IDC-CG 1

Solid Modeling

Chapter 11-12 in FVD, Chapter 10 in Hearn & Baker

IDC-CG 2

3D Object Representation

• Solid Modeling attempts to develop methods and algorithms to model and represent real objects by computers.

• Since there are many different kinds of objects (trees, flowers,clouds, rocks, buildings, water, fire,...), modeling techniques are diverse.

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Object Representation• The need to represent objects

arises in two cases:– Modeling of existing objects.

• Approximate modeling.

– Sculpting an object "from scratch".• modeling is precise.• interactive sculpting capabilities.

• Three types of objects in 3D:– 1D curves.– 2D surfaces.– 3D objects.

IDC-CG 4

General Techniques for Modeling

• Primitive Based: A composition of “simple” components.– Not precise.– Efficient and Simple.

• Free Form: Global representation, curved manifolds.– Precise.– Complicated.

• Statistical Form: Modeling of statistical phenomena, such as: fog, trees, rocks.

IDC-CG 5

Lecture Plan • 1D Curves in 3D

– Primitive based: line segments.– Free form:

• Implicit, Explicit, Parametric (polynomials, splines)

• 2D Surfaces in 3D– Primitive Based: Polygon mesh.– Free Form: As above

• 3D volumes in 3D– Volume Rep.

• Sweep volumes• Spatial Occupancy (voxels, Octree, BSP)• Constructive Solid Geometry

– Boundary Rep.• Polyhedra• Free form: as above

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Curve Representation

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Curves: Primitive Based

• Line segments: A curve is approximated by a collection of connected line segments.

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• Implicit form: f1(x,y,z)=0 & f2(x,y,z)=0

– Not efficient for drawing

• Parametric: x(u), y(u), z(u) – A mapping from [0,1]®R3

u ® [x(u),y(u),y(u)] ; uÎ[0,1]

– Common for drawing.

Curves: Free Form

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Example: A Circle of radius R

• Implicit: x2+y2+z2-R2=0 & z=0

• Parametric: x(u)=Rcos(q)y(u)=Rsin(q)z(u)=0

• How do we deal with transformations?

q

x

y

z

IDC-CG 10

Approximation/Interpolation by Parametric Curves

• Given a set of control points Pi known to be on the curve, find a parametric curve that interpolates/approximates the points.

P0

P1

P2

P3

P4

P0

P1

P2

P3

P4

Interpolating Curve

Approximating Curve

IDC-CG 11

Parametric Polynomials• For interpolating n points we need a

polynomial of degree n-1.

• Polynomial interpolation has several disadvantages:– Basis functions with wide supports,

coefficients are geometrically meaningless, global effects.

– Polynomial degree is high ®unwanted wiggles are introduced.

– Polynomial degree is low ®too little flexibility.

p0

p1

p2

p3

p4

!+++= 2321)( uauaaux

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The solution:Spline Representation

• Piecewise, low degree, polynomial curves, with continuous joints.

• Advantages:– Rich representation.– Geometrically meaning coefficients.– Local effects (local supported B.F.)– Interactive sculpting capabilities.

(Will be covered at later lessons)

p0

p1

p2

p3

p4c1(u)

c2(u) c3(u) c4(u)

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Surface (2D) Representation

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Primitive Based: Polygon Mesh• A polygonal mesh is a collection of

polygons that defines a 2D surface.• Polygon entities:

– Vertex • shared by at least two edges

– Edge • Connects two vertices.• Can be shared by two polygons

– Face• Defined by a polygon.• Has two sides: inside and outside

– Normal• Attached to each vertex (!)

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• Curved surfaces can be approximated by polygonal meshes:

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Properties of Meshes

• Connectivity: A mesh is connectedif there is an path of edges between any two vertices.

• Simplicity: A mesh is simple if the mesh is connected and has no holes in it.

• Planarity: A mesh is planar if every face of it is a planar polygon.

IDC-CG 17

Representing Polygon Meshes

v1

v2

v4

v3

v5

• Explicit (vertex list):F1=(V1 V2 V3)F2=(V2 V5 V4 V3)

• Pointers to a vertex listV=(V1 V2 V3 V4 V5)

F1=(1 2 3)F2=(2 5 4 3)

• Pointers to an edge listV=(V1 V2 V3 V4 V5)

E1=(1 2 F1)E2=(3 1 F1)E3=(2 3 F1 F2)E4=(4 3 F2)E5=(5 4 F2)E6=(2 5 F2)

F1=(1 3 2)F2=(3 6 5 4)

E1

E2

E3

E4

E5

E6

F1F2

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Properties:• Explicit:

– Space effective for single polygons– Not very informative– Redundant (issues: efficiency, correctness)

• Pointers to a vertex list:– Saves spaces for shared vertices– Easy to modify a single vertex– Difficult to find neighboring polygons– Multiple drawing of shared edges

• Pointers to an edge list– Solves multiple drawing and clipping

problem– Still hard to find all edges sharing a vertex

IDC-CG 19

Freeform Surface Representation

• Explicit form: z=f(x,y)– f(x,y) must be a function.– Not a rotation invariant representation.– Difficult to represent vertical tangents.

• Implicit form: f(x,y,z)=0– Difficult to connect two surfaces in a

smooth manner.– Not efficient for drawing.

– Useful for testing object inside/outside.

• Parametric: x(u,v), y(u,v), z(u,v)– A mapping from [0,1]2®R3.– Very common in modeling.

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• Example: Sphere of radius R

• Explicit: z=sqrt(R2-x2-y2) z=-sqrt(R2-x2-y2)

• Implicit: x2+y2+z2-R2=0

• Parametric: (x,y,z)=(R cosq cosf,R sinq cosf, R sinf)

qÎ[0 2p] , f Î[-p/2 p/2 ]

fq

IDC-CG 21

Approximated/Interpolated Parametric Surfaces

• Given a set of control points Pi known to be on the surface, find a parametric surface that interpolates/approximates the points.

IDC-CG 22

( )( )( ) ( ) ( )uvSuSvvuS

vuuCDuuSuBAuuS

21

2

1

)1(,]1,0[,)1(

)1(

+-=Î+-=

+-=

Bilinear Interpolation

u

v

A B

CD

0 1

1

• An interpolating parametric surface s(u,v), passing through 4 points.

• The surface is linear for a fixed u or v.• Can be performed for x(u,v), y(u,v), or

z(u,v).

S1

S2

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Bilinear example

z=15 z=7

z=2 z=3

u

v

1

1.5

2

1

1.5

20

5

10

15

z

u v

0

10

IDC-CG 24

Parametric Surface Splines

( )åå=i j

ij vucvuc ,),(

• In Bilinear interpolation the “seams” between patches are non differentiable.

• Solution: A Spline surface. Piecewise low-degree polynomial surface patches with continuous seams.

IDC-CG 25

3D VolumeRepresentation

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• Boundary Rep.– Polyhedra– Free form.

• Volume Rep.• Sweep volumes• Spatial Occupancy(voxels,

Octree, BSP)• Constructive Solid Geometry

3D VolumeRepresentation

IDC-CG 27

Boundary Representation

• A closed 2D surface defines a 3D object.

• At each point on the boundary there is "in" side and "out" side.

• B-reps can be defined in two ways:– Primitive based. A collection of

primitives forming the boundary.– Freeform based (splines, parametric

surface, implicit forms).

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Primitive Based: Polyhedra• A polyhedron is a solid bounded by a set

of polygons.• A polyhedron is constructed from:

– Vertices V– Edges E– Faces F

• Each edge must connect two vertices and be shared by exactly two faces.

• At least three edges must meet at each vertex

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• A simple polyhedron is one that can be deformed into a sphere (contains no holes).

• A simple polyhedron must satisfies Euler's formula:

V=8E=12F=6

V=5E=8F=5

V-E+F=2

V=6E=12F=8

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Free Form Boundary Representation

• To define a volume, its boundary should be closed.

• With parametric spline we define a closed surface by cyclic-connecting the surface points.

IDC-CG 31

IDC-CG 32

Sweep VolumesSweep Volume = 1. 2D Cross section area2. Sweeping rule

Translational Sweep = 2D area swept along a linear trajectory normal to the 2D plane.

Tapered Sweep = scale area while sweeping.

Volume Representation

IDC-CG 33

Slanted Sweep = trajectory is not normal to the 2D plane.

Rotational Sweep = 2D area is rotated about an axis.

General Sweep = any object along any trajectory with any transformation along the sweep.

IDC-CG 34

Translational and rotational sweep volumes.

IDC-CG 35

Spatial Occupancy Enumeration

• Space is described as a regular array of cells (usually cubes). Each cell is called a Voxel.

• A 3D object is represented as filled voxels.

• An object = a list of filled voxels.

IDC-CG 36

• Pros: – Easy to verify if a point (a voxel)

is inside or outside an object.– Boolean operations are easy to

apply.• Cons:

– Memory costs are high.– Resolution is limited to size and

shape of voxel.

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Quadtrees

• A Quadtree is a data structure enabling efficient storage of coherent 2D data.

• The idea behind it is to represent large uniform (full or empty) regions by one cell, using a recursive subdivision method.

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Quad Tree representation

Image

1 2

3 4 1 2 3 4

1 23 4

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Octrees

• An Octree is essentially the same idea as Quadtree, however, each node in an Octree has eight children rather than four.

• This makes Octrees good candidates for representing 3D objects.

1 23 4

6

7 8

1 2 3 4 5 6 7 8

IDC-CG 41

E

E

P

P

F

Binary Operations On Quad/Oct Trees

Notation: P FInternal Node(Partially full)

Full Leaf Node

Empty Leaf Node

E

Union: P P Recurse Union on descendentsÈ

P È = F

EP È =

F

Intersection: P P Recurse Intersect on descendentsÇ

P Ç =

EP Ç =

EF

Complement: P Recurse Complement on descendents

=

E =

C

C

CF

P

F

Difference: P P Recurse Difference on descendents-

P - =

EP - =

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F

P

EP P

FE E E F EE E

P

P

F E

F EE

P F E

FF

F

A B

A È B A Ç B

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An example where a quad-tree representation is not efficient.

IDC-CG 44

a j

b

f

cd

g

kh

ie g

j

k outoutin

h

iinoutin

out

f

in out

ec

d out

outin

b

a

BSP - Binary Space Partition Tree

• Each internal node represents a plane in 3D space.

• Each node has 2 children pointers one for each side of the plane.

• A leaf node represents a homogeneous portion of space - either “in” or “out”.

• Easy to determine if a point is inside or outside an object (recurse down the BSP tree).

• Questions: – How many parameters we have to store at each

node?– How can we determine whether a point (x,y) is

inside/outside an object?

IDC-CG 45

Constructive Solid Geometry

• Combine simple primitives using Boolean Operations and represent as a binary tree.

• To generate the object the tree is processed in a depth-first pass.

• Cons: many different CSG can represent an object (not unique).

A

B

A

B

A-B A ÈB

IDC-CG 46

- -

È

An object defined by a binary CSG tree

IDC-CG 47

World Coord.

Body Coord.

Cockpit Rear Wheels

Front Wheels

Left

Wheel

Right

Wheel

Left

Wheel

Right

Wheel

Trans(P0)Rot(q,n)

Trans(*)Rot(steer)

Trans(*)

Trans(*)

Trans(*)Trans(*)Trans(*)Trans(*)

Hierarchical Modeling Transformation

Representation in

World Coord.

yw

zw

xw

world

Front-Wheel System

x

y

z

p0

IDC-CG 48

push;Translate(p0);Rotate(q,n)DrawTractor(steer)pop;

DrawTractor(steer)DrawBody;

push;Translate(-1,0,0);DrawCockpit; pop;

push;Translate(-1.5,-1,0);DrawRearWheels; pop;

push;Translate(0,-1,0);Rotate(steer,0,1,0);DrawFrontWheels; pop;

DrawBodypush;scale(2,1,1);DrawCube; pop;

DrawFrontWheelspush;Translate(0,0,0.5);Scale(0.5,0.5,0.5);DrawWheel;pop; push;Translate(0,0,-0.5);Scale(0.5,0.5,0.5);DrawWheel; pop;

yw

zw

xw

world

Front-Wheel System

x

y

z

p0

IDC-CG 49

THE END

IDC-CG 50

• Given V1,V2,V3, the plane normal is: S=(V1-V2)x(V3-V2) and N=S/|S|.

• If P is any point on the plane and VT=(x,y,z), then V is in the plane if:

(V-P) • N=V•N-P•N=0

• If a polygon is defined with more than 3 vertices:– Plane approximation.– Triangular subdivision.

Finding the Normal Vectors

V1

V2

V3

N=[A,B,C]

D

IDC-CG 51

[ ] 0

1

0 =

úúúú

û

ù

êêêê

ë

é

=+++zyx

DCBAorDCzByAx

• N=[A,B,C] is the plane normal (|N|=1).• D= -P•N• |D| is the plane distance from the origin.• If N points "outside”:

» Ax+By+Cz+D<0 , the point (x,y,z) is in the inside part.

» Ax+By+Cz+D>0 , the point (x,y,z) is in the outside part.

V1

V2

V3

N=[A,B,C]

D

The Plane Equation