Www.sccg.sk/~zimanyi/lectures/PG Representation and modeling of 3D objects (basic concepts) FMFI UK Katedra počítačovej grafiky a spracovania obrazu Marek

www.sccg.sk/~zimanyi/lectures/PG

Representation and modeling of 3D objects

(basic concepts)

FMFI UKKatedra počítačovej grafiky

a spracovania obrazu

Marek Zimányi

Computer Graphics 2Computer Graphics 2

2Marek Zimányi,Department of Computer Graphics

and Image Processing, FMPI, Comenius University, Bratislava Marek Zimányi, Andrej Ferkowww.uniba.sk/~kpgso

Contents

Polygonal representation Boundary rep. Sweep

Constructive Solid Geometry

Implicit representation F-rep

Spatial subdivision techniques Procedural modeling



Polygonal representation

Boundary representation (B-Rep) Topological description

Specifies: vertices, edges, faces Geometrical description

Specifies: equation of the surface

Sweep



Boundary representation

Polygon and Polygonal Mesh Polyhedron Normals Euler’s formula DCEL

Polygonal representationPolygonal representation



Boundary representation

Describing of boundary points set

faces,edges, vertices region, loop, shell,

hole,handle vertices :

V1 = (x1, y1, z1), V2 = (x2, y2, z2), … edges :

h1 = (1, 2), h2 = (2, 3), … faces :

P1 = (1, 2, 3), P2 = (4, 2, 1), …




Boundary representation - Mesh

Collection of faces (polygons) Advance: shading process Parameters:

Vertices, faces, normals Vertex normals versus face

normals ! This it not the same !





Definition: Set of vertices, Normals Faces,

With Indexes of vertices With Indexes of normals

Be careful -> left or right hand CS




Normal computing:

1

0)()( ))((

N

iinextiinextix zzyym

1

0)()( ))((

N

iinextiinextiy xxzzm

1

0)()( ))((

N

iinextiinextiz yyxxm

)()( 2321 VVVVm

Martin Newell method:




Properties Solidity – enclose a positive and finite

amount of space Connectedness – exist unbroken path

between any two vertices Simplicity – solid with no holes Planarity – every face is a planar

polygon Convexity – line between any two

point lies inside the object



Boundary rep. - Polyhedron

Def.: A polyhedron is a connected mesh of

simple planar polygons that enclose a finite amount of space


Prop. Every edge is shared by exactly two

faces At least 3 edges meets at each

vertex Faces do not interpenetrate:



Manifolds

2-manifolds: point from manifold has small neighborhood of points around, topologically the same as a disk in the plane

Orientable <-> NonOrientable manifoldMobius strip, Klein bottle

Many b-rep systems support only solid whose boundaries are 2-manifold

Polygonal representation - B-repPolygonal representation - B-rep

Unreal objects => Non-Manifolds



Manifolds

Klein bottle http://www.math.ohio-state.edu/~fiedorow/math655/Klein2.html http://www.kleinbottle.com/

Mobius strip http://www.mobiusproductsandservices.com/tms.html http://www.cut-the-knot.org/do_you_know/moebius.shtml

Another :-) http://pbskids.org/zoom/phenom/index.html




Non-Manifolds



The Winged-Edge Representation

Double-connected Edge List (DCEL)For each edge is given info:1. Incident vertices2. Left and right adjacent face3. Preceding and succeeding edge in clockwise

order4. Preceding and succeeding edge in

counterclockwise order




B-Rep

Important facts in the B-Rep: Classification of edges

sharp and auxiliary Normals in vertexes Boundary of face –

find all edges of a face Position of a point in the space

Point is IN or OUT of object

Polygonal representationPolygonal representationPolygonal representation - B-repPolygonal representation - B-rep



Sweep

Sweeping an object along a trajectory Translation sweep Rotational sweep Conical sweep Sphere sweep General Cylinder

Polygonal rep ?




Translational Sweep

Method:Define the object with the contour (2D) and the

path.




Rotational SweepMethod:

Define the object by rotating of the contour (2D) with the arbitrary axis.




Conical SweepMethod:

Define the object by the contour (2D) and a 3D-point (top of the pyramide).




Sphere SweepMethod:

Define the object by the sphere with the varying radius and a path.




General CylinderMethod:

Define the object by the set of „control“-contours and by the path.




Polygonal rep ?

General sweeps – difficult to model efficiency

General sweeps do not always generate solids

It is a polygonal rep?


Yes and No.

Trajectory and object shape – may swept object intersect

Volume calculations complicated

Sweeping a 2D area in its own plane

another 2D area



Polygonal rep ?

It is a difficult to apply regularized Boolean operations(after BOOL op. it wouldn’t be SWEEP obj)

BUT sweeps are a natural and intuitive way to construct of the objects

Many systems allow user construct sweepsBUT


store the objects in other representation



Constructive Solid Geometry

CSG Composing of Primitives: Sphere, Cone, Cube, Cylinder, ... Operations: , , -, ... The leafs of CSG-tree are primitives

and the operations are in the rest of the tree nodes. The CSG-tree represents the solid. (Scene graph idea!)



Example

Polygonal representation - CSGPolygonal representation - CSG





Notes

Pros Low memory consumption Simple combining Exact representation of complex

surfaces 12 Mantylas operators and his proof

Cons Slow and difficult processing

? Not unique representation




CSG processing

Rendering Algorithms ray casting (tracing) extended depth buffer algorithm

Evaluation/Conversion Algorithms boundary representation octree constructive cubes




Implicit representation

Jim Blinn 1982 Implicit surfaces

Blobby objects Soft objects Meta balls



Spreadsheet RenderingImplicit representationImplicit representation



Animation Path in t1t2 PlaneImplicit representationImplicit representation



Spatial subdivision techniques

Octrees Iterative space subdivision Nodes have property: V, F or M Subdividing P-nodes until the desired

Approximation Quality The dividing planes do not depend on

objects in the scene (differs from BSP tree!)

Analogy in 2D: „Quad Tree“ BSP trees



Octree

Sort form of space Disadvantage: sampling

V – void

F – full

M – mixed

(FFFFF(FFFVFVFV)VV)

Spatial subdivision techniquesSpatial subdivision techniques



Octree Boolean set operations and

transformations Traverse both trees top-down in

parallel Rotations (*90), scaling (*2) General transformations – severe

problem(aliasing)

Neighbor finding Not trivial

Linear notation




Binary Space-Partitioning Trees (BSP trees)

octrees divide space uniformly - drawback BSP divide space recursively into pairs of

subspaces: each subspace has the same object complexity

(# of objects) Usual division step:

1. sort the objects according to actual coordinate (x,y,z)

2. Choose median3. Place halfplane (halfline in 2D) between median

and next object4. Shift halfline if too many objects intersects

the halfplane




Procedural modeling

Alternative to slow user modeling Useful for complex, hard-to-manipulate,

organic, fuzzy, scalable, large, … objects Software designer prepares a procedure,

implementing object-production algorithm Software user specifies only the input

parameters for the procedure Stochastic principle is often applied

(no 2 trees are exactly identical)

Documents

Www.sccg.sk/~zimanyi/lectures/PG Representation and modeling of 3D objects (basic concepts) FMFI UK Katedra počítačovej grafiky a spracovania obrazu Marek