geometric modeling 91.580.201 mortenson chapter 1 (including some material from farin ch. 2)

Geometric Modeling91.580.201

Mortenson Chapter 1

(including some material from Farin Ch. 2)

Vector Properties

zpypxp 222 p

] [ zyx ppppVector:

zyx ppp 222 pMagnitude:

Direction cosines:

pxp

cosp

ypcos

pzp

cos

1coscoscos 222 b

scalar

(dot)

product:

vector

(cross)

product:

source: Mortensonsource: Mortenson

zyx rrr

Matrix Properties

44434241

34333231

24232221

14131211

aaaa

aaaa

aaaa

aaaa

A

Special square matrices:- Diagonal: 0 elements except on main diagonal- Identity matrix- Symmetric matrix: symmetric about main diagonal- Antisymmetric (skew symmetric): aij = - aji

Transpose: AT

(AAT)T =AATInverse of square matrix: A-1

AA-1 = A-1A=IElements of A-1:

where |A’ji| is determinant of cofactor that deletes row j and column i from A.

A

A jiji

ija')1(

1

Determinant properties: |A|- |A|= |AT|- |AB|= |A||B|


Matrix Properties (continued)

0p

Eigenvalues and Eigenvectors:- Let p be a point and A be a transformation matrix.- Suppose that scalar exists such that: Ap= p.- Each vector for which Ap= p is true is an eigenvector of A.- Each scalar for which Ap= p is true is an eigenvalue of A corresponding to p.- Characteristic equation ( if ): |A-I|=0. Solutions are eigenvalues of A.- Finding eigenvectors: (see Introduction to Linear Algebra by Krause)

-Theorem: A linear transformation has a diagonal matrix there is a basis of eigenvectors of that matrix.

As we’ll see in Ch. 2 of Farin, a linear transformation preserves linear combinations:

R, ;)( AgAfgfA

Also in Ch. 2 of Farin: Eigenvectors can be used to determine axes of norm ellipse for a 2D set of points whose centroid is at origin. Eigenvalues give lengths of ellipse’s axes. This helps to characterize the point set.

sources: Mortenson, Farinsources: Mortenson, Farin

Matrix Properties (continued)



Farin Chapters 2,3

(including some 91.504 material from O’Rourke)

Points and VectorsBarycentric combination of points:

Note: some ’s may be negative.Example: “center of gravity”

Special case: convex combination of points: ’s must all be positive.

1 , 103

0

njj

n

jj Ebbb

Convex Hull of a set of points is the set of all convex combinations of points in the set.

nonconvex polygonnonconvex polygonconvex hull of a point setconvex hull of a point set

sources: Farin, O’Rourkesources: Farin, O’Rourke

Barycentric Coordinates in the Plane

source: Farin source: Farin

Linear Interpolation

. and between is point 10For baxt

Rbaxx tttt ;)1()(


Affine Maps

• A map that maps E3 into itself is an affine map if it leaves barycentric combinations invariant.

• So, if

and is an affine map, then

• Formulation for a point x in a coordinate system with a 3x3 matrix A and vector from R3:

1 , 103

0

njj

n

jj Ebbb

,, 3

0

Ebxbb

jj

n

jj

vAxx


Affine Maps (continued)• Examples of affine transformations (can be

composed using matrix multiplication to form chains of transformations) – Identity: A = I and v = 0. x is unchanged.

– Translation: A = I and v = translation vector. x is shifted by v.

– Scaling: A is a diagonal matrix and v = 0. x is “stretched”.

– Rotation: A is a rotation matrix (for rotation about an axis) and v = 0. x is rotated about the associated axis.

– Shear: A is I with some 0 elements replaced by non-0 value(s). v = 0. Effect on x is like “tilting” an axis.

– Parallel projection: center of projection is at infinity, so that there is a single viewing direction. Primary types are oblique and orthographic.

• Rigid Body Motions preserve lengths and angles. ATA = I.– Translations and Rotations are examples.

vAxx


Linear Interpolation

• Linear interpolation is affinely invariant:

babax tttt )1())1((

Rbaxx tttt ;)1()(


More on Affine Maps…

• Every map that takes straight lines to straight lines and is ratio preserving is an affine map.– For 3 collinear points: a, b, c:

where vol1 denotes one-dimensional volume.

),(vol

),(vol),,(ratio

1

1

cb

bacba

),(vol

),(vol),,(ratio

1

1

cb

bacba


Function Spaces

• This material will simplify notation later.• Let C[a,b] be set of all real-valued continuous functions

defined over [a,b ].• C[a,b] forms a linear space over the reals if

– For

• n functions are linearly independent if

],[, baCgf ][ )()())(( a,bttgtftgf

],[,,, 21 baCfff n

0],[ 0 21 nii cccbatfc


Function Spaces (continued)• Some useful subspaces of C[a,b]:

– Polynomials of degree n:– Monomials are linearly independent and form a basis for

the polynomials.– A linear transformation preserves linear combinations:

– Piecewise linear functions over a fixed partition of [a,b]. Hat functions form a basis.

R, ;)( AgAfgfA

bttta n 10

],[ )( 10 battataatp nn

n

,,,,1 2 nttt

jitH

tH

ji

ii

if 0)(

1)(:functionHat

n

jjj tHftf

0

)()(

210 23 HHHf


Variation Diminishing Property


Menelaos’ Theorem

210 )]1()1[()1)(1(],[],[ bbbbb stststststts

10)1(],0[ bbb ttt

],1[],0[)1(],[ ststts bbb

useful in proof of many algorithms

],1[],0[)1(],[ tstsst bbb

21)1(],1[ bbb sss 21)1(],1[ bbb ttt

10)1(],0[ bbb sss


Blossoms210 )]1()1[()1)(1(],[],[ bbbbb stststststts

1 ,*];[,*][),*][( srsr bbb

][],,[ nttt bb

Example of a blossom:

In general, a blossom is an n-variate function b[t1,…, tn] from Rn to E2 or E3 with properties:-Symmetry:

-Multiaffinity:

-Diagonality:

where denotes permutation)],,([],,[ 11 nn tttt bb

],[])[(0

iniini

n

i

n sri

nsr bb

where indicates same arguments on both sides

This determines a polynomial curve.

Special case has Leibniz formula:

where argument is repeated n times


Common Computational Geometry Structures

Voronoi DiagramVoronoi DiagramConvex HullConvex Hull

New PointNew Point

source: O’Rourke, Computational Geometry in Csource: O’Rourke, Computational Geometry in C

Delaunay TriangulationDelaunay Triangulation


OpenGL

OpenGL Overview

• Open-Source Graphics API Specification– Hardware-independent– Operating system independent– Vendor-neutral– Bindings for C and some other languages– Client/server paradigm

• Managed by OpenGL Architecture Review Board

source:http://web.cs.wpi.edu/~matt/courses/cs563/talks/OpenGL_Presentation/OpenGL_Presentation.htmlsource:http://web.cs.wpi.edu/~matt/courses/cs563/talks/OpenGL_Presentation/OpenGL_Presentation.html

OpenGL Graphics Pipeline


store commands for later use

evaluate polynomial functions

transformation,lighting, clipping, projection, viewport selection

produce fragments (series of framebuffer addresses and values)

hidden surface removal via depth (z)-buffering

stores pixel values for bit planes

Window and Viewport

source: Hill / Kelleysource: Hill / Kelley

OpenGL Overview

• State-Based: stacks maintain state– Push saves current state– Pop restores previous state– Matrix stack for transformations– Attribute stack for properties such as color


OpenGL Geometric Primitives

source:on-line OpenGL Programming Guide (see course web site for link)source:on-line OpenGL Programming Guide (see course web site for link)

OpenGL GLU Evaluators for Curves and Surfaces

source:http://hal.chem.uwm.edu/library/SGI_bookshelves/SGI_Developer/books/OpenGL_PG/sgi_html/ch13.htmlsource:http://hal.chem.uwm.edu/library/SGI_bookshelves/SGI_Developer/books/OpenGL_PG/sgi_html/ch13.html

Bezier curve with 4 control points

wireframe Bezier surface

lit, shaded Bezier surface using mesh

NURBS surface

GL Utility Toolkit (GLUT)

• Event-driven, interactive windowing support– call-back functions

source:http://www.avl.iu.edu/~ewernert/b581/lectures/12.1/index.htmlsource:http://www.avl.iu.edu/~ewernert/b581/lectures/12.1/index.html

OpenGL and GLUT part of HW#1

• Starting point: articulated robot by Michelle Daniels

OpenGL and GLUT part of HW#1• Add some GLUT objects to the scene.


Polyhedra

source: O’Rourke, Computational Geometry in Csource: O’Rourke, Computational Geometry in C

Polyhedra: What are they?• Polyhedron generalizes 2D polygon to 3D• Consists of flat polygonal faces• Boundary/surface contains

– 0D vertices 1D edges 2D faces• Components intersect “properly.” Each face pair:

– disjoint or have vertex in common or have vertex/edge/vertex in common

• Local topology is “proper”– at every point neighborhood is homeomorphic to disk

• Global topology is “proper”– connected, closed, bounded – may have holes

notnot polyhedra! polyhedra!

polyhedrapolyhedra

for more examples, seefor more examples, see http://www.ScienceU.com/geometry/facts/solids/handson.html http://www.ScienceU.com/geometry/facts/solids/handson.html

Polyhedra: Regular Polytopes

• Convex polyhedra are polytopes

• Regular polyhedra are polytopes that have: – regular faces, = faces, = solid (dihedral) angles

• There are exactly 5 regular polytopes

Excellent math references by H.S.M. Coxeter:Excellent math references by H.S.M. Coxeter:- - Introduction to GeometryIntroduction to Geometry (2nd edition), Wiley &Sons, 1969 (2nd edition), Wiley &Sons, 1969- - Regular PolytopesRegular Polytopes, Dover Publications, 1973, Dover Publications, 1973

p v (p-2)(v-2) Name Description 3 3 1 Tetrahedron 3 triangles at each vertex 4 3 2 Cube 3 squares at each vertex 3 4 2 Octahedron 4 triangles at each vertex 5 3 3 Dodecahedron 3 pentagons at each vertex 3 5 3 Icosahedron 5 triangles at each vertex

Polyhedra: Euler’s Formula

Proof has 3 parts:

1) Convert polyhedron surface to planar graph

2) Tree theorem

3) Proof by induction

V - E + F = 2V - E + F = 2

Polyhedral Boundary Representations

winged edgewinged edge

ff00

ff11ee

ee11++

ee11--

ee00++

ee00--

vv11

vv00

twin edge twin edge [[DCEL: doubly connected edge listDCEL: doubly connected edge list]]

ff00

ff11ee

ee4,04,0

vv11

vv00

- focus is on edgefocus is on edge

- edge orientation is arbitraryedge orientation is arbitrary

- represent edge as 2 halvesrepresent edge as 2 halves- lists: vertex, face, edge/twinlists: vertex, face, edge/twin- more storage space more storage space - facilitates face traversalfacilitates face traversal- can represent holes with face can represent holes with face inner/outer edge pointerinner/outer edge pointer

e’s e’s twintwin

vv22

vv77

vv66

vv55

vv33

vv44

ee0,10,1

Polyhedral Boundary Representations: Quad-Edge

- general: subdivision of oriented 2D general: subdivision of oriented 2D manifoldmanifold- edge record is part of:edge record is part of:

- endpoint 1 listendpoint 1 listendpoint 2 listendpoint 2 listface A listface A listface B listface B list

ee0,10,1ff00

ff11

ee4,04,0

vv11

vv00

vv22

vv77

vv66

vv55

vv33

vv44

ee0,10,1

twin edge twin edge [DCEL: doubly connected edge list][DCEL: doubly connected edge list]

ee0,00,0

ee1,01,0

ee2,02,0

ee3,03,0ee5,15,1

ee6,16,1

ee7,17,1

ee1,11,1

ee5,15,1

ee6,16,1ee7,17,1

ee1,01,0

ee0,00,0ee2,02,0

ee3,03,0 ee4,04,0

ee1,11,1

geometric modeling 91.580.201 mortenson chapter 1 (including some material from farin ch. 2)

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