Une Introduction aux Algorithmes de la Conception Géométrique · Une Introduction aux Algorithmes...

Une Introduction aux AlgorithmesUne Introduction aux Algorithmesde la Conception Géométriquede la Conception Géométrique

Geometry Design Tutorial 1Instituto Tecnológico de Veracruz 21-25 April 2008

Fathi El-YafiProject and Software Development Manager

Engineering Simulation

Géométries: Aperçu Général

• Géométrie� Définitions

� Data

� Sémantique

• Topologie� Mathématiques


� Mathématiques� Hiérarchie

• Approche CSG• Approche BREP

� Courbes

� Surfaces

Géométrie: Concepts

Sommets:Position(x,y,z)



arêtes:reliées par deux sommets



surfaces:ensemble d’arêtes fermées










volumes: ensemble de surfaces fermées






Corps: ensemble de volumesvolumes: ensemble de

surfaces fermées







Corps: ensemble de volumes

boucle: ensemble d’arêtes ordonnés

coedges: orientationd’arête respectant la boucle


















shell: ensemble orienté de surfaces comportant un volume

coface: surface







surfaceorienté respectantla shell




shell: ensemble orienté de surfaces comportant un volume


Volume 1

Surface 11

Géométrie Manifold : Chaque volume maintient son propre ensemble de surfaces uniques


Volume 1

Surface 1 Surface 2 Surface 3 Surface 4 Surface 5 Surface 6

Volume 2

Surface 8 Surface 9 Surface 10 Surface 11

Surface 7

Volume 2

Surface 7

surfaces uniques


Volume 1

Géométrie Non Manifold : Les volumes partagent des surfaces coïncidentes


Volume 1

Surface 1 Surface 2 Surface 3 Surface 4 Surface 5 Surface 6

Volume 2

Surface 8 Surface 9 Surface 10

Surface 7

Volume 2

Surface 7

Geometry: Data

• Modèle– Filaire– Surface– Volume

• Séparation des Attributs des Données Géométriques


• Séparation des Attributs des Données Géométriques• Attributs

– Couleurs– Paramètres– Etc.

• Objets Graphiques• Parts Visibles

Sémantique: Surface-Volume-Déflection-Défauts


Sémantique: Surfaces et Lignes Caractéristiques


Sémantique: Détails


Sémantique: Détail-Maillage-Boite Englobante


Sémantique: Décomposition–Courbure-STL-Maillage FEM


Modèle Filaire: Limites


Eléments de Topologie

Topologie??

• Notion d’intérieur et d’extérieur

• Orientation


• Calcul fiable de paramètres élémentaires:volume, centre de gravité, axes d’inertie …

• Contour = surface orientable et d'aire finie

• Arbre de construction

Relations: (Géométrie, Topologie)

face

SolideVolume

Surface


face

Contour

Surface

Courbe

Sommet

Arête

Point

Un contour "adhère" partout à son intérieur

Soit S un ensemble de R3

Adhérence : A(S)

P ∈ A( S) : tout voisinage de P contient un point de S

Topologie: Mathématiques


P ∈ A( S) : tout voisinage de P contient un point de S

Intérieur : I(S)

P ∈ I( S) si ∃ V(P) ⊂ S

Frontière : B(S)

P ∈ B( S) si P ∈ A( S) et P ∈ A( C(S)) où C(S) désigne le complémentaire de S dans R3

Ouvert : S = A(S)

Fermé : S = I(S)



Solide régulier S = A(I(S)) : adhérence de son intérieur = R(S)

A ∩∩∩∩ B = R(A∩B)A ∪∪∪∪ B = R(A∪B)A - B = R(A-B)C(A) = R(C(A))

Opérateur de Régularisation



A

B

A ∩∩∩∩ B

R(A ∩∩∩∩ B)A

A - B

R(A – B)

A B

Opérateur de Régularisation



A ∪∪∪∪ B

A B

C(A)/B

R(C(A)/B)

Formule d’Euler: S + F = A + 2


Polyèdre Type des FacesF S A

Tétraèdre Triangles équilatéraux 4 4 6

Octaèdre Triangles équilatéraux 8 6 12


F = nombre de faces, S = nombre de sommetsA = nombre d’arêtes

Cube Carrés 6 8 12

Dodécaèdre Pentagones12 20 30

Icosaèdre Triangles équilatéraux 20 12 30

Démonstration de Cauchy (1789-1857)

V+F=E+2 V+F=E+1 V+F+1=(E+1)+1 V+F=E+1



V+F=E+2 V+F=E+1 V+F+1=(E+1)+1 V+F=E+1

V+F=E+1 V+F=E+1 (V-2)+F=(E-2)+1 V+F=E+1

3+1=3+1!!

Examples de Maillage

F = 3844S = 1924A = S + F –2 = 5766



F = 47566S = 23793A = S + F –2 = 71357

Topologie : Hiérarchie


Représentation constructivePrimitives volumiques paramétréesTransformationsOpérations booléennes : Union, intersection, différence entre des objets

CSG : Constructive Solid Geometry


Notion de grapheAvantage :Description simple

Simulation « Usinage » des objets

CSG: Primitives


U

CSG: Opérateurs Booléens

Union


U

Union

U



U

Soustraction



U

-

∩

Commun



Congés



Géométrie: Courbes

Point de rebroussement

us(u)

x, y, z fonction de u, dérivée première continue


Point multiple

Longueur d’arc, abscisse curviligne

0

2 2 2 2

2 2 2

ds = dx + dy + dz

( )u

u

dx dy dzs u du

du du du = + +

∫

u

u=u0

s(u)

τ(s) τ(s+ds)

Géométrie : Courbure


O

n

Rayon duCercle Osculateur

=Rayon de Courbure

nOM

ρ=



M

ρ

Courbe paramétrique:

Equation de courbe y = f(x):

Géométrie : Repère de Frenet

2

2

2 2

2 2

2

2

( ) et n( )

d xdxdsds

dOM dy d OM d yu u

ds ds ds dsdz d zds ds

τ

= =


n = normale Principale

t

OM(s+dh1)

OM(s+dh2)

OM(s)

Géométrie : Torsion

M(s)

M(s+ds)


Plan Osculateur at M(s)

Torsion

Géométrie : Arbre de Transmission

2

2

2

2

2

2

2 2 2 2 2 2 2 2

2 2

= -R c o s = -R s inx = R c o s

y = R s in , = R c o s , = -R s in

z = p = p = 0

a in s i e t d o n c s =

R = - s in

d xd xdd

d y d y

d dd z d zd d

d s d x d y d z R p R p

d x

d s R p

θθ θθθθ θ θ

θ θθ

θ θ

θ

θ

= + + = + + ×

+

2

2 2 2

R = - c o s

d x

d s R pθ

+


2 2

R =

d s R p

d y

d s Rτ

+ 2 2 2

2

2 2 2 2 22 2

2

22 2

2 2

2 22 2

2 2

= - c o s

R Rco s e t = - s in a v ec =

p = 0 =

ps in

-p p c o s e t d o n c

R

d s R p

d d y

d s d s R p R pp

d zd z

d sd s R p

R p

d bb n T T

d s R pR p

R p

τθ θ ρ

θ

τ θ

+

+ ++ +

+= × = =

++ +

-1-0.5

00.5

1

-1-0.5

00.5

1

3-1

-0.500.5

1 -1-0.5

00.5

1

-1

-0.5

00.5

1

3

-1

-0.5

00.5

1

Discrétisation

Géométrie : Arbre de Transmission


0

1

2

0

1

2

Géométrie : Frenet – SerretRéférence, Equations, Courbure, Torsion

Frenet Reference:

The Frenet - Serret equations are a convenient framework for analyzing curvature. T(s) is the unit tangent to the curve as a function of path length s.N(s) is the unit normal to the curve B(s) is the unit binormal; the vector cross product of T(s) and N(s).


Frenet Equations:

Frenet Reference:

For any parametric function f(t), the expression of the curvature and the torsion are the following:

Géométrie : Courbure (Gaussian, Average)

Courbure Gaussienne


Courbure Moyenne

Courbure Cmap

Courbure Gaussienne = 2

1

R

Sphère



R

Courbure Moyenne = 1_R

Tore



Gaussienne Moyenne


Tore


Géométrie : Surfaces de Révolution

-1-0.5

00.5

1

-1-0.5

00.5

1

2

3

4-1-0.5

00.5

1

2

4

-1-0.5

00.51


0

1 -4

-2

0

2

4-4

-2

0-1

-0.5

-4

-2

0

2

4

-1-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1-0.5

0

0.5

1

-1

-0.5

0

0.5

1

Géométrie : Surfaces Particulières

20

40

-100

10

Ruban de Möbius


-25

0

25

50

-40

-20

0

20-10

-25

0

25

50

Bouteille de Klein



Bouteille de Klein: Courbure



Surface de Kuen

Géométrie Surfaces Particulières


x=2*(cos(u)+u*sin(u))*sin(v)/(1+u*u*sin(v)*sin(v))

y=2*(sin(u)-u*cos(u))*sin(v)/(1+u*u*sin(v)*sin(v))

z=log(tan(v/2))+2*cos(v)/(1+u*u*sin(v)*sin(v))

Surfaces de Dini



x=a*cos(u)*sin(v)y=a*sin(u)*sin(v) z=a*(cos(v)+log(tan((v/2))))+b*ua=1,b=0.2,u={ 0,4*pi},v={0.001,2}

Astéroïde



x= pow (a*cos(u)*cos(v),3) y= pow (b*sin(u)*cos(v),3) z= pow (c*sin(v),3)

Surface «Derviche»



Géométrie : Courbes

Lagrange Interpolating Polynomial


The Lagrange interpolating polynomial is the polynomial P(x) of degree <= (n-1)that passes through the n points (x1,y1= f(x1)), x2,y2= f(x2)), ..., xn,yn= f(xn)),and is given by:

Lagrange Interpolating Polynomial

Where:



Written explicitly:

Cubic Spline Interpolating Polynomial



A cubic spline is a spline constructed of piecewisethird-order polynomials which pass through a set of m control points. The second derivative of each polynomial is commonly set to zero at the endpoints, since this provides a boundary condition that completes the system of m -2 equations.This produces a so-called "natural" cubic spline and leads to a simple tridiagonal systemwhich can be solved easily to give the coefficients of the polynomials.However, this choice is not the only one possible, and other boundary conditions can be used instead.

Consider 1-dimensional spline for a set of n+1 points (y1, y2, .., yn),let the ith piece of the spline be represented by:

Where t is a parameter and i = 0, …, n-1 then

Cubic Spline Interpolating Polynomial



Rearranging all these equations,leads to the following beautifullysymmetric tridiagonal system:

If the curve is instead closed, the system becomes

Cubic Spline/Lagrange

1

2

1

2



1 2 3 4 5 6 7

-3

-2

-1

1 2 3 4 5 6 7

-5

-4

-3

-2

-1

Bézier

Given a set of n + 1 control points P0, P1, .., Pn, the corresponding Bèzier curve (or Bernstein- Bèzier curve) is given by:



Where Bi,n(t) is a Bernstein polynomial and .

A "rational" Bézier curve is defined by:

where p is the order, Bi,p are the Bernstein polynomials,Pi are control points, and the weight Wi of Pi is the last ordinate of the homogeneousPoint Pi

w. These curves are closed under perspective transformations, and can representconic sections exactly.

Bézier:Properties

� The Bézier curve always passes through the first and last control points.



� The curve is tangent to P1–P0 and Pn–Pn-1 at the endpoints.

� The curve lies within the convex hull of the control points.

Bézier:Properties

� A desirable property is that the curve can be translated and rotated by performing these Operations on the control points.



� Undesirable properties of Bézier curves are their numerical instability for large numbers of control points, and the fact that moving a single

control point changes the global shape of the curve .

Bézier: Bernstein Polynomials

Bin(t) = Cn

i (t-1)n-iti , Cni = n! / i!(n-i)!

Bi j(t) = (1-t)Bi

j-1(t) + t Bi-1j-1(t)



B0 1(t) = (1-t)B0

0(t) + t B-10(t)

B1 1(t) = (1-t)B1

0(t) + t B00(t)

B0 2(t) = (1-t)B0

1(t) + t B-11(t)

Bézier: Bernstein Polynomials

•Unit Partition: Σi=0,n Bin(t) = 1

Bi j(t) = (1-t)Bi

j-1(t) + t Bi-1j-1(t)



•0<=B in(t)<= 1

•Bin(0) = 0 et Bi

n(1) = 0

•B0n(0) = 1

•Bnn(1) = 1

B-Spline

A B-Spline is a generalization of the Bézier curve.Let a vector known as the knot vector be defined

T = {t0, t1, …, tm},



where T is a no decreasing sequence with ,and define control points P0, ..., Pn.Define the degree as: p = m-n-1

The "knots“ tp+1, ..., tm-p-1 are called internal knots.

B-Spline

�Define the basis functions as:



�Then the curve defined by: is a B-spline.

�Specific types include the non periodic B-spline(first p+1 knots equal 0 and last p+1 equal to 1; illustrated above)and uniform B-spline (internal knots are equally spaced).

�A curve is p - k times differentiable at a point where k duplicate knot values occur.

�A B-spline with no internal knots is a Bézier curve.

NURBS-Curve

A non uniform rational B-spline curve defined by:



where p is the order, Ni,p are the B-Spline basis functions, Pi are control points,and the weight Wi of Pi is the last ordinate of the homogeneous point Pi

w.These curves are closed under perspective transformations, and can representconic sections exactly.

Conics

P(t) = w N t P t

w N t

i i i

i

i

i i

i

i

×

=

=

×

=

=

∑

∑

,

,

( ) ( )

( )

2

0

3

2

0

3



Parabola w=1

Hyperbole w=4 Ellipse w=1/4

i =0

(0,0,0,1,1,1)

•NURBS of degree 2•Control points (Isosceles triangle)•Knot vector (0,0,0,1,1,1)

Arcs



Circles

(0, 0, 0, 1/3, 1/3, 2/3, 2/3, 1, 1, 1)



(0, 0, 0, 1/4, 1/4, 1/2, 1/2, 3/4, 3/4, 1, 1, 1)

Géométrie : Surfaces

A given Bézier surface of order (n, m) is defined bya set of (n + 1)(m + 1) control points ki,j.

Bézier

evaluated over the unit square, where:


evaluated over the unit square, where:

is a Bernstein polynomial, and

is the binomial coefficient.

S(u,v) = Σi=0,nΣj=0,m Bin(u) Bj

m(v) Pij

(n+1)(n+1) points Pij

Bézier



Pi(v) = Σj=0,m Bjm(v)Pij

NURBS



NURBS are nearly ubiquitous for computer-aided design (CAD), manufacturing (CAM), and engineering (CAE)and are part of numerous industry wide used standards,such as IGES, STEP, ACIS, Parasolid.

NURBS: PropertiesNURBS curves and surfaces are useful for a number of reasons:

�They are invariant under affine as well as perspective transformations.

�They offer one common mathematical form for both standard analytical shapes(e.g., conics) and free-form shapes.



�They are generalizations of non-rational B-Splines and non-rational and rational Béziercurves and surfaces.

(e.g., conics) and free-form shapes.

�They provide the flexibility to design a large variety of shapes.

�They reduce the memory consumption when storing shapes (compared to simpler methods).

�They can be evaluated reasonably quickly by numerically stable and accurate algorithms.

Une Introduction aux Algorithmes de la Conception Géométrique · Une Introduction aux Algorithmes...

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