une introduction aux algorithmes de la conception géométrique · une introduction aux algorithmes...
TRANSCRIPT
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
Geometry Design Tutorial 2Instituto Tecnológico de Veracruz 21-25 April 2008
� Mathématiques� Hiérarchie
• Approche CSG• Approche BREP
� Courbes
� Surfaces
Géométrie: Concepts
Sommets:Position(x,y,z)
Geometry Design Tutorial 3Instituto Tecnológico de Veracruz 21-25 April 2008
Sommets:Position(x,y,z)
arêtes:reliées par deux sommets
Géométrie: Concepts
Geometry Design Tutorial 4Instituto Tecnológico de Veracruz 21-25 April 2008
surfaces:ensemble d’arêtes fermées
Sommets:Position(x,y,z)
arêtes:reliées par deux sommets
Géométrie: Concepts
Geometry Design Tutorial 5Instituto Tecnológico de Veracruz 21-25 April 2008
surfaces:ensemble d’arêtes fermées
Sommets:Position(x,y,z)
arêtes:reliées par deux sommets
Géométrie: Concepts
Geometry Design Tutorial 6Instituto Tecnológico de Veracruz 21-25 April 2008
volumes: ensemble de surfaces fermées
surfaces:ensemble d’arêtes fermées
Sommets:Position(x,y,z)
arêtes:reliées par deux sommets
Géométrie: Concepts
Geometry Design Tutorial 7Instituto Tecnológico de Veracruz 21-25 April 2008
Corps: ensemble de volumesvolumes: ensemble de
surfaces fermées
surfaces:ensemble d’arêtes fermées
Sommets:Position(x,y,z)
arêtes:reliées par deux sommets
Géométrie: Concepts
Geometry Design Tutorial 8Instituto Tecnológico de Veracruz 21-25 April 2008
volumes: ensemble de surfaces fermées
Corps: ensemble de volumes
boucle: ensemble d’arêtes ordonnés
coedges: orientationd’arête respectant la boucle
surfaces:ensemble d’arêtes fermées
Sommets:Position(x,y,z)
arêtes:reliées par deux sommets
Géométrie: Concepts
Geometry Design Tutorial 9Instituto Tecnológico de Veracruz 21-25 April 2008
volumes: ensemble de surfaces fermées
Corps: ensemble de volumes
boucle: ensemble d’arêtes ordonnés
coedges: orientationd’arête respectant la boucle
surfaces:ensemble d’arêtes fermées
Sommets:Position(x,y,z)
arêtes:reliées par deux sommets
Géométrie: Concepts
Geometry Design Tutorial 10Instituto Tecnológico de Veracruz 21-25 April 2008
volumes: ensemble de surfaces fermées
Corps: ensemble de volumes
boucle: ensemble d’arêtes ordonnés
shell: ensemble orienté de surfaces comportant un volume
coface: surface
surfaces:ensemble d’arêtes fermées
coedges: orientationd’arête respectant la boucle
Sommets:Position(x,y,z)
arêtes:reliées par deux sommets
Géométrie: Concepts
Geometry Design Tutorial 11Instituto Tecnológico de Veracruz 21-25 April 2008
surfaceorienté respectantla shell
volumes: ensemble de surfaces fermées
Corps: ensemble de volumes
boucle: ensemble d’arêtes ordonnés
shell: ensemble orienté de surfaces comportant un volume
Géométrie: Concepts
Volume 1
Surface 11
Géométrie Manifold : Chaque volume maintient son propre ensemble de surfaces uniques
Geometry Design Tutorial 12Instituto Tecnológico de Veracruz 21-25 April 2008
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
Géométrie: Concepts
Volume 1
Géométrie Non Manifold : Les volumes partagent des surfaces coïncidentes
Geometry Design Tutorial 13Instituto Tecnológico de Veracruz 21-25 April 2008
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
Geometry Design Tutorial 14Instituto Tecnológico de Veracruz 21-25 April 2008
• Séparation des Attributs des Données Géométriques• Attributs
– Couleurs– Paramètres– Etc.
• Objets Graphiques• Parts Visibles
Sémantique: Surface-Volume-Déflection-Défauts
Geometry Design Tutorial 15Instituto Tecnológico de Veracruz 21-25 April 2008
Sémantique: Surfaces et Lignes Caractéristiques
Geometry Design Tutorial 16Instituto Tecnológico de Veracruz 21-25 April 2008
Sémantique: Détails
Geometry Design Tutorial 17Instituto Tecnológico de Veracruz 21-25 April 2008
Sémantique: Détail-Maillage-Boite Englobante
Geometry Design Tutorial 18Instituto Tecnológico de Veracruz 21-25 April 2008
Sémantique: Décomposition–Courbure-STL-Maillage FEM
Geometry Design Tutorial 19Instituto Tecnológico de Veracruz 21-25 April 2008
Modèle Filaire: Limites
Geometry Design Tutorial 20Instituto Tecnológico de Veracruz 21-25 April 2008
Eléments de Topologie
Topologie??
• Notion d’intérieur et d’extérieur
• Orientation
Geometry Design Tutorial 21Instituto Tecnológico de Veracruz 21-25 April 2008
• 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
Geometry Design Tutorial 22Instituto Tecnológico de Veracruz 21-25 April 2008
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
Geometry Design Tutorial 23Instituto Tecnológico de Veracruz 21-25 April 2008
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)
Topologie: Mathématiques
Geometry Design Tutorial 24Instituto Tecnológico de Veracruz 21-25 April 2008
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
Topologie: Mathématiques
Geometry Design Tutorial 25Instituto Tecnológico de Veracruz 21-25 April 2008
A
B
A ∩∩∩∩ B
R(A ∩∩∩∩ B)A
A - B
R(A – B)
A B
Opérateur de Régularisation
Topologie: Mathématiques
Geometry Design Tutorial 26Instituto Tecnológico de Veracruz 21-25 April 2008
A ∪∪∪∪ B
A B
C(A)/B
R(C(A)/B)
Formule d’Euler: S + F = A + 2
Topologie: Mathématiques
Polyèdre Type des FacesF S A
Tétraèdre Triangles équilatéraux 4 4 6
Octaèdre Triangles équilatéraux 8 6 12
Geometry Design Tutorial 27Instituto Tecnológico de Veracruz 21-25 April 2008
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
Topologie: Mathématiques
Geometry Design Tutorial 28Instituto Tecnológico de Veracruz 21-25 April 2008
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
Topologie: Mathématiques
Geometry Design Tutorial 29Instituto Tecnológico de Veracruz 21-25 April 2008
F = 47566S = 23793A = S + F –2 = 71357
Topologie : Hiérarchie
Geometry Design Tutorial 30Instituto Tecnológico de Veracruz 21-25 April 2008
Représentation constructivePrimitives volumiques paramétréesTransformationsOpérations booléennes : Union, intersection, différence entre des objets
CSG : Constructive Solid Geometry
Geometry Design Tutorial 31Instituto Tecnológico de Veracruz 21-25 April 2008
Notion de grapheAvantage :Description simple
Simulation « Usinage » des objets
CSG: Primitives
Geometry Design Tutorial 32Instituto Tecnológico de Veracruz 21-25 April 2008
U
CSG: Opérateurs Booléens
Union
Geometry Design Tutorial 33Instituto Tecnológico de Veracruz 21-25 April 2008
U
Union
U
CSG: Opérateurs Booléens
Geometry Design Tutorial 34Instituto Tecnológico de Veracruz 21-25 April 2008
U
Soustraction
CSG: Opérateurs Booléens
Geometry Design Tutorial 35Instituto Tecnológico de Veracruz 21-25 April 2008
U
-
∩
Commun
CSG: Opérateurs Booléens
Geometry Design Tutorial 36Instituto Tecnológico de Veracruz 21-25 April 2008
Congés
CSG: Opérateurs Booléens
Geometry Design Tutorial 37Instituto Tecnológico de Veracruz 21-25 April 2008
Géométrie: Courbes
Point de rebroussement
us(u)
x, y, z fonction de u, dérivée première continue
Geometry Design Tutorial 38Instituto Tecnológico de Veracruz 21-25 April 2008
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
Geometry Design Tutorial 39Instituto Tecnológico de Veracruz 21-25 April 2008
Géométrie : Courbure
Geometry Design Tutorial 40Instituto Tecnológico de Veracruz 21-25 April 2008
O
n
Rayon duCercle Osculateur
=Rayon de Courbure
nOM
ρ=
Géométrie : Courbure
Geometry Design Tutorial 41Instituto Tecnológico de Veracruz 21-25 April 2008
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
= =
Geometry Design Tutorial 42Instituto Tecnológico de Veracruz 21-25 April 2008
n = normale Principale
t
OM(s+dh1)
OM(s+dh2)
OM(s)
Géométrie : Torsion
M(s)
M(s+ds)
Geometry Design Tutorial 43Instituto Tecnológico de Veracruz 21-25 April 2008
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
+
Geometry Design Tutorial 44Instituto Tecnológico de Veracruz 21-25 April 2008
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
Geometry Design Tutorial 45Instituto Tecnológico de Veracruz 21-25 April 2008
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).
Geometry Design Tutorial 46Instituto Tecnológico de Veracruz 21-25 April 2008
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
Geometry Design Tutorial 47Instituto Tecnológico de Veracruz 21-25 April 2008
Courbure Moyenne
Courbure Cmap
Courbure Gaussienne = 2
1
R
Sphère
Géométrie : Courbure (Gaussian, Average)
Geometry Design Tutorial 48Instituto Tecnológico de Veracruz 21-25 April 2008
R
Courbure Moyenne = 1_R
Tore
Géométrie : Courbure (Gaussian, Average)
Geometry Design Tutorial 49Instituto Tecnológico de Veracruz 21-25 April 2008
Gaussienne Moyenne
Géométrie : Courbure
Tore
Geometry Design Tutorial 50Instituto Tecnológico de Veracruz 21-25 April 2008
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
Geometry Design Tutorial 51Instituto Tecnológico de Veracruz 21-25 April 2008
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
Geometry Design Tutorial 52Instituto Tecnológico de Veracruz 21-25 April 2008
-25
0
25
50
-40
-20
0
20-10
-25
0
25
50
Bouteille de Klein
Géométrie : Surfaces Particulières
Geometry Design Tutorial 53Instituto Tecnológico de Veracruz 21-25 April 2008
Bouteille de Klein
Géométrie : Surfaces Particulières
Geometry Design Tutorial 54Instituto Tecnológico de Veracruz 21-25 April 2008
Bouteille de Klein: Courbure
Géométrie : Surfaces Particulières
Geometry Design Tutorial 55Instituto Tecnológico de Veracruz 21-25 April 2008
Surface de Kuen
Géométrie Surfaces Particulières
Geometry Design Tutorial 56Instituto Tecnológico de Veracruz 21-25 April 2008
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
Géométrie : Surfaces Particulières
Geometry Design Tutorial 57Instituto Tecnológico de Veracruz 21-25 April 2008
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
Géométrie : Surfaces Particulières
Geometry Design Tutorial 58Instituto Tecnológico de Veracruz 21-25 April 2008
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 : Surfaces Particulières
Geometry Design Tutorial 59Instituto Tecnológico de Veracruz 21-25 April 2008
Géométrie : Courbes
Lagrange Interpolating Polynomial
Geometry Design Tutorial 60Instituto Tecnológico de Veracruz 21-25 April 2008
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:
Géométrie : Courbes
Geometry Design Tutorial 61Instituto Tecnológico de Veracruz 21-25 April 2008
Written explicitly:
Cubic Spline Interpolating Polynomial
Géométrie : Courbes
Geometry Design Tutorial 62Instituto Tecnológico de Veracruz 21-25 April 2008
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
Géométrie : Courbes
Geometry Design Tutorial 63Instituto Tecnológico de Veracruz 21-25 April 2008
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
Géométrie : Courbes
Geometry Design Tutorial 64Instituto Tecnológico de Veracruz 21-25 April 2008
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:
Géométrie : Courbes
Geometry Design Tutorial 65Instituto Tecnológico de Veracruz 21-25 April 2008
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.
Géométrie : Courbes
Geometry Design Tutorial 66Instituto Tecnológico de Veracruz 21-25 April 2008
� 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.
Géométrie : Courbes
Geometry Design Tutorial 67Instituto Tecnológico de Veracruz 21-25 April 2008
� 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)
Géométrie : Courbes
Geometry Design Tutorial 68Instituto Tecnológico de Veracruz 21-25 April 2008
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)
Géométrie : Courbes
Geometry Design Tutorial 69Instituto Tecnológico de Veracruz 21-25 April 2008
•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},
Géométrie : Courbes
Geometry Design Tutorial 70Instituto Tecnológico de Veracruz 21-25 April 2008
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:
Géométrie : Courbes
Geometry Design Tutorial 71Instituto Tecnológico de Veracruz 21-25 April 2008
�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:
Géométrie : Courbes
Geometry Design Tutorial 72Instituto Tecnológico de Veracruz 21-25 April 2008
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
Géométrie : Courbes
Geometry Design Tutorial 73Instituto Tecnológico de Veracruz 21-25 April 2008
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
Géométrie : Courbes
Geometry Design Tutorial 74Instituto Tecnológico de Veracruz 21-25 April 2008
Circles
(0, 0, 0, 1/3, 1/3, 2/3, 2/3, 1, 1, 1)
Géométrie : Courbes
Geometry Design Tutorial 75Instituto Tecnológico de Veracruz 21-25 April 2008
(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:
Geometry Design Tutorial 76Instituto Tecnológico de Veracruz 21-25 April 2008
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
Géométrie : Surfaces
Geometry Design Tutorial 77Instituto Tecnológico de Veracruz 21-25 April 2008
Pi(v) = Σj=0,m Bjm(v)Pij
NURBS
Géométrie : Surfaces
Geometry Design Tutorial 78Instituto Tecnológico de Veracruz 21-25 April 2008
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.
Géométrie : Surfaces
Geometry Design Tutorial 79Instituto Tecnológico de Veracruz 21-25 April 2008
�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.