cad course master in aerospace and mechanical engineering · master in aerospace and mechanical...

1

Computer Aided Design

CAD course

Master in Aerospace and Mechanical Engineering

2


Service...

Eric Béchet (it's me !) Engineering studies in Nancy (Fr.) Ph.D in Montréal (Can.) Academic career in Nantes as Post-doc then ass. Prof. in

Metz (Fr.)then Liège...

Christophe Leblanc Assistant ( Ph.D. student ) at our university

Website http://www.cgeo.ulg.ac.be/CAO

Contacts : [email protected]

3


Procedures

12 Ex-Cathedra lectures (~2h except for the 1st) Practical sessions

On the NX CAD software (5 4h-sessions) Programming (8 2h-sessions)

Practical assessment : practical works (pw) are rated.

Final test about theory (tt) – Jan./Sept. The practical sessions, the project and the final

test are all mandatory, with over 8/20 to pass. The final mark m=pw/3 + 2tt/3, over 10 to pass & get the credits for the course.

4


Procedures Lectures are given on Tuesday, 2 PM – Lecture

room is O1 ( B37 bldg.) Begins today (exceptionally lasts 4h !), then

Sept. 26, Oct. 3, 10, 17, 24; Nov. 7, 14, 21, 28, Dec. 5, 14, 19

Practical work begins Sept. 26th after the course (check website for more dates & info) It is organized by groups of around 20 students

formed in accordance with availabilities. Any question : I am available mostly Fridays,

appointment via email or phone : 04-366-9165Office : Bldg B52 +2/438

5


Outline (1st few classes)

Introduction Brief history Some basics about solid modeling

Parametric forms Curves Surfaces

Implicit forms Plane curves et surfaces Implicit volume modeling

6


Outline (whole course)

Interpolation and polynomial approximation Cubic splines Bézier curves B-Splines Curves and rational surfaces In-depth B-REP models

7


Introduction

CAD/CAM – Computer Aided Design / Computer Aided Manufacturing Development since the 1950s-1960s

Iso Schoenberg (at U-Wisconsin) – splines (40's) First « interactive » CAD graphic engine at the MIT

(I. Sutherland, Sketchpad) – 60's James Ferguson (at Boeing) – splines (60's) Paul De Casteljau (at Citroën) – Bézier curves (50's) Pierre Bézier (at Renault) – '' '' Carl De Boor (at GM / U-Wisconsin) – B-Splines (70's)

Catia, Unigraphics (now Siemens NX) date back to the mid-70’s, emergence of 3d modeling kernels on mainframes.

Huge expansion in the 80’s with cheap personal computers becoming ubiquitous.

8


Introduction

Sketchpad (Ivan Sutherland) – 1963

9


Introduction

Old mathematical and theoretical bases Practical framework has been developed

because of industrial needs. In return, it has contributed at the theoretical

level.

10


Basics

11


Solid modeling

Many types of CAD systems First systems (such as Sutherland's) were aides for

drafting in 2D With the evolution of computers, 3D capabilities

were added Wireframe representation Surface modeling Then, real 3D modelers came over.

Today, most CAD system incorporate all capabilities.

One can start with surface design, from which a 3D model can be built, which yields at the end the 2D technical drawings that may be printed on paper.

12


Solid modeling

More generally, what are the geometric queries that a solid modeler is expected to respond to ?

Eg. « Is a given point inside or outside a volume, on a

face or an edge, or even on an existing vertex ? » « Generate a set of points located on the boundary

of the solid » « Compute the union of two solids » « Build one solid from a set of intersecting

surfaces »

...

13


Solid modeling

3D modelers, many types of representation for volumes Two families

Volumetric models By construction

Half spaces (level-sets) Composition of primitive objects (CSG)

By decomposition Exhaustive enumeration

and structured meshes Models defined by boundaries

Definition of the volume by the enumeration of its boundariesB-Rep

14


Solid modelling

Fundamental properties of solid models Power of expression

Degree in the capability to represent all real solids, without approximation.

ValidityAmong the possible representations within the model, are some representations invalid ?

15


Solid modelling

Fundamental properties of solid models Non-ambiguity and uniqueness

Does a representation in the model correspond to one or several real solids ? / Can a real solid be represented in different ways ?

ConcisionTo represent a particular solid, do we need many primitives or operations ?

The wireframerepresentationis ambiguous

16


Solid modelling

Fundamental properties of solid models Is the set of operations closed or open with respect to the

model ?During the application of an operation onto a model, can we obtain a non-valid solid, (are every operations valid ?) or a solid that is non representable ?

Calculation cost and applicabilityIs the construction of a solid within the selected model costly ?

Does the selected model address the issue raised by the user ?

17


CSG model

Description of a solid with the help of simpler primitives and elementary operations is ubiquitous in CAD modeling : CSG model (Constructive Solid Geometry) Internal representation

is in the form of a binary tree Leaves are geometric primitives Nodes are operations

« union », « difference »,« intersection »

Possible to :« move », « complement », « change scale » etc...

Wik

iped

ia

18


Solid modeling

Characteristics of the modeling by half-spaces / CSG

Power of expression: depends on the set of available half-spaces (planes, quadratic surfaces, etc.) or primitivesIt is not possible to represent a single surface limited by 4 curves (a patch) , for example.

Validity : The half spaces are infinite (limited by planes for example). A combination can give an infinite volume i.e. not acceptable under the assumption that solids are finite and “closed” volumes.

Ambiguity/uniqueness : each valid combination of half-spaces determines an unique solid → non ambiguous.A given solid can be represented by several different combination of half-spaces (non uniqueness)

19


Solid modeling

Concision : the model by half-spaces is relatively concise.

Closedness of the set of operations : we focus here on boolean operations of union and intersection, these always give a valid description → closed set of operations

Calculation cost and applicability

The operation of modeling itself is not costly (update of a tree). The extraction of useful information can be much more difficult e.g. calculation of surfaces... or very simple such as collision detection (inside / outside test for a point)

20


Solid modeling CSG models

User-friendliness Widely used in the interface shown by the CAD software

to the user In general many CAD software

integrate some sort of history of the constructionof the model (cf model tree in Catia)

21


Solid modeling

Modeling by spatial decomposition Division in regular cells Exhaustive enumeration of non empty cells

M. M

änty

lä

22


Solid modeling

Modeling by spatial decomposition Cells of variable size Recursive division of a cell containing the whole

solid in smaller cells located along the boundary of the solid, until a prescribed error is reached.

23


Solid modeling

Modelling by spatial decomposition Quadtree (2D)

Each node has 4 children. Each children has a different representation in function of

the “filling” of the area

1 2

3 4

24


Solid modeling

Modelling by spatial decomposition Quadtree (2D)

Each node has 4 children. Each children has a different representation in function of

the “filling” of the area

1

2.1

4

2.2

2.3 2.4

3.1 3.2

3.3 3.4

25


Solid modeling

Modelling by spatial decomposition Boolean operations : union, complement,

difference, intersection

U =

U =

U =

U =

U =

U =

Union

U

=

=

=

=

=

=

Intersection

U

U

U

U

U

=

=

=

=

=

=

difference

\

\

\

\

\

\

=\

=

=

=

complement

_

_

_=\

26


Solid modeling

Modelling by spatial decomposition Octree (3D) – analogy : voxels

Each node has 8 children. Often used in medical images as means of efficient

storage Effective means of geometric

location in 3D space

27


Solid modeling

Characteristics of the modelling by spatial decomposition

Power of expression: By definition, it's an approximation as soon as the boundaries of the object are not aligned with the cells

Validity : always valid Ambiguity/uniqueness

Every description corresponds to an unique solid → non ambiguous.

A given solid is represented by a single way for the same root cell (uniqueness)

→ However, since the representation is approximative, we can consider that many real solids can lead to the same representation.

28


Solid modeling

Concision : For a precise representation, the model can become very heavy thus it is not very concise

Closedness of the set of operations : we speak here about boolean operations of union and intersection, that systematically give a valid description → closed set of operations


Such models are easy to manipulate but with a limited applicability due to high calculation costs (large quantity of data). However some operations are efficient , collision (inside/outside) test for instance.

29


Solid modeling

Description by meshes It is a variant of the exhaustive decomposition that

is respecting to a certain extent the boundaries of the object

Cells are not regularly organized anymore Used mainly for scientific computations

30


Solid modeling

B-Rep model « Boundary representation » Model based on the representation of surfaces Model of exchange (STEP format) and definition The “natural” set of operators is richer than for CSG

Extrusion, chamfer etc ... Does not inherently carry the history of construction

of the model (whereas CSG usually does)

31


Solid modeling

B-Rep model Consists of two type of information

GeometricGeometric information is used for defining the spatial position, the curvatures, etc...

That's what we will see in the next couple of lectures – Nurbs curves and surfaces

TopologicalThis allows to make links between geometrical entities.

Two types of entities Geometric entities: surface, curve, point Topological entities : volume, face, edge, vertex

A topological entity “lies on” a geometric entity, which is its geometrical support

32


Solid modeling

B-Rep model Simplified hierarchical model vertex-edge-face-

volume

ff

f

v

ev v

33


Solid modeling

B-Rep model Simplified hierarchic model vertex-edge-face-volume

v1

v2

v3

v4

a1

v8

v5

v6

v7

e2

e3

e4 f

1

f1 e

1 e

2 e

3 e

4

f2 … … … ...

f3 … … … ...

...

e1 v

1 v

2

e2 v

2 v

3

e3 v

3 v

4

e4 v

4 v

1

...

v1 x

1 y

1 z

1

v2 x

2 y

2 z

2

v3 x

3 y

3 z

3

...

Composed of

Composed of

Volume 1

Composed of

34


Solid modeling

B-Rep model Complete hierarchical structure found in most solid

modelers

Volume « shells » Boundary Face

face « loops » Boundaryedge

edge Boundary

vertex

vertex G.S. Expressed inparametric coordinates u,v or t

If it exists, the geometric support (G.S.) is expressed in cartesian coordinates x,y,z

« Composed of »

« is related to »

1:n 1:n

1:{1 if manifold , 2 otherwise }

1:n 1:n

1:{2 if manifold, n otherwise }

1:2

1:n

35


Solid modeling

Characteristics of the B-REP modelling Power of expression: Allows to represent any solid if we

dispose of appropriate surfaces/curves (NURBS) Validity : Every B-REP is not necessarily valid

Topological validity (see the Euler Poincaré relations in the sequel)

Geometric validity Ambiguity/uniqueness : Every valid B-REP represents an

unique solid → non ambiguous.There are many different ways to represent a given solid (non uniqueness)

36


Solid modeling

Concision : Generally concise Closedness of the set of operations : In general,

operations made on B-REPs do not necessarily yield regular models of real solids (in particular for boolean operations)

Euler operators do maintain the topological validity, but not necessary the geometric validity


boolean operators are costly, but other modelling operators may be easier to implement using a B-REP representation. In general, modelling operators are more rich than for CSG

37


Solid modeling

Validity of the model The boundary of a face is made up of edges that are not

allowed to intersect each other

invalid

38


Solid modeling

Validity of the model The faces of a model can only intersect in common

edges or vertices.

Invalid

Pyramid on base

39


Solid modeling

How to represent individual entities ? CSG

Primitives (Leaves of the CSG tree) → equations ??? Operators (Nodes " " " ) → How ?

B-REP Vertices → trivial (coordinates) Edges → straight line or ??? Faces → Planar or ??? Volumes → Indirectly Need to define a geometric support.

40


Point

Point Defined in an euclidean space (2D or 3D)

Notion of distance between two points is trivial. Sometimes defined in a non euclidean space (e.g.

parametric space, in 1D = curve or 2D=surface ) Ex : points of a curve localized on a parametric surface. Notion of distance between two points is not trivial on a

curve or a surface. Sometimes, use of homogeneous coordinates (4th

coordinate) 4 dimensions + one equivalence relation → 3D Perspective division to come back to euclidean 3D space. Rational curves and surfaces.

41


Point

Homogeneous coordinates and class of equivalence Coordinates x, y, z and w (weight) All points (4D) located on a line passing through the

origin O(0,0,0,0) are equivalent. Particularly, a point located on the plane w=1.

Change from homogeneous coordinates to euclidean coordinates by perspective division:

(xyzw)≡(

x /wy /wz /w

1)≡(

x /wy /wz /w )

42


Curve

Curve : 1D manifold in a space of higher dimension Plane curves (2D)

Position, tangent, curvature Skew curves (3D)

Position, tangent, curvature, torsion

43


Curve

Meaning of curvature k Deviation rate of the curve in comparison with a

straight line. Radius of curvature is the inverse of the curvature We can define a vector n that is normal at the

tangent and point to the center of curvature : it is the normal vector

n

t

44


Curve

Meaning of torsion t Deviation of the curve in comparison with a plane

formed by the tangent and the normal vector... For a plane curve, torsion is zero. We can define a vector normal to normal vector and

also to the tangent : it is the binormal vector b b points to the torsion center

nt

b

45


Continuity

Geometrical continuity : Given two « independent » curves : They are discontinuous

of continuity G0 (position p)

of continuity G1 (p+tangent t)

of continuity G2 (p+t+curvatures)

... G

46


Surfaces

2D manifold in 3D space Tangent plane (defined by two linearly independent

vectors tangent to the surface) Normal vector Several curvatures may be defined

Gaussian (or total) curvature and average curvature Principal curvatures

47


Different representations of curves and surfaces

48


Modes of representation

There are many Parametric form Explicit form Implicit form Etc. (eg. solution of partial differential equations –

PDEs … )

49


Parametric representation

50



Use of scalar parameters to “walk” on the curve/surface The number of those scalars determine the dimension

of the entity and the mapping from one space (the parametric space to the other (usually cartesian) space.

0 – Point , 1 (u) – Curve , 2 (u,v) – Surface

3 (u,v,w) – Volume ( transformation used for finite elements) We associate to those scalars some function for each

of the space coordinates ( x,y in the plane; x,y,z in 3D)

P⃗ (u)={x= f (u)y=g (u)z=h(u)

or P⃗ (u , v)={x= f (u , v)y=g (u , v)z=h(u , v)

51



Parametric curves

52



For a curve, the representation takes the following form:

u is a real parameter. We obtain all the points of the curve by varying u. The parametrization is not unique ! The parameter u can be limited (limits of the curve)

P u={x= f uy=g uz=h u x

y0

1

2 3 4

5

67

8

53



Continuity of the parametrization A parametric curve is said to be C

k if the

parametrization P(u) is Ck (i.e. the kth derivative is

continuous) However : a parametric curve of class C

may have an

angular point... (i.e. some continuity of the parametrization does not necessarily involve the same geometric continuity of the resulting curve !)

P u={x=u3

y=u2

x

y

P 'u=0 ={00

54



Regularity of the parametrization A parametric curve P(u) is regular if the first

derivative P'(u) does not vanish at any place on all the interval of definition.

On the contrary, points where P'(u) vanishes, are called singular points.

A given curve can admit two parametric forms such that one is regular and the other one isn't...

Example : curve y=x2 with two parametric forms P(u) and Q(v)

P u={ x=uy=u2

Q v={x=v3

y=v6P 'u=0 ={10 Q '

v=0 ={00

55



Regularity and continuity of the parametrization A curve of geometric continuity G

k can be described by

a parametrization that is non- Ck

A curve of parametrization Ck may not be G

k

A curve admitting singular points can be Gk and/or C

k.

A regular curve is not necessary Ck and/or G

k.

To conclude : Parametric continuity, geometric continuity, and

regularity are all relatively independent notions!

56



Ck -equivalent parametric forms

Two parametric forms P(u) and Q(v) of the same curve are C

k - equivalents if there is a bijective

function j of class Ck whose reciprocal is also of

class Ck and such that:

v=u P u=Q u

57



Length of a regular curve It is a measure that is independent of the

parametrization (provided that those are C1-

equivalent)

L=∫a

b

∣P 'u∣du= ∫

c=a

d=b

∣Q 'v∣dv

58



Natural parametrization It is a parametrization s such that :

s is also named curvilinear abscissa along the curve.

It can be built from any regular parametrization u of the curve :

L AB=s B−sA x

yA

s=0.1

B

su =∫A

u

∣P ' t ∣dt

59



Parametrization of a circle in the plane :

Natural parametrization of the same circle...

r{x=r cos uy=r sin u

, u∈[0, 2 [

{x=r cos

sr

y=r sinsr

, s∈[0, 2 r [ s=r u

60



Differential geometry for parametric curves Position P :

Tangent vector T :

Normal vector N :

Binormal vector B :

T u=dPds

=dPdu

⋅duds

=P '

∣P '∣

P u=x u y u z u P '

=dPdu

P ''=

d 2 P

du2

N (u)=Norm(u)

∣Norm(u)∣ with Norm(u)=P ''

−( P ''⋅T )T

B u=T u×N u

x

y

TN

z B

(It is not defined when Norm(u) = 0 !)

61



Frenet's frame and equations

2D

3D [d T s

dsd N s

dsd Bs

ds]=[

0 s 0−s 0 s

0 −s 0 ]⋅[T sN sBs ]

T s

N s

B s

s

[d T s

dsd N s

ds]=[ 0 s

− s 0 ]⋅[T sN s]

Works only with s : the curvilinear abscissa !

62



Curvature k :

Torsion t:

= d 2 T

ds2 ,dTds

,T

2 = P ''' , P '' , P '

∣P '×P ''∣

2

|k|=|dTds |=

|P '×P ''|

|P '|3

u=T 'u⋅N u

∣P 'u ∣

u=N '

u⋅B u

∣P 'u∣

Curvature vector

=N u u

63



Parametric surfaces

64



A parametric surface is represented as :

u,v are two real parameters All the points of the surface can be

obtained by varying u and v. Again, the parametrization is not unique ! The interval of definition is limited (limits the

surface)

P u , v ={x= f u , v y=g u , v z=hu , v

x

y z

u,v

u=cst

v=cst

65



A curve can be defined on the surfaceParametric space « Ambient » space

The equations seen for parametric curves are valid in this particular case. x

y z

t

u=cst

v=cst

P u , v : {x= f u , v y=g u , v z=hu , v

t :{x= f u t , v t y=g u t , v t z=hu t , v t

uvt : {u=u t

v=v t

66



Regularity and continuity of the parametrization A parametric surface is of class C

k if the mapping

P(u,v) onto R3 is of class Ck.

A parametrization is regular if and only if (iff)

The points on the parametric space for which it is not the case are called singular points.

Ck–equivalent parametric forms follow the same

definitions as for curves

∂ P∂u

u0 , v0×∂ P∂ v

u0 , v0≠0 ∀u0 , v0∈D⊂ℝ2

67



Differential geometry for parametric surfaces

Position P :

Unit tangent vectors T u and T v :

These vectors are, in general, not orthogonal ! Tangent plane (parametric form)

or

T uu , v=

∂ P∂u

⋅∣∂ P∂ u ∣

−1

=P u

∣P u∣T v

u , v =∂ P∂ v

⋅∣∂ P∂ v ∣

−1

=Pv

∣Pv∣

P u , v=[x u , vy u , v z u , v ] Pu

=∂ P∂ u

Puv=

∂2 P

∂u∂v⋯

Pt u0 , v0a ,b=P u0 , v0a⋅T u

u0 , v0b⋅T vu0 , v0

a ,b∈ℝ2

P⃗t(u0 , v0)(a ,b)=P⃗ (u0 , v0)+a⋅P⃗u

(u0 , v0)+b⋅P⃗v(u0 , v0)

68



Normal vector N :

N (u , v)=Norm(u , v)

|Norm(u , v)| with Norm(u , v)=T u

×T v or Pu×P v

u ,v

u=cst

v=cst

T u

T v

N

69



Area of a surface

1st fundamental form Other notation of the area of a surface

du P u

dv P v

dS

A=∬Ω

dS

dS=∣du⋅Pu×dv⋅Pv

∣=∣Pu×P v

∣dudv

∣a×b∣2=a⋅a⋅b⋅b−a⋅b2

A=∬D

eg− f 2dudv

Lagrange's identity

dS=√(e g− f 2)dudv with e=Pu

⋅Pu , f =Pu⋅Pv , g=Pv

⋅Pv

70



Length of a curve on a surface

Γ⃗uv(t): {u=u(t )

v=v (t )Γ⃗(t ):{

x= f (u(t ) , v (t ))y=g (u(t ) , v (t ))z=h(u(t) , v (t ))

Γ'=

dP (u(t ) , v (t ))dt

P u , v : {x= f u , v y=g u , v z=hu , v

Pu=∂ P u , v

∂u

Pv=∂ P u , v

∂ v

't =u'

t Puu t , v t v '

t Pvut , v t

Γ'=∂ P (u(t ) , v (t ))

∂ududt

+∂ P (u(t) , v (t ))

∂ vdvdt

Derivative of the curve expressed in the parametric space of the surface Derivative of the surface (in 3D)

71



L=∫a

b

∣ ' t ∣dt=∫

a

b

∣ 't ∣

2

dt

't =u'

t Puu t , v t v '

t Pvut , v t

∣'t ∣2

=e u' t 22 f u '

t v 't g v '

t 2

with e=Pu⋅Pu , f =Pu

⋅Pv , g=P v⋅Pv

72



If we set

that is equivalent to :

( we have )

, we get a quadratic form :

ds=√e u'(t )2

+2 f u'(t )v '

(t)+g v '(t)2 dt

L=∫sa

s b

ds=sb −sa

ds=√e du2+2 f dudv+g dv2

e u't 2

2 f u't v '

t g v 't 2=u'

t v ' t e f

f g u't

v 't

L=∫a

b

u't v '

t e ff g u

't

v 't dt

73



Angle between two curves ...

cosα=

(u1'(t 1) v1

'(t1))( e f

f g )(u2

'(t 2)

v2'(t 2))

√(u1'(t1) v1

'(t1))( e f

f g )(u1

'(t1)

v1'(t1))(u2

'(t 2) v2

'(t 2))( e f

f g )(u2

'(t2)

v2'(t 2))

Γ1'(t1)⋅Γ2

'(t 2)=∣Γ1

'(t1)∣∣Γ2

'(t 2)∣cosα=(u1

'(t 1) v1

'(t 1))( e f

f g )(u2

'(t 2)

v2'(t 2))

74



The first fundamental form is the application :

It is a symmetric bilinear form that allows to measure real distances from variations in the parametric space...

The matrix M1 is a representation of the metric tensor,

often noted

M1 also is related to the Jacobian matrix

of the transformation (u,v) → (x,y,z) (it is JTJ ).

j1(d Γ1uv , d Γ2

uv)=(du1 dv1)( e f

f g )(du2

dv2)=(du1 dv1 ) M1(

du2

dv2)

L=∫a

b

√j(d Γuv , d Γ

uv)dt cosα=

j(d Γ1uv , d Γ2

uv)

√j(d Γ1uv , d Γ1

uv)j(d Γ2

uv , d Γ2uv)

A=∬D√det M1 dudv ( =∬

D

det J dudv under some conditions)

with e=Pu⋅Pu , f =Pu

⋅P v , g=P v⋅Pv

J=(∂ x /∂u ∂ x /∂ v∂ y /∂u ∂ y /∂ v∂ z /∂ u ∂ z /∂ v )

g ij , i , j=u , v

75



Curvatures...

Normal curvature :

Geodesic curvatureor relative curvature w/r to the surface :

Simple or total curvature of the 3D curve :

kn

kg

k

76



Curvature vectors

Necessarily, we do have :

Differentiating yields :

N

P(u,v)

n

k

kn

kg

T

k=d Tds

=k⋅n=kn+k g

Center of curvature

(e.g. from Frenet's relations)

N⋅T=0

d Tds

⋅N +T⋅d Nds

=0

kn=d Tds

⋅N=−T⋅d Nds

=−d P (u , v)

ds⋅

d Nds

=−d P⋅d N

ds2 =l du2

+2 mdudv+n dv2

e du2+2 f dudv+g dv2

l=−dPdu

⋅d Ndu

m=−12(

dPdu

⋅d Ndv

+dPdv

⋅d Ndu

) n=−dPdv

⋅d Ndv

with

G=N×Tkn=kn⋅N k g=kg⋅G

(kg=d Tds

⋅G ) (ds=

√e du2+2 f dudv+g dv2)

77



This can be rewritten by noting that

and

, and integrating by parts

where e.g. for l :

l=−dPdu

⋅d Ndu

m=−12(

dPdu

⋅d Ndv

+dPdv

⋅d Ndu

) n=−dPdv

⋅d Ndv

dPdu

⋅N=0dPdv

⋅N=0

l=d 2 P

du2 ⋅N m=d 2 Pdu dv

⋅N n=d 2 P

dv2 ⋅N

UV=∫U dV +∫V dU

U=dPdu

dV=dNdu

⋅du

78



Curvatures and the second fundamental form The second fundamental form is the application

Normal curvature of a curve on a surface :

Geodesic curvature (curvature in the tangent plane) Measures the deviation of the curve in comparison with a

geodesic

j2(d Γ1uv , d Γ2

uv)=(du1 dv1)( l m

m n )(du2

dv2)=(du1 dv1 ) M2(

du2

dv2)

with l=N⋅Puu , n=N⋅Pvv , m=N⋅Puv

kn=j2(d Γ

uv , d Γuv)

j1(d Γuv , d Γ

uv)

kn=j2(d Γ

uv , d Γuv)

ds2

79



Definition of the geodesic It is a curve with a minimal length on a surface.

In the plane, those are lines On a sphere, great circles By definition, the geodesic has no geodesic curvature.

Relations between « simple » curvature, geodesic curvature and normal curvature is obtained from the curvature vectors

The geodesic curvature is therefore readily obtained from the others (except for the sign, which is same as )

k2=kn

2+kg

2

kg2=k

2−kn

2

k=kn+k g kn=kn⋅N k=k⋅n k g=kg⋅(N×T )=kg⋅G

d Tdu

⋅G

80



A circle of radius R in the plane

Curvature k = 1/R

Geodesic curvature kg= 1/R

Normal curvature kn= 0

A big circle on a sphere or radius RCurvature k

= 1/R

Geodesic curvature kg= 0

Normal curvature kn= 1/R

In the previous slides, k is the curvature of the curve in 3D (not in the parametric space !!!)

81



Curvatures of the surface

The normal curvature kn at a point only depends on the

local orientation of Γ on the surface. If we sweep through all the orientations, this curvature passes by a minimum k

min then a maximum k

max for two perpendicular directions.

These directions and the particular values of kn are

respectively called principal directions and principal curvatures of the surface. They do not depend on the parametrization of the surface...

They are the solution of an eigenvalue problem. The Shape operator (also called Weingarten Map)

plays a key role here, see what follows.

kn=j2(d Γ

uv , d Γuv)

j1(d Γuv , d Γ

uv)

82



Shape operator

Here, N is the normal to the surface, p is a point on the surface, w is a tangent vector to the surface at p. is a directional derivative along a unit vector w (in the surface) :

it is equal to .This operator gives back a vector in the tangent plane.

If one works in an orthonormal basis, eigenvectors are easy to compute as the operator is symmetric...

Let's build such an orthonormal basis from the natural one :

W p(w )=−∇w⋅N

∇w

P u=∂ P∂u

P v=∂ P∂ v

N=N (u , v )=P u

×Pv

‖P u×Pv‖

t1=Pu

‖Pu‖

t 2=N×t1=P v

−(Pv⋅t 1)t1

‖P v−(Pv

⋅t 1)t1‖

wu∂∂u

+w v∂∂ v

w=wu⋅Pu+wv⋅Pv

83



It is easy to show that one can switch from one basis to the other using coefficients of the 1st fundamental form …

One can also switch the other way back, working on coefficients rather than on vectors e.g. for any vector

One gets

t1=Pu

‖Pu‖=

Pu

√et 2=

e P v− f Pu

√e √eg− f 2=

e Pv− f P u

√e √h

[ t1

t 2]=1

√e√h [ √h 0− f e ][P

u

P v]

w⃗=w1 t1+w2 t 2=wu P u+wv Pv

[wu

wv]=

1

√e √h [√h − f0 e ][w1

w2]=Φ [w1

w2]

84



Now back to the shape operator : we will start with a vector expressed in the orthonormal frame (w

1,w

2) , convert it to the parametric basis (w

u,w

v),

take the directional derivative of the

normal n and then project back to the orthonormal frame.

Here, is precisely the matrix of the

coefficients of the second fundamental form...

wu∂∂u

+w v∂∂ v

W p(w )=−[ t1

t 2][nu nv]Φ [w1

w2]=−Φ

T [Pu

P v][nu nv]Φ [w1

w2]−[P

u

P v ][nu nv ]

85



In the orthonormal basis

In the original basis

(Weingarten Equation)

W p(w)=ΦT [ l m

m n ]Φ[w1

w2]=

1e h [ √h 0

− f e ][l mm n ][√h − f

0 e ][w1

w2]

=1eh [ h l (e m− f l)√h

(e m− f l )√h f 2 l+e2 n−2e f m][w1

w2]

~W p(w)=ΦW p(w)Φ−1 [wu

wv]=Φ(Φ

T [ l mm n ]Φ)Φ

−1[wu

wv]

=ΦΦT [ l m

m n ][wu

wv ]=1h [

g − f− f e ][ l m

m n ][wu

wv ]=[ e ff g ]

−1

⋅[ l mm n ][wu

wv ]

86



Eigenvalues and eigenvectors of the shape operator

The shape operator gives the normal curvature for a given orientation (i.e. the derivative of the unit normal vector with respect to a small increment along that orientation on the surface)

One can work either in the original frame (non orthonormal)

Or the orthonormal frame, for which the matrix is symmetric and the 2D eigenvectors are orthogonal

(L−k I )(dudv)=(00) with L=M 1

−1 M 2=~W p(w ) det (L−k I )=0

(W p(w)−k I )(dt1

dt 2)=(00) det (W p(w)−k I )=0

87



The eigenvectors are obtained using any of the relations:

Just use the other relation if the basis contains a null vector for one of the values of :

(dudv )⋅(

l−k em−k f )=0 (du

dv )=α⋅(m−k fk e−l ) , α>0: du2

+dv2=1

(dudv )⋅(

m−k fl−k g )=0 (du

dv )=α⋅( l−k gk f −m) , α>0: du2

+dv2=1

k

(M 2−k M 1)(dudv )=(00) ( l−k e m−k f

m−k f n−k g )(dudv)=(00)

88



ExampleP u , v=

uv

cos uPu

u , v=10

−sin u Pvu , v=

010

N (u , v)=Pu

×P v

∣Pu×P v∣

=(−sin u

01 )⋅ 1

√1+sin2 u

Puuu , v=

00

−cos uPvv

u , v =000

Puvu ,v=

000

89


M1=Pu⋅Pu Pu

⋅Pv

Pu⋅Pv Pv

⋅P v=1sin2 u 00 1

(L−k I )(dudv)=(00) with L=M 1

−1 M 2

M2=(N⋅Puu N⋅Puv

N⋅Puv N⋅Pvv)=(−cos u

√1+sin2 u0

0 0)


90



Numerical application : u=0 and v=0

M1−1=1 0

0 1

L=M 1−1 M 2=−1 0

0 0

M1=1sin2 u 00 1=1 0

0 1

M2=(−cos u

√1+sin2 u0

0 0)=(−1 00 0)

1=−1 c1=102=0 c2=01

c1

c2

o

C13D=(Pu , Pv)⋅(10)

C 23D=(Pu , Pv)⋅(01)

91



Umbilical points Points whose normal curvature k

n is independent of

the orientation of Γ . Locally, the surface looks like a plane or a sphere.

In this case, the following relations hold for the matrices M

1 and M

2 :

le=

mf=

ng

if f ≠0

le=

ng

if f =0 and m=0

kn=le=kmin=kmax= const.

Principal directions are arbitrary.

92



Gaussian curvature and average curvature Invariants (do not depend on the parametrization)

M=minmax

2=

12

trace L=12

e ng l−2 f m

e g− f 2

kT=kminkmax=det (L)=det (M 2)

det (M 1)=

l n−m2

e g− f 2

L=M 1−1 M 2

93



Local nature of the surface k

T>0 the surface is locally an ellipsoid – elliptic point

kT<0 : hyperbolic paraboloid – saddle point –

hyperbolic point

kT=0 and k

M<>0 : parabolic cylinder – parabolic

point

kT=0 and k

M=0 : no information – locally plane

94


Christoffel symbols

Complete developments. Christoffel symbols Expression of the first derivatives along a

curve...

Lets differentiate that expression again...

Γ ' (t )=u ' Pu+v ' P v

u'=

du(t )dtPu=

∂ P (u , v)∂u

Pv=∂ P (u , v)

∂ vv '=

dv (t)dt

95


Christoffel symbols

The terms are in the tangent plane The terms can be expressed as

linear combinations of tangential and normal components :

The are the Christoffel symbols (often written in textbooks ) and the are the coefficients of the second fundamental form.

(Γ ' (t)=u ' Pu+v ' P v) '

Γ" (t )=u" Pu+u ' (u ' Puu+v ' Puv)+v " Pv+v ' (u ' P vu+v ' P vv)

=u" Pu+v " Pv+u ' 2 Puu+u ' v ' Puv+u ' v ' P vu+v ' 2 P vv

u" Pu+v " P v

P ij , i , j={u , v }

P ij=C iju Pu+C ij

v P v+Lij N

C ijk

Lij

Gauss formula

Γijk

96


Christoffel symbols

Using Einstein's notation (summation over repeated indices ):

How to compute and ? Multiply by

P ij=C ijk P k+Lij N , i , j , k=u , v

C ijk Lij

P l , l=u , v

P ij⋅P l=C ijk P k⋅P l+Lij N⋅P l

0

Coefficients of the 1st fundamental form = = metric tensorg kl

P ij⋅P l⋅(g kl )−1=C ij

k g kl⋅g kl−1

C ijk=P ij⋅P l⋅g lkC ij

k=P ij⋅P l⋅(g kl )

−1

Conventional notation of the inverse of the metric tensor

97


Christoffel symbols

Now, multiply by

One recovers the formula seen before:

P ij=C ijk P k+Lij N

N

P ij⋅N=C ijk P k⋅N +Lij N⋅N

0

1

Lij=P ij⋅N

Lij≡M 2 M2=(N⋅Puu N⋅Puv

N⋅Puv N⋅Pvv)

98


Christoffel symbols

Use of Christoffel symbols and the second fundamental form Let us return to the expression

Now replace by and use the Einstein notation again :

Now substitute

Γ" (t )=u" Pu+v " Pv+u ' 2 Puu+u ' v ' Puv+u ' v ' P vu+v ' 2 P vv

u , v ui , i=1,2

Γ" (t )=(ui)

" P i+(ui)

'⋅(u j

)' P ij

P ij=C ijk P k+Lij N

Γ" (t )=((uk)"+C ij

k(ui

)'⋅(u j

)')P k+(u

i)

'⋅(u j

)' Lij N

Total tangential component Normal component

99


Christoffel symbols

Used to express curvatures (or second derivatives) in the 3D euclidean space from curvatures (or second derivatives) expressed in the parametric space.

Normal curvature: based on the normal component

If we have an arc-length parametrization :

if not : need to adjust

One recovers the formula :

kn=Lij(u

i)

'(u j

)'

ds2 =Lij(u

i)'(u j

)'

g kl (uk)'(ul

)'

kn=Γ"⋅N =Lij(ui)'(u j

)'

( =j2(d Γuv , d Γ

uv))

(ui)

'⋅(u j

)' Lij N

kN=j2(d Γ

uv , d Γuv)

j1(d Γuv , d Γ

uv)

100


Christoffel symbols

Geodesic curvature : based on the tangential components

If the curve is parametrized by arc-length :

Therefore, the tangential component of is orthogonal to and to N as well. There is a proportionality factor between and .This factor is the geodesic curvature

((uk)

"+C ij

k(ui

)'⋅(u j

)')P k

Γ"⋅Γ '=0=((uk)

"+C ij

k(ui

)'⋅(u j

)')P k⋅Γ '+(ui

)'⋅(u j

)' Lij N⋅Γ '

Γ '⋅Γ '=1 Γ '⋅Γ"=0=0

Γ"Γ '

Γt "

Γt " N×Γ '

101


Christoffel symbols

Therefore, , valid only if the parametrization of is normal (arc-length)

If not, need to adjust ...

Some developments* leads to the following identity :

Again, this expression makes only use of derivatives in the parametric space...

kg=[N ,Γ ' ,Γ" ]Γ

kg=√det (M 1)[−C11

2 u ' 3+C 22

1 v ' 3−(2C12

2−C 11

1)u ' 2 v '+(2C12

1−C22

2)u ' v ' 2

+u " v '−v " u ' ]

((u ' v ' )M1(u 'v ' ))

3/2

Γt "=kg (N ×Γ ' ) Γt "⋅(N×Γ ' )=kg(N×Γ ' )⋅(N×Γ ' )

kg=(N×Γ ' )⋅Γt "=(N×Γ ' )⋅Γ"

kg=[N ,Γ ' ,Γ" ]

ds3 =[N ,Γ ' ,Γ " ]

(g kl (uk)

'(ul

)' )

3/2

* see e.g.“Modern Differential Geometry of Curves and Surfaces with Mathematica”, 2nd ed. Boca Raton, FL: CRC Press, pp. 501-518, 1997.

102


Christoffel symbols

Equations of a geodesic

Need to set that the geodesic curvature is equal to zero.

One obtains a set of differential equations May be used to actually compute an approximation

of the geodesic since it is usually impossible to solve that coupled system of ODEs algebraically.

103



Darboux frame and equations

s is the curvilinear abscissa

T s={∂ x s∂ s

,∂ y s∂ s

,∂ z s∂ s }

Su , v

s

uvs:{u=u s

v=v s s: {

x s= f u s , v sy s=g u s , v sz s=hu s , v s

P⃗ (u , v): {x= f (u , v)y=g (u , v)z=h(u , v)

N s= N u s , v s

G⃗ (s)=N⃗×T⃗

Darboux frame

T s

G s N s

104



Darboux frame and equations

[d T s

dsd G s

dsd N s

ds]=[

0 g s n s−g s 0 − g s−n s g s 0 ]⋅[

T sG sN s]

S u , v

s

Darboux frame

T s

G sN s

Geodesic torsion (or relative torsion)

105



Geodesic torsion

is identically zero iff the curve is along a direction of principal curvature (line of curvature)

Geodesic curvature

is identically zero iff the curve is a geodesic

106



In an euclidean space: The shortest line between two points has simple

curvature k equal to zero (= straight line) The minimal surface carried by a curve is of

average curvature kM equal to 0 everywhere

In a non euclidean space (on a surface): The (or one of the) shortest line between two points

on the surface has a geodesic curvature kg equal to

zero (it's a geodesic)

107



A good reference on « shape interrogation » Shape interrogation is the ability to get information

from geometric models

N. M. Patrikalakis and T. Maekawa, Shape Interrogation for Computer Aided Design and Manufacturing, Springer Verlag, February 2002. ISBN 3-540-42454-7

(Available as a PDF electronic copy at the library)

108



Some results of differential geometry

Gauss-Bonnet's theorem

is Euler's characteristic of surface S

∫S

kT dA+∫∂S

kg ds=2π χ(S )

S

=2 =0=−2

=−4=1

109



Application of Gauss-Bonnet's theorem For a polyhedron, we have (Euler's formula) :

We want to approximate a sphere with hexagons ( H ) and pentagons ( P ).

Each pentagon (hexagon) has 5 (6) nodes, shared by 3 faces Each pentagon (hexagon) has 5 (6) edges, shared by 2 faces So the total number of nodes N is ( 5P+6H )/3 Total number of edges E is ( 5P+6H )/2, total number of faces F

is P+H

The sphere is compact and its gaussian curvature is 1/R2

so P=12 … 12 pentagons are necessary and as many hexagons as we want to make soccer ball.

S =N−EF

S =N−EF=5P6H /3−5P6H /2PH=P /6

∫S

T dA∫∂ S

G ds=1

R2∫S

dA=4=2 S

110



111


Explicit curves and surfaces

112


Explicit forms

Curve as a function

We can swap x, y and z according to needs

y= f x z= f x , y

xOK !

y= f x

y

x

y

xOK !

x= f y

y

KO !

113


Explicit forms

Polar coordinates Sometimes used. Circle is readily represented :

r =1

114


Explicit forms

Definition of an explicit surface Depend on reference axis No multiple definitions i.e. unique value of z for a

position (x,y) For a given axes system, only one definition No closed curves /surfaces in one time

Unit circle – we must split it up at least in :

and y=1−x2 y=−1−x2

115


Implicit representation

116


Implicit forms

Ex. Unit circle : Infinity of representations of a

given curve Invariance by the multiplication by a constant

We can take absolute value too

... but the form is then not differentiable etc...

f x , y=0

x2 y2

−1=0

k⋅ f x , y=0 ⇔ f x , y=0 ∀ k≠0

∣ f x , y ∣=0 ⇔ f x , y =0

f =0

117


Implicit forms

Advantages of an implicit form

For a point we know : If it is on the curve : If it is not on the curve : In some cases, we may know if it is inside or

outside the volume delimited by the curve:

If , P is inside the circle

But : has no meaning Multiplication by negative k : interior exterior

P [ x p , y p]

f x p , y p=0f x p , y p≠0

f ( x p , y p)<0 or f ( x p , y p)>0

x p2 y p

2−10

∣x p2 y p

2−1∣0

118


Implicit forms

Normal forms / Contour line / level sets Particular case of implicit forms such that :

, is the signed distance to the curve In this case, for a point ,

is the distance from the point to the curve. d<0 means usually that the point is inside.

Case of the unit circle :is the distance to the circle.

Unique representation !(except the sign of the distance function – but it indicateswhere is the interior)

x , y =d dx p , y pP [ x p , y p]

x , y= x2 y2

−1

0

0

=0

119


Implicit forms

Normal forms A normal form is a distance function, we have :

For the circle :

∣∇∣=∣∂

∂ x∂

∂ y∂

∂ z∣=∣

2 x

2 x2 y2

2 y

2 x2 y2

0∣= x2

x2 y2

y2

x2 y2

=1

∣∇ x , y ∣=1

120


Implicit forms

Normal forms Ellipse ? We can « flatten » a circle :

Set

x* , y*= x*2

y*2−1

y*=2 y ; x*

=x

x , y= x22 y 2−1

0

0

=0

∣∇e∣=∣x

x24 y2

4y

x24 y2

0∣= x2

16 y2

x24 y2 ≠1

121


Implicit forms

The normal form associated with a curve (or a surface in 3D) is usually nontrivial.

To get it nevertheless: Use of a numerical approximation (spatial discretization) We know the position of the curve or the surface: The normal form is reconstructed in the considered

domain by solving the « Eikonal » equation (from greek « image »):

{|⃗∇ j|=1 in Ωj=0 on Γ

j=0 on Γ

int

ext

122


Implicit forms

Resolution of the Eikonal equation It is a non-linear, hyperbolic equation « Fast Marching » method

Cf . J.A. Sethian, Fast Marching Methods, SIAM Review (41) 2, pp 199-235, 1999

Consist in a step-by-step construction of the distance field, from the initial front where d = 0.

J.A

. S

etia

n

123


Implicit forms

Resolution algorithm at first order on a regular grid

Known value in a point = 0 a) Calculation of test points J.A

. S

etia

n

124


Implicit forms

First order resolution algorithm on a regular grid

b) Among test points, wechoose the one of lowest value

c) We fix it and (re)calculate new test points

J.A

. S

etia

n

125


Implicit forms

Resolution algorithm at first order on a regular grid

c) We fix it and we recalculate... Here a test point has been computedtwice : we keep the lowest value

b) Among test points, wechoose the one of lowest value

J.A

. S

etia

n

126


Implicit forms

The former procedure is repeated until all grid points have been fixed.

The result is a (possibly signed) distance field known at each grid point. Interpolation inside square patches makes it continuous.

One can extract an iso-value (e.g. for 0) and consider it as the surface of interest

Also, one may actually modify the field (with transport equations for instance) to change the shape and position of the surface.

127


Implicit forms

There are algorithms of order 2 or higher It is also possible to solve that equation efficiently

on non regular grids ( unstructured meshes) Example : mesh

128


Implicit forms

There are algorithms of order 2 or higher It is also possible to solve that equation efficiently

on non regular grids ( unstructured meshes) Example : iso-distance curves

129


Implicit forms

There are high-performance algorithms to update discretized level-sets Resolution of partial differential equations

cf. J.A. Sethian « Level-Sets Methods and Fast Marching Methods, Evolving interfaces in computational geometry ... », Cambridge University Press, 1999.

Alternative representation of the geometry really useful for real objects (i.e. non perfect)

eg. Medical imaging, « historical » artifacts , etc... But also evolving interfaces (fluid surface, cracks in

solids)

130


Implicit forms

Transport equation of a level-set

We solve that using explicit time-stepping Normal speed is given (known) Below, the speed is depending on the density ...

∂j

∂ t+V n⋅∇ j⋅⃗n=0

N.

Par

agio

s

131


Implicit forms

Example of geometries based on « level-sets »

132


Implicit forms

Example of geometries based on « level-sets »

M.Y

. W

ang

133


Implicit forms

Some words on differential geometry...

134


Implicit forms

Normal (ext.) to the curve / surface We use the gradient

Tangent vector for a planar implicit curve Use of a permutation

of the gradient

∇=[∂x , y , z

∂ x∂x , y , z

∂ y⋮

]N =

∇

∣∇∣

is equal to 1 if j is a signed distance ( a normal form )

T =Tan

∣∇∣

Tan =[−∂x , y , z

∂ y∂x , y , z

∂ x]

135


Implicit forms

Curvature of a planar implicit curve Use of the Hessian matrix

Alternative

notation

H =[∂

2

∂ x2

∂2

∂ x∂ y

∂2

∂ y∂ x∂

2

∂ y2 ]=−T

T∗H ∗T

∣∇∣=

1Ro

=0

Ro

Osculating circle

=−Tan

T∗∇ Tan ×Tan ⋅n3

∣Tan ∣3

Basis vector (along z)

136


Implicit forms

Demonstration of the equation of the curvature From Frenet equations :

we have :

and so :

now,

=−d N s

ds⋅T s [

d T sds

d N sds

]=[ 0 s−s 0 ]⋅[T s

N s]

=−∂N s∂ x

dxds

∂N s∂ y

dyds ⋅T s

∇ N=[∂Nx

∂ x∂N y

∂ x∂Nx

∂ y

∂N y

∂ y] T s =[

dxdsdyds

]=−T sT

∗∇ N s∗T s

137


Implicit forms

We have :

Quod Erat Demonstrandum, QED

=−T sT∗∇ N s∗T s=−T

T∗H ∗T

∣∇∣

∇ N=∇ ∇

∣∇∣=∣∇∣∇ ∇ −∇ ∣∇∣∇

∣∇∣2

∇ ∇ =H ∇⋅T=∣∇∣⋅N⋅T=0

138


Implicit forms

Curvature of a surface There are 2 curvatures :

Gaussian curvature ( = total curvature ) kT

Average curvature kM .

These are related to principal curvatures kmin

and

kmax

(curvature measured along principal

directions) :

Along principal directions T1 and T

2 , curvatures are

extrema ( kmin

or kmax

)

min ,max=M±M2−T M=

minmax

2T=minmax

143


Implicit forms

Direct expression of principal directions Complicated but explicit formulas exist

Cf.Erich Hartmann, On the curvature of curves and surfaces defined by

normalforms, Computer Aided Geometric Design 16, pp. 355-376, 1999.

Numerical expression of the curvatures Cf. sequel...

144


Implicit forms

Numerical expression of the curvatures Conjugate transpose Hessian matrix (matrix of the

cofactors) :

H =[∂

2

∂ x2

∂2

∂ x∂ y∂

2

∂ x∂ z

∂2

∂ y∂ x∂

2

∂ y2

∂2

∂ y∂ z

∂2

∂ z ∂ x∂

2

∂ z ∂ y∂

2

∂ z2]=[

, xx , xy , xz

, yx , yy , yz

, zx , zy , zz]

H*=[

, yy , zz− , zy , yz , yz , zx− , yx , zz , yx , zy− , yy , zx

, xz , zy− , xy , zz xx , zz− , xz , zx , xy , zx− , xx , zy

, xy , yz− , xz , yy , yx , xz− , xx , yz , xx , yy− , xy , yx]

145


Implicit forms

Expression of curvatures of an implicit surface

T=∇

T∗H*

∗∇

∣∇∣4

M=∇

T∗H ∗∇−∣∇∣

2

⋅traceH

2∣∇∣3

146


Implicit forms

Implicit curves in 3D

147


Implicit forms

Curve in 3D Intersection of two implicit surfaces

TN

B

{ x , y , z =0 } ∩ { x , y , z =0 }

148


Implicit forms

Tangent

Curvature For planar curves : By substituting ... (we must take the norm , because

that vector is not along n3 anymore)

Tan ,=∇ x , y , z ×∇ x , y , z

T ,=Tan ,∣Tan ,∣

=1 if we deal with two normal forms that are “orthogonal”

=∣Tan ,T∗∇ Tan , ×Tan ,∣

∣Tan ,∣3

=− Tan

T∗∇ Tan ×Tan ⋅n3

∣Tan ∣3

149


Implicit forms

Torsion The formulas are obtained from the definition of the

torsion for the parametric curves We set :

Note : the curvature, written with the same kind of notation, is :

T *=Tan ,=∇x , y , z ×∇ x , y , z

T **=T *

∗∇ T *

T ***=T *

∗∇ ∇ T * ∗T * TT *

∗∇ T *∗∇ T *

=T * ,T ** ,T ***

∣T *×T **∣

2

=∣T *

×T **∣

∣T *∣3

150


Implicit forms

Volume operations on contour lines (level-sets) We can represent a volume as the set of points

such as

Manipulations of volumes Union, intersection, etc...

≤0

151


Implicit forms

Compounds of level-sets Union

Intersection

Complement

Difference

u=1∪2

i=1∩2

d=1∖2=1∩2

u x , y =min [1 x , y ,2 x , y]

i x , y=max [1 x , y ,2 x , y]

d x , y=max [1 x , y ,−2 x , y]

c=

c x , y =−x , y

152


Implicit forms

Disc

Square 4 half planes !

d x , y= x2 y2

−1

1

2

3 4

c1 x , y= y

2 x , y =− y2

3 x , y=−x0.2

4 x , y =x−1.8

Region of interest

153


Implicit forms

1

2

3 4

c

Region of interest

Use of intersections for the square :

Finally, difference between and

sq=1∩2∩3∩4

d x , y= x2 y2

−1

sq x , y=max [1,2,3,4]

dsq

f x , y=max [sq ,−d ]

f=sq∖d

f x , y =max [1,2,3,4,−d ]

154


Implicit forms

Disadvantges of an implicit representation Obtaining points on the curve/surface :

Resolution of a set of non-linear equations Not very robust, and slow No parametrization , no « explicit » orientation

Discretization of the curve/surface(« rasterization ») From a point, use of specific algorithms (“marching

cubes”) Not always possible and/or robust

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