An Introduction to Geometry Design Algorithms · An Introduction to Geometry Design Algorithms Instituto Tecnológico de Veracruz Geometry Design Tutorial 21-25 April 2008 1 Fathi

An Introduction to Geometry Design An Introduction to Geometry Design AlgorithmsAlgorithms

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

Fathi El-YafiProject and Software Development Manager

Engineering Simulation

Geometry: Overview

• Geometry Basics� Definitions

� Data

� Semantic

• Topology� Mathematics


� Mathematics� Hierarchy

• CSG Approach• BREP Approach

� Curves

� Surfaces

vertices:x,y,z location

Geometry: Concept Basics




curves:bounded by two vertices




surfaces:closed set of curves








volumes: closed set of surfaces






body: collection of volumesvolumes: closed

set of surfaces







set of surfaces

loops: ordered set of curves on surface


coedges: orientation of curve w.r.t. loop


surfaces:closed set of curves (loops)




set of surfaces








body: collection of volumes


shell: oriented set of surfaces comprising a volume volumes: closed set

of surfaces (shells)







body: collection of volumesvolumes: closed set

of surfaces (shells)


shell: oriented set of surfaces comprising a volume



coface: oriented





body: collection of volumes


shell: oriented set of surfaces comprising a volume

oriented surface w.r.t. shell

volumes: closed set of surfaces (shells)

Volume 1

Surface 11

Manifold Geometry: Each volume maintains its own set of unique surfaces



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

Volume 1

Non-Manifold Geometry: Volumes share matching surfaces



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

• Model– Wireframe– Surface– Volume

• Geometrical Data Separated from Attributes


• Geometrical Data Separated from Attributes• Attributes

– Colors– Parameters– Etc.

• Graphical Objects• Visible Parts

Semantic: Surface-Volume-Deflection- Defects


Semantic: Surfaces and Features


Semantic: Detail


Semantic: Detail-Mesh–Bounding Box


Semantic: Decomposition–CurvatureMesh STL-Mesh FEM


Wireframe Model: Limits


Basics of Topology

Topology??

•Concept of interior and exterior

•Orientation


•Reliable calculation of basic parameters:volume, center of gravity, axis of inertia …

•Contour = oriented surface and limited area

•Tree construction

Relations: (Geometry, Topology)

Volume Solid

TopologyGeometry


Surface

Curve

Vertex

face

Contour

Edge

Point

A contour “grips" to its interior

Either S is a Setof R3

Adhesion : A(S)

P ∈ A( S) : any neighborhood of P contains a point of S

Topology: Mathematics


P ∈ A( S) : any neighborhood of P contains a point of S

Interior : I(S)

P ∈ I( S) if ∃ V(P) ⊂ S

Boundary : B(S)

P ∈ B( S) if P ∈ A( S) and P ∈ A( C(S)) where C(S) refers to the complementary of S in R3


Open : S = A(S)

Closed : S = I(S)


Regular SolidS = A(I(S)) : adhesion of its interior = R(S)


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

Operator of Regularization


A

B

A ∩∩∩∩ B

R(A ∩∩∩∩ B)A

A - B

R(A – B)


A B

Operator of Regularization


A ∪∪∪∪ B

A B

C(A)/B

R(C(A)/B)


Euler Formula: V+F = E+2

Polyhedron Type of Faces F V E

Tetrahedron Equilateral Triangles 4 4 6

Octahedron Equilateral Triangles 8 6 12


F = number of faces, V = number of vertices E = number of edges

Cube Squares 6 8 12

Dodecahedron Pentagons 12 20 30

Icosahedrons Equilateral Triangles 20 12 30


Cauchy Proof (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!!


Mesh Examples

F = 3844V = 1924E = V + F –2 = 5766


F = 47566V = 23793E = S + F –2 = 71357

Topology : Hierarchy


Constructive RepresentationParametric Volume PrimitivesTransformationsBoolean Operators:

Union, common, subtract

CSG : Constructive Solid Geometry


Concept of GraphAdvantage :

Simple DescriptionSimulation of Object « Manufacturing »

CSG: Primitive Components


U

CSG: Boolean Operators

Union


U

Union


U


U

Subtract



U

-

∩

Common




Fillet


Geometry: Curves

Lap back point

us(u)

x, y, z function of u, continuous first order


Multiple point

Arc length, abscissa curvilinear

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)

Geometry: Curvature


Geometry: Curvature


Geometry: Curvature

O

n

Ray ofOsculator Circle

=Curvature Ray

nOM

ρ=


M

ρ

Parametric curve:

For the curve with the equation y = f(x):

Geometry: Vector and TangentialFrenet Reference

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 = Principal normal

t

OM(s+dh1)

OM(s+dh2)

OM(s)

Geometry: Torsion

M(s)

M(s+ds)


Osculator Plane at M(s)

Torsion

Geometry: Propeller Circular

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

+

+ ++ +

+= × = =

++ +

Geometry: Propeller Circular

-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

Discretizing


0

1

2

0

1

2

Geometry: Frenet - SerretReference, Equations, Curvature, 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:

Geometry: CurvatureGaussian, Average

Gaussian Curvature


Average Curvature

Curvature Cmap


Gaussian Curvature = 2

1

R

Sphere


R

Average Curvature = 1

_R


Torus


Gaussian Average

Geometry: Curvature

Torus


Geometry: Surfaces of Revolution

-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

Geometry: Particular Surfaces

20

40

-100

10

Mobius Strip: 'Endless Ribbon'


-25

0

25

50

-40

-20

0

20-10

-25

0

25

50


The Klein Bottle



The Klein Bottle



The Klein Bottle: Curvature



The Kuen Surfaces


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))


The Dini Surfaces


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}


Asteroid


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


The «Derviche»


Geometry: Curves

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:

Geometry: Curves

Lagrange Interpolating Polynomial

Where:


Written explicitly:

Geometry: Curves

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.

Geometry: Curves

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

Geometry: Curves

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

Geometry: Curves

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.

Geometry: Curves

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.

Geometry: Curves

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 .

Geometry: Curves

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)

Geometry: Curves

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<=Bin(t)<= 1

•Bin(0) = 0 et Bi

n(1) = 0

•B0n(0) = 1

•Bnn(1) = 1

Geometry: Curves

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.

Geometry: Curves

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.

Geometry: Curves

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.

Geometry: Curves

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)

Geometry: Curves

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

Arcs


Geometry: Curves

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)

Geometry: 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.

Geometry: Surfaces

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

Geometry: Surfaces

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.

Geometry: Surfaces

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.

Documents

An Introduction to Geometry Design Algorithms · An Introduction to Geometry Design Algorithms Instituto Tecnológico de Veracruz Geometry Design Tutorial 21-25 April 2008 1 Fathi