reader nds a brief introduction to additional properties which are notstudied in full detail in this thesis, but are important nonetheless. At the end ofthis chapter the reader nds an example which covers the calculations requiredto nd the strain energy at a point of a Bézier surface. This example serves asa complement to the results in section 4.1.

3.1 Bézier Curves

As already mentioned in the introduction, a Bézier curve is dened by its as-sociated control polygon and the Bernstein basis polynomials. This means thatthe components of a Bézier curve of degree n are linear combinations of theBernstein basis polynomials of degree ≤ n.

Denition 3.1.1 (Bernstein basis polynomials). Let n ∈ N. The polynomial

Bni (t) =

(ni

)ti(1− t)n−i if i = 0, . . . , n

0 otherwise

is called the ith Bernstein basis polynomial of degree n. Together the Bernstein

Bråmå, 2018. 21

22 Chapter 3. Bézier Surfaces

basis polynomials Bni form a basis for the vector space Pn of polynomials ofdegree ≤ n.Proposition 3.1.2 (Properties of Bernstein basis polynomials). The Bernsteinbasis polynomials has the following properties:

n∑i

Bni (t) = 1, t ∈ [0, 1] (3.1)

Bni (t) ≥ 0, t ∈ [0, 1] (3.2)

Bnn−i(t) = Bni (1− t) (3.3)

Bni (t) = (1− t)Bn−1i (t) + tBn−1i−1 (t) (3.4)

Example 3.1.3 (Bernstein polynomials as a basis). The Bernstein polynomialsof degree 0,1,2 and 3 are given below.

B00(t) = 1

B10(t) = 1− t, B1

1(t) = t

B20(t) = (1− t)2, B2

1(t) = 2t(1− t), B22(t) = t2

B30(t) = (1− t)3, B3

1(t) = 3t(1− t)2, B32(t) = 3t2(1− t), B3

3(t) = t3

As a consequence of (3.4), the derivative of a Bernstein polynomial of degreen is recursively dened by Bernstein polynomials of degree n− 1.

Proposition 3.1.4 (Derivatives of Bernstein polynomials). The rst derivativeof a Bernstein polynomial of degree n is recursively dened as

(Bni )′(t) = n(Bn−1i−1 (t)−Bn−1i (t))

We will illustrate proposition 3.1.4 with an example.

Example 3.1.5 (Derivatives of cubic Bernstein basis polynomials). The rstderivatives of the cubic Bernstein basis polynomials are

(B30)′(t) = −3(1− t)2 = −3B2

0(t) (3.5)

(B31)′(t) = 3(1− t)2 − 6t(1− t) = 3(B2

0(t)−B21(t)) (3.6)

(B32)′(t) = 6t(1− t)− 3t2 = 3(B2

1(t)−B22(t)) (3.7)

(B33)′(t) = 3t2 = 3B2

2(t). (3.8)

3.1. Bézier Curves 23

Note that we in equations 3.5 and 3.8 used the fact that B2−1(t) = 0 and

B23(t) = 0 as per denition 3.1.1.

Denition 3.1.6 (Bézier curve). Let n ∈ N and let b0, . . . ,bn be n+ 1 pointsin R3. The parametric curve B dened by

B : [0, 1]→ R3

t 7→ B(t) =

n∑i=0

biBni (t)

is called the Bézier curve of degree n, with control points b0, . . . ,bn. Thepolygon with vertices b0, . . . ,bn is called the control polygon of B and Bni (t) isthe ith Bernstein basis polynomial of degree n.

Example 3.1.7 (Cubic Bézier curve). Let b0,b1,b2,b3 be points in R2. Theparametrized curve B given by

B(t) =

3∑i=0

biB3i (t), t ∈ [0, 1],

is then the cubic Bézier curve with control points b0,b1,b2 and b3. If we, forexample, let b0 = (−1, 0), b1 = (− 1

2 ,12 ), b2 = ( 1

2 ,−12 ) and b3 = (1, 0), then

B(t) =

3∑i=0

biB3i (t)

= (−1, 0)(1− t)3 + 3(− 12 ,

12 )(1− t)2t + 3( 1

2 ,−12 )(1− t)t2 + (1, 0)t3

= (−t3 + 32 t

2 + 32 t− 1 , 3t3 − 9

2 t2 + 3

2 t).

Note that B(0) = (−1, 0) = b0 and B(1) = (1, 0) = b3.

The following algorithm is yet another consequence of the recursive propertyof Bernstein polynomials.

Proposition 3.1.8 (The de Casteljau Algorithm). Let B(t), t ∈ [0, 1], be aBézier curve of degree n with control points b0, . . . ,bn. If we dene

b0i (t) = bi, i = 0, . . . , n

and

bri (t) = (1− t)br−1i (t) + tbr−1i+1 (t), 1 ≤ r ≤ n, i = 0, . . . , n− r

then B(t) = bn0 (t).


Example 3.1.9 (The de Casteljau algorithm). Let B(t), t ∈ [0, 1], be a Béziercurve of degree 2 with control points (b0,b1,b2).

B(t) =

2∑i=0

biB2i (t) = (1− t)2b0 + 2t(1− t)b1 + t2b2.

Following the de Casteljau algorithm we get the same result:

b0i = bi, i = 0, 1, 2

b20(t) = (1− t)b1

0 + tb11

= (1− t)((1− t)b0 + tb1) + t((1− t)b1 + tb2)

= (1− t)2b0 + 2b1t(1− t) + b2t2

= B(t)

The gure to the left illustrates the de Castiljau algorithm when evaluating thepoint B(0.5) = b2

0(0.5). The gure to the right shows the quadric Bézier curveB(t), t ∈ [0, 1].

AAAAAAAAA

b0

b1

b2

b10 b1

1

b20

r

r

rr rr

AAAAAAAAA

b0

b1

b2

B(t)

r

r

rProposition 3.1.10 (Properties of Bézier curves). Let B(t) be a Bézier curvewith control points b0, . . . ,bn and let t ∈ [0, 1].

• (Convex hull property) B(t) is contained within the convex hull of itscontrol polygon, i.e.

B(t) ∈ CHb0, . . . ,bn, (3.9)

see A.0.6.

3.1. Bézier Curves 25

• (Invariance under ane transformation) If T is an ane transformation,then

T( n∑i=0

biBni (t)

)=

n∑i=0

T (bi)Bni (t). (3.10)

• (Endpoint interpolation property) The endpoints B(0) and B(1) of B(t)coincide with b0 and bn, respectively. That is,

B(0) = b0 and B(1) = bn (3.11)

• (Endpoint tangent property) The tangent of B(t) in its endpoints is givenby

B(0) = n(b1 − b0) and B(1) = n(bn − bn−1) (3.12)

Remark 3.1.11. When calculating the strain energy of a few dierent Béziersurfaces in chapter 4, we make use of property (3.10) in order to manipulate theshape of the surfaces.

Example 3.1.12 (Derivatives of a Cubic Bézier Curve). If B is a cubic Béziercurve then the rst- and second derivative of B are given by:

B(t) =

3∑i=0

biB3i (t)

=

3∑i=0

3bi(B2i−1(t)−B2

i (t))

= −3b0B20(t) + 3b1(B2

0(t)−B21(t)) + 3b2(B2

1(t)−B22(t)) + 3b3B

22(t)

= 3(b1 − b0)B20(t) + 3(b2 − b1)B2

1(t) + 3(b3 − b2)B22(t)

B(t) = 6(b0 − 2b1 + b2)B10(t) + 6(b1 − 2b2 + b3)B1

1(t).

Now, if we let b(1)0 = 3(b1 − b0), b

(1)1 = 3(b2 − b1), b

(1)2 = 3(b3 − b2) and

b(2)0 = 2(b

(1)1 − b

(1)0 ), b

(2)1 = 2(b

(1)2 − b

(1)1 ), it follows that

B(t) =

2∑i=0

b(1)i B2

i (t) and B(t) =

1∑i=0

b(2)i B1

i (t).

The results in example 3.1.12 suggests the following proposition.


Proposition 3.1.13 (Derivatives of Bézier curves). Let B(t) be a Bézier curveof degree n with control points b0, . . . ,bn. The rth derivative of B(t) is givenby

B(r)(t) =

n−r∑i=0

b(r)i Bn−ri (t), (3.13)

where b(r)i = n(n− 1) · · · (n− r − 1)

∑rj=0(−1)r−j

(rj

)bi+j .

This means that the derivatives of a Bézier curve are recursively dened byits control points by the use of the de Castiljau algorithm. In other words, forany given control polygon the appropriate Bézier curve and its derivatives arealready given.

3.2 Bézier Surfaces

A Bézier surface of degree (m,n) can be seen as a "product" of two Bézier curvesof degree m and n, respectively, and is dened by its control polygon consistingof points in R3.

Denition 3.2.1 (Bézier surface). Let (bi,j), i = 0, . . . ,m, j = 0, . . . , n, be(m + 1)(n + 1) points in R3 and let s, t ∈ [0, 1]. The parametric surface Xdened by

X : [0, 1]× [0, 1]→ R3

(s, t) 7→ X(s, t) =

m∑i=0

n∑j=0

bi,jBmi (s)Bnj (t)

is a Bézier surface of degree (m,n), with control points (bi,j). Bmi (s) and Bnj (t)

are the ith and jth Bernstein basis polynomials of degree m and n, respectively.

Remark 3.2.2. Strictly speaking, a Bézier surface is a tensor product of twoBézier curves. This notion is, however, not vital for the material in this thesisand will therefore be left out.

Some of the interesting properties of Bézier curves mentioned in proposition3.1.10, such as the convex hull property, is inherited by the Bézier surfaces. Thismeans that a point of a Bézier surface can be evaluated using the de Castiljaualgorithm, as the following example illustrates.

Example 3.2.3 (The de Castiljau algorithm for Bézier surface). Let m = 1,n = 2 and let (bi,j), i = 0, 1, j = 0, 1, 2, be the 6 control points of the Béziersurface X(s, t) of degree (1, 2), s, t ∈ [0, 1]. The gure below illustrates howthe de Casteljau algorithm can be used to evaluate the points of X; here ats = t = 0.5.

3.2. Bézier Surfaces 27

@@

@@@

@@

@@@

@@

@@@

@@

@@@

rr r

rr

rr

r

rrr

rr

b0,0

b0,1

b0,2

b1,0

b1,1 b1,2

b(0,1)0,0

b(0,1)0,1

b(0,1)1,0

b(0,1)1,1

b(0,2)1,0

b(0,2)0,0

b(1,2)0,0

Proposition 3.2.4 (Derivative of a Bézier surface). Let X(s, t), with s, t ∈[0, 1], be a Bézier surface of degree (m,n) with control points bi,j . The partialderivative of order α and β with respect to s and t, respectively, is given by

Xαβ(s, t) =

m−α∑i=0

n−β∑j=0

b(α,β)i,j Bm−αi (s)Bn−βj (t), (3.14)

where

b(α,β)i,j =

n!

(n− α)!

m!

(m− β)!

α∑k=0

β∑l=0

(−1)k+l(α

k

)(β

l

)bi+α−k,j+β−l. (3.15)

Using denition 2.3.2 and 2.4.1 regarding the rst and second fundamentalforms, as well as proposition 3.2.4 stated above, we can formulate the funda-mental forms of a Bézier surface.

Proposition 3.2.5 (Fundamental forms of a Bézier surface). The fundamentalforms of a Bézier surface can be written in matrix form as

FI =

(X10 ·X10 X10 ·X01

X10 ·X01 X01 ·X01

)and FII =

(X20 ·N X11 ·NX11 ·N X02 ·N

),

respectively, where N =X10 ×X01

‖X10 ×X01‖is the standard unit normal.

We end this chapter with an extensive example where the strain energy at apoint of a Bézier surface is calculated. The plot of the surface, along with theassociated strain energy surface, can be seen in chapter 4.


Example 3.2.6 (Strain energy of a Bézier surface). Let X(s, t), s, t ∈ [0, 1], bea Bézier surface of degree (3, 3) such that X : [0, 1] × [0, 1] → R3. Let (bi,j),where i = 0, . . . 3 and j = 0, . . . 3, be the 16 control points of X listed in CP .

CP =

b0,0 b0,1 · · · b1,0 b1,1 · · · b3,0 b3,1 · · · b3,3

=

0 0 0 0 0.2 0.2 0.2 0.2 0.80 0.2 0.8 1 0 0.2 0.8 1 00 0.2 0.2 0 0.2 0.2 0.2 0.2 0.2

0.8 0.8 0.8 1 1 1 10.2 0.8 1 0 0.2 0.8 10.2 0.2 0.2 0 0.2 0.2 0

We calculate the strain energy in the point X(0.1, 0.1).

X(s, t) =

3∑i=0

3∑j=0

bi,jB3i (s)B3

j (t) = b0,0B30(s)B3

0(t) + b0,1B30(s)B3

1(t) + . . .

. . .+ b1,0B31(s)B3

0(t) + b1,1B31(s)B3

1(t) + . . .

. . .+ b3,0B33(s)B3

0(t) + b3,1B33(s)B3

1(t) + · · ·+ b3,3B33(s)B3

3(t).

X(0.1, 0.1) = · · · =

0.07120.07120.0934

X10(s, t) =

2∑i=0

3∑j=0

b(1,0)i−1,jB

2i (s)B3

j (t) =

2∑i=0

3∑j=0

3(bi+1,j − bi,j)B2i (s)B3

j (t)

X01(s, t) =3∑i=0

2∑j=0

b(0,1)i,j−1B

3i (s)B2

j (t) =

3∑i=0

2∑j=0

3(bi,j+1 − bi,j)B3i (s)B2

j (t)

X10(0.1, 0.1) = · · · =

0.81600

0.3504

X01(0.1, 0.1) = · · · =

00.81600.3504

N(0.1, 0.1) =X10(0.1, 0.1)×X01(0.1, 0.1)

‖X10(0.1, 0.1)×X01(0.1, 0.1)‖=

−0.3670−0.36700.8547

3.2. Bézier Surfaces 29

X20(s, t) =

1∑i=0

3∑j=0

b(2,0)i,j B1

i (s)B3j (t) =

1∑i=0

3∑j=0

6(bi+2,j − 2bi+1,j + bi,j)B1i (s)B3

j (t)

X02(s, t) =

3∑i=0

1∑j=0

b(0,2)i,j B3

i (s)B1j (t) =

3∑i=0

1∑j=0

6(bi,j+2 − 2bi,j+1 + bi,j)B3i (s)B1

j (t)

X11(s, t) =

2∑i=0

2∑j=0

b(1,1)i,j B2

i (s)B2j (t)

=

2∑i=0

2∑j=0

9(bi+1,j+1 − bi+1,j − bi,j+1 + bi,j)B3i (s)B1

j (t)

X20(0.1, 0.1) = · · · =

1.92000

−0.8760

X02(0.1, 0.1) = · · · =

01.9200−0.8760

X11(0.1, 0.1) = · · · =

00

−1.1520

FI(0.1, 0.1) =

(X10(0.1, 0.1) ·X10(0.1, 0.1) X10(0.1, 0.1) ·X01(0.1, 0.1)X10(0.1, 0.1) ·X01(0.1, 0.1) X01(0.1, 0.1) ·X01(0.1, 0.1)

)=

(0.7886 0.12280.1228 0.7886

)FII(0.1, 0.1) =

(X20(0.1, 0.1) ·N(0.1, 0.1) X11(0.1, 0.1) ·N(0.1, 0.1)X11(0.1, 0.1) ·N(0.1, 0.1) X02(0.1, 0.1) ·N(0.1, 0.1)

)=

(−1.4535 −0.9847−0.9847 −1.4535

)

H =(X20 ·N)(X01 ·X01)− 2(X11 ·N)(X10 ·X01) + (X02 ·N)(X10 ·X10)

det(FI)

K =det(FII)

det(FI)

W = (4H2 − 2K)√det(FI)


W (0.1, 0.1) = · · · = 5.9603

A plot of this Bézier surface can be found in gure 4.1. In table 4.1 the strainenergy in three dierent points on the surface can be found. In this case a = 0.2and according to the table, the strain energy in the point X(0.1, 0.1) is 0.2980.Note that there is a scaling factor 1/20 applied to the values i the table, so inthis case 1

20W (0.1, 0.1) ≈ 0.2980.

Chapter 4

Results and Future Work

One rather important aspect to consider when constructing for example anairplane, is the strain energy in the material. If the strain energy is too great insome part of the airplane body it might break, causing a disaster. It is thereforedesireable to be able to calculate the strain energy. In this chapter we do this forfour Bézier surfaces of dierent shapes to illustrate how the shape contributesto the overall durability of, for example, a segment of an airplane body.

4.1 Strain Energy of a Bézier Surface

We analyze the strain energy of Bézier surfaces of degree (3, 3) with controlpoints bi,j(a), 0 < a < 0.5, represented in CP (a) below.

CP (a) =

b0,0(a) b0,1(a) · · · b3,0(a) b3,1(a) · · · b3,3(a)

=

0 0 0 0 a a a a 1− a0 a 1− a 1 0 a 1− a 1 00 a a 0 a a a a a

1− a 1− a 1− a 1 1 1 1a 1− a 1 0 a 1− a 1a a a 0 a a 0

The parameter a determines the shape of the Bézier surface. If X : [0, 1] ×

[0, 1] → R3 is the Bézier surface, the strain energy W (s, t) is plotted as thesurface given by X(s, t) + 1

wW (s, t)N(s, t). However, the fact that the standardunit normal of the surface is undened in the points at the very edges, i.e. fors = 0, 1 and t = 0, 1, makes plotting the strain energy there impossible. Hence,the strain energy surface is redened as X + 1

wWN : (0, 1)× (0, 1)→ R3.

Bråmå, 2018. 31

32 Chapter 4. Results and Future Work

0

1.4

0.05

1.2

0.1

1 1.4

0.15

0.8 1.21

0.2

0.60.8

0.25

0.4 0.6

0.3

0.2 0.40.20

0-0.2

-0.2-0.4 -0.4

Figure 4.1: Strain energy surface of Bézier surface with parameter values a = 0.1and w = 200

0

1.4

0.05

0.1

1.2

0.15

1 1.4

0.2

0.8 1.2

0.25

1

0.3

0.6

0.35

0.80.4

0.4

0.6

0.45

0.2 0.40.20

0-0.2

-0.2-0.4 -0.4


4.1. Strain Energy of a Bézier Surface 33

0

1.2

0.05

0.1

1

0.15

1.20.8

0.2

1

0.25

0.60.8

0.3

0.4

0.35

0.6

0.4

0.40.20.2

00

-0.2 -0.2


0

1.2

0.1

1

0.2

1.2

0.3

0.8

0.4

10.6

0.5

0.8

0.6

0.4 0.6

0.7

0.40.20.2

00

-0.2 -0.2


34 Chapter 4. Results and Future Work

Figures 4.1-4.4 illustrates the resulting Bézier surfaces, along with the ap-propriate strain energy surface, for a = 0.1, a = 0.2, a = 0.3 and a = 0.4,respectively. Since W ≥ 0, this means that the strain energy surface will, withrespect to the tangent plane, be situated above the Bézier surface and if W = 0for some point the two surfaces are tangent at this point. The factor 1

w is ap-plied to scale the strain energy and is chosen in such a way as to make theillustrations easier to interpret. The coloring of the strain energy surfaces rangefrom purple (low strain energy) to orange (high strain energy) and the color ofthe Bézier surface is uniformly purple. Note that, for the sake of comparability,the value of w is 10 times larger in gure 4.1 in order to compensate for therelatively high strain energy in the corners in this case compared to the otherthree.

For each one of the four cases, the strain energy is measured in the same setof three dierent points; one close to the edge in a corner, one in the middle,close to the edge, and one in the center of the surface. The results are listed intable 4.1. Note that w = 20 for all the listed values.

Table 4.1: Strain energyX(s, t) a = 0.1 a = 0.2 a = 0.3 a = 0.4

X(0.1, 0.1) 0.3362 0.2980 0.1844 0.1528X(0.5, 0.5) 0.0012 0.0062 0.0184 0.0444X(0.9, 0.5) 0.0451 0.0534 0.0861 0.2292

As the value of a increases from 0.1 to 0.4 the curvature in the four cornersis decreased while the curvature along the sides is increased. The increase instrain energy at the point X(0.1, 0.1) for the increasing values of a illustratesthis phenomenon. The fact that the strain energy in X(0.9, 0.5), for increasingvalues of a, is increased, further illustrate this.

For small values of a the center of the surface is almost at, as the strainenergy at X(0.5, 0.5) indicates. As the value of a increases, however, the overall shape of the surface becomes more curved and as a result the strain energyat the center of the surface is increased.

4.2 Future work

In the examples in section 4.1 we changed the shape of the surfaces by changingthe associated control polygon. There is, however, a more practical way ofdoing this, namely by using a rational Bézier surface. A rational Bézier surface

4.2. Future work 35

is similar to a non-rational one with the exception that each point comes witha corresponding weight, giving it the following parametrization:

X(s, t) =

∑mi=0

∑nj=0 ωi,jbi,jB

mi (s)Bnj (t)∑m

i=0

∑nj=0 ωi,jB

mi (s)Bnj (t)

Changing the values of the weights ωi,j alters the shape of the surface whilethe control polygon remains unchanged. Increasing the value of one of theweights would result in the surface being "dragged" towards the correspondingcontrol point. This way we can nd the optimal values of the weights such that,for example, the strain energy of the surface is minimized.

While it is theoretically possible to model a Bézier surface into any desiredshape, most of the time it requires a Bézier surface of a high degree, whichisn't desireable from a computational complexity point of view. However, bythe use of the end point tangent property stated in (3.12), it is possible to use amultiude of lower degree Bézier surfaces that together make up the whole of thedesired surface. With each Bézier surface being of low degree, the computationalcomplexity decreases dramatically. This, along with the recursive property, iswhat makes Bézier surfaces the highly versatile and powerful tool that it is.

Bibliography

[Pressley] A. Pressley: Elementary Dierential Geometry second edition,Springer, (2010)

[Marsh] D. Marsh: Applied Geometry for Computer Graphics and CAD secondedition, Springer, (2005)

[Do Carmo] Manfredo P. Do Carmo: Dierential Geometry of Curves and Sur-

faces, Prentice-Hall (1976)

[Bonneau] G.P. Bonneau: Variational Design of Rational Bézier Curves and

Surfaces, Univ. Kaiserslautern (1993)

[Laur-Sabl] Pierre-Jean Laurent, Paul Sablonnière: Pierre Bézier: An engineer

and a mathematician, Computer Aided Geometric Design 18 (2001): 609-617

Bråmå, 2018. 37

Appendix A

Fundamental Concepts

Here follows some denitions of mathematical concepts used in this thesis.

Denition A.0.1 (Smooth map). A map f : Rm → Rn is smooth if all of itscomponents have continuous partial derivatives of all orders.

Denition A.0.2 (Homeomorphism). A map Φ : U→ U is a homeomorphism

if Φ and Φ−1 are continuous and Φ is bijective. The sets U and U are thensaid to be homeomorphic.

Theorem A.0.3 (Inverse function theorem). Let f : U ⊂ Rn → Rn be adierential mapping and suppose that the dierential dfp : Rn → Rn at a pointp ∈ U is an isomorphism. Then there exists a neighborhood V of p in U and aneighborhood W of f(p) in Rn such that f : V →W has a dierential inversef−1 : W→ V.

Denition A.0.4 (Bilinear form). Let V be a vector space of nite dimensionn over R. A map

V× V→ R(u,w) 7→ 〈u,w〉

is called a bilinear form if, ∀λ1, λ2 ∈ R, and u1,u2,w ∈ V, we have

〈λ1u1 + λ2u2,w〉 = λ1〈u1,w〉+ λ2〈u2,w〉〈w, λ1u1 + λ2u2〉 = λ1〈w,u1〉+ λ2〈w,u2〉

Thus, 〈u,w〉 is a linear function of u for each xed w, and a linear function ofw for each xed u. If v1, . . .vn is a basis of V, any bilinear form 〈 , 〉 on V is

Bråmå, 2018. 39

40 Appendix A. Fundamental Concepts

determined by the n× n matrix whose (i, j)-entry is 〈vi,vj〉 for i, j = 1, . . . , n.For, if

u =

n∑i=1

λivi, w =

n∑j=1

µjvj

are any two vectors in V, then

〈u,w〉 =

n∑i,j=1

λiµj〈vi,vj〉.

A bilinear form 〈 , 〉 is called symmetric if

〈u,w〉 = 〈w,u〉 ∀ u,w ∈ V.

Equivalently, the matrix of 〈 , 〉 with respect to any basis of V is symmetric.

Theorem A.0.5 (Self-adjoint map). Let M : V → V be a self-adjoint linearmap. Then, V has a basis u1, . . . ,un consisting of eigenvectors of M. More-over, if ui and uj are eigenvectors corresponding to distinct eigenvalues, thenui and uj are orthogonal.

Denition A.0.6 (Convex Hull). Let X = x0, . . . ,xn be a nite set of points.The convex hull of X is dened to be

CHX =α0x0 + · · ·+ αnxn

∣∣∣ n∑i=0

αi = 1, αi ≥ 0.

Appendix B

Matlab

Code 1: Binomial Coecients

function f = Binomial(n,i)

f = factorial(n)/((factorial(n-i))*factorial(i));

end

Code 2: Bernstein Polynomial

function g = Bernstein_Polynom(n,i,t)

g = Binomial(n,i)*((1-t)^(n-i))*t^i;

end

Code 3: Control Polygon

function cp=CP_matrix(a)

p00=[0; 0; 0]; %fixed

p01=[0; a; a];

p02=[0; (1-a); a];

p03=[0; 1; 0]; %fixed

p10=[a; 0; a];

p11=[a; a; a];

p12=[a; (1-a); a];

Bråmå, 2018. 41

42 Appendix B. Matlab

p13=[a; 1; a];

p20=[(1-a); 0; a];

p21=[(1-a); a; a];

p22=[(1-a); (1-a); a];

p23=[(1-a); 1; a];

p30=[1; 0; 0]; %fixed

p31=[1; a; a];

p32=[1; (1-a); a];

p33=[1; 1; 0]; %fixed

cp=

[p00 p01 p02 p03

p10 p11 p12 p13

p20 p21 p22 p23

p30 p31 p32 p33

];

end

Code 4: A function used by the Bézier Coordinates function.

function cpx=CP_coord(r,m,n,cp,i,j,dx,dy)

cpx=0;

for k=0:dx

for l=0:dy

cpx = cpx

+ (factorial(m)/factorial(m-dx))

* (factorial(n)/factorial(n-dy))

* ((-1)^(k+l))

* Binomial(dx,k)

* Binomial(dy,l)

* cp(r,((i+dx-k)*(n+1)+(j+dy-l)+1));

end

end

end

Code 5: Bézier Coordinates

43

function xr=Bezier_Coordinates(r,m,n,cp,s,t,dx,dy)

xr=0;

for i = 0:(m-dx)

for j = 0:(n-dy)

xr = xr

+ CP_coord(r,m,n,cp,i,j,dx,dy)

* Bernstein_Polynom((m-dx),i,s)

* Bernstein_Polynom((n-dy),j,t);

end

end

end

Code 6: First Fundamental Form

function f=I_Form(m,n,cp,s,t)

f=[];

Xu=[

Bezier_Coordinates(1,m,n,cp,s,t,1,0)



];

Xv=[




];

f=[

sum(Xu.*Xu) sum(Xu.*Xv);

sum(Xu.*Xv) sum(Xv.*Xv)

];

end

Code 7: Standard Unit Normal

function un=Standard_Unit_Normal(m,n,cp,s,t)


un=[];

N=[];

Na=[




];

Nb=[




];

N=cross(Na,Nb);

un=(1/sqrt(sum(N.*N))).*N;

end

Code 8: Second Fundamental Form

function ff=II_Form(m,n,cp,s,t)

ff=[];

Xuu=[




];

Xvv=[




];

Xuv=[



45


];

ff=[

sum(Xuu.*Standard_Unit_Normal(m,n,cp,s,t))

sum(Xuv.*Standard_Unit_Normal(m,n,cp,s,t));

sum(Xuv.*Standard_Unit_Normal(m,n,cp,s,t))

sum(Xvv.*Standard_Unit_Normal(m,n,cp,s,t))

];

end

Code 9: Strain Energy

function E=Strain_Energy(m,n,cp,s,t,w)

FI=I_Form(m,n,cp,s,t);

FII=II_Form(m,n,cp,s,t);

detFI=(FI(1,1)*FI(2,2))-(FI(1,2)*FI(1,2));

detFII=(FII(1,1)*FII(2,2))-(FII(1,2)*FII(1,2));

q=(FI(1,1)*FII(2,2))-(2*FI(1,2)*FII(1,2))+(FI(2,2)*FII(1,1));

E=(1/w)*(((q/detFI)^2)-(2*(detFII/detFI)))*sqrt(detFI);

end

Code 10: A function used by the Surface Coloring function.

function scx=s_c_x(m,n,cp,G,w,a)

scx=[];

lx=(1/G);

ly=(1/G);

for i=1:(ly-1)

scx=[scx Strain_Energy(m,n,cp,G*i,G*a,w)];

end

end

Code 11: Surface Coloring generates a matrix consisting of strain energy valueswhich are plotted as the strain energy surface in the Bézier Surface function.


function sc=Surface_Colouring(m,n,cp,G,w)

sc=[];

lx=(1/G);

ly=(1/G);

for j=1:(ly-1)

sc=[sc; s_c_x(m,n,cp,G,w,j)];

end

end

Code 12: A function used by the z-coordinate function.

function zx=zx_coord(m,n,cp,G,a,e,w,lx)

zx=[];

if e==1

for k=1:(lx-1)

zx=[

zx

(

Bezier_Coordinates(3,m,n,cp,G*k,G*a,0,0)

+ sum(

[0 0 Strain_Energy(m,n,cp,G*k,G*a,w)]

.* (Standard_Unit_Normal(m,n,cp,G*k,G*a))

)

)

];

end

else

for i=0:lx

zx=[zx Bezier_Coordinates(3,m,n,cp,G*i,G*a,0,0)];

end

end

end

Codes 13, 14 and 15 are functions which generate the x, y and z-coordinates ofthe strain energy surface.

Code 13: x-coord

47

function x=x_coord(m,n,cp,G,e,w)

x=[];

if e==1

for k=G:G:1-G

x=[

x

(

Bezier_Coordinates(1,m,n,cp,k,0,0,0)

+ sum(

[Strain_Energy(m,n,cp,k,0,w) 0 0]

.* (Standard_Unit_Normal(m,n,cp,k,0))

)

)

];

end

else

for j=0:G:1

x=[x (Bezier_Coordinates(1,m,n,cp,j,0,0,0))];

end

end

end

Code 14: y-coord

function y=y_coord(m,n,cp,G,e,w)

y=[];

if e==1

for k=G:G:1-G

y=[

y

(

Bezier_Coordinates(2,m,n,cp,0,k,0,0)

+ sum(

[0 Strain_Energy(m,n,cp,0,k,w) 0]

.* (Standard_Unit_Normal(m,n,cp,0,k))

)

)


];

end

else

for j=0:G:1

y=[y Bezier_Coordinates(2,m,n,cp,0,j,0,0)];

end

end

Code 15: z-coord

function z=z_coord(m,n,cp,G,e,w)

z=[];

lx=(1/G);

ly=(1/G);

if e==1

for j=1:(ly-1)

z=[z; zx_coord(m,n,cp,G,j,e,w,lx)];

end

else

for j=0:ly

z=[z; zx_coord(m,n,cp,G,j,e,w,lx)];

end

end

Code 16: Bézier Surface and Strain Energy Surface

function S = Bezier_Surface(m,n,a,G,e,w)

%Parameter a: Determines the shape of the Bézier surface.

% 0<a<1

%Parameter e:

% *If e=0, plot only Bézier surface.

% *If e=1, plot Bézier surface and the associated

% strain energy surface.

%Parameter w: Weight associated with the strain energy.

% w>1

cp=CP_matrix(a);

C=zeros(((1/G)+1),((1/G)+1));

49

Sx=x_coord(m,n,cp,G,0,w);

Sy=y_coord(m,n,cp,G,0,w);

Sz=z_coord(m,n,cp,G,0,w);

if e==1

SEx=x_coord(m,n,cp,G,e,w);

SEy=y_coord(m,n,cp,G,e,w);

SEz=z_coord(m,n,cp,G,e,w);

SC=Surface_Colouring(m,n,cp,G,w)

S=surf(Sx,Sy,Sz,C); hold on

SE=surf(SEx,SEy,SEz,SC)

print -deps graph.eps

elseif e==0

S=surf(Sx,Sy,Sz,C);

print -deps graph.eps

else

disp 'Error: e must be 0 or 1'

end

end

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c© 2018, Erik Bråmå

http://www.ep.liu.se/

http://www.ep.liu.se/

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