Object Modeling Recap

Computer GraphicsLecture 6

Dr. Marc Eduard Fr̂ıncuWest University of Timisoara

April 7, 2020

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Object Modeling Recap

Outline

1 Object ModelingProcedural ModelingSolid Modeling

2 Recap

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Object Modeling Recap

Object Modeling

Why & how

need a method of representing/manipulating 3D objects in a computer (3D →2D image via projections)

surface (frontiers) & solid modeling (union of elementary volumes)

construction:interactive modeling tools (e.g., CAD programs. . . )scanning tools (e.g., CAT, MRI, . . . )computer vision (e.g., stereo, vision)procedural generation (e.g., sweeps, fractals, grammars)

How much information is needed?lines & curves not always enough for reconstruction → ambiguous, extra informationnot necessarily useful

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Object Modeling Recap

Object Modeling

Why & how

need a method of representing/manipulating 3D objects in a computer (3D →2D image via projections)

surface (frontiers) & solid modeling (union of elementary volumes)

construction:interactive modeling tools (e.g., CAD programs. . . )scanning tools (e.g., CAT, MRI, . . . )computer vision (e.g., stereo, vision)procedural generation (e.g., sweeps, fractals, grammars)

How much information is needed?

lines & curves not always enough for reconstruction → ambiguous, extra informationnot necessarily useful

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Object Modeling Recap

Object Modeling

Why & how

need a method of representing/manipulating 3D objects in a computer (3D →2D image via projections)

surface (frontiers) & solid modeling (union of elementary volumes)

construction:interactive modeling tools (e.g., CAD programs. . . )scanning tools (e.g., CAT, MRI, . . . )computer vision (e.g., stereo, vision)procedural generation (e.g., sweeps, fractals, grammars)

How much information is needed?lines & curves not always enough for reconstruction → ambiguous, extra informationnot necessarily useful

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Object Modeling Recap

Object Modeling

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Object Modeling Recap

Procedural Modeling

Procedural Modeling

Procedural modeling

– describe 3D models algorithmically

repeated models

self-similar processes

random processes

advantages:automatic generationconcise representationparameterized classes of models

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Object Modeling Recap

Procedural Modeling

Procedural Modeling

Sweeps

Example: shell generation

θi+1 = ti + ∆t = θi + ∆θ(arithmetic progression)

ri+1 = r0ξtir ξ

∆tr = riλr (geometric

progression)

zi+1 = z0ξtiz ξ

∆tz = ziλz

(geometric progression)

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Object Modeling Recap

Procedural Modeling

Procedural Modeling

Fractals

deterministic: parts are scaledcopies of original

nice shapes

statistical: parts have samestatistical properties as original

3D terrains3D plants

Figure: Deterministic fractalhttp://www.cs.princeton.edu/courses/archive/spr03/cs426/#Coursework

Figure: Statistical fractalhttp://www.cs.princeton.edu/courses/archive/spr03/cs426/#Coursework

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Object Modeling Recap

Procedural Modeling

Procedural Modeling

Grammars

– useful for generating plantsExample: tree generation

Tree → Branch Tree | LeafBranch→ Cylinder | [Tree]

Figure: Tree generationhttp://www.cs.princeton.edu/courses/archive/spr03/cs426/#Coursework

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Object Modeling Recap

Solid Modeling

Solid Modeling

Representation rules

1 the representation field should be large enough

2 the representation should be non-ambiguous and accurate

3 no invalid representations should be allowed

4 valid representations should always be easy to create

5 efficient boolean operations: union, intersection, difference

6 closeness (invariant) to an affine transformation

7 compact (memory efficient) representation

8 fast algorithms for computing the physical properties of the representation

∃ representation following ALL rules above?NO ⇒ compromise

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Object Modeling Recap

Solid Modeling

Solid Modeling

Representation rules

1 the representation field should be large enough

2 the representation should be non-ambiguous and accurate

3 no invalid representations should be allowed

4 valid representations should always be easy to create

5 efficient boolean operations: union, intersection, difference

6 closeness (invariant) to an affine transformation

7 compact (memory efficient) representation

8 fast algorithms for computing the physical properties of the representation

∃ representation following ALL rules above?

NO ⇒ compromise

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Object Modeling Recap

Solid Modeling

Solid Modeling

Representation rules

1 the representation field should be large enough

2 the representation should be non-ambiguous and accurate

3 no invalid representations should be allowed

4 valid representations should always be easy to create

5 efficient boolean operations: union, intersection, difference

6 closeness (invariant) to an affine transformation

7 compact (memory efficient) representation

8 fast algorithms for computing the physical properties of the representation

∃ representation following ALL rules above?NO ⇒ compromise

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Object Modeling Recap

Solid Modeling

Solid Modeling

Representation schemes

1 boundary representation (reducing the dimension)

2 pure primitive instantiation (parameterized prototypes)

3 sweeping method (moving objects, trajectory)

4 enumerating the spatial occupancy (voxels = volumetric pixels)

5 decomposing in cells (gluing elementary volumes)

6 constructive solid geometry (operations with elementary volumes)

7 multiple representations (particle systems)

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Object Modeling Recap

Solid Modeling

Solid Modeling

Boundary representations

point list

lines:transversal sections: points from the same sections are joined to form section contourwireframe: object represented by line and curve segments (good for for animations, badfor computing areas, volumes, mass, . . . )

explicit segments: extremities of the coordinatesimplicit segments: pair of indices to a list of verticespolygonal linestransversal sections & longitudinal lines

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Object Modeling Recap

Solid Modeling

Solid Modeling

Boundary representations

polygonsexplicit: store vertices in the order they appear: V = {V1,V2, . . . ,Vn}

duplicate coordinates, double drawing of edges, problem finding concurrent edgespointers to a list of vertex indices: VP1 = {1, 2, 4}, . . . , P = {VP1, . . . ,VPk}double drawing of edges, problem finding concurrent edgespointers to a list of edges: E1 = {V1,V2,P1} ,. . . , P = {E1, . . . , Em}problem finding concurrent edges (we draw lines NOT polygons)

find interior?store vertices in a given order (clock-wise, counter clock-wise)draw edge only from a vertex with a smaller index to one with a greater index value

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Object Modeling Recap

Solid Modeling

Solid Modeling

Boundary representations

polygonsexplicit: store vertices in the order they appear: V = {V1,V2, . . . ,Vn}duplicate coordinates, double drawing of edges, problem finding concurrent edges

pointers to a list of vertex indices: VP1 = {1, 2, 4}, . . . , P = {VP1, . . . ,VPk}double drawing of edges, problem finding concurrent edgespointers to a list of edges: E1 = {V1,V2,P1} ,. . . , P = {E1, . . . , Em}problem finding concurrent edges (we draw lines NOT polygons)

find interior?store vertices in a given order (clock-wise, counter clock-wise)draw edge only from a vertex with a smaller index to one with a greater index value

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Object Modeling Recap

Solid Modeling

Solid Modeling

Boundary representations

polygonsexplicit: store vertices in the order they appear: V = {V1,V2, . . . ,Vn}duplicate coordinates, double drawing of edges, problem finding concurrent edgespointers to a list of vertex indices: VP1 = {1, 2, 4}, . . . , P = {VP1, . . . ,VPk}

double drawing of edges, problem finding concurrent edgespointers to a list of edges: E1 = {V1,V2,P1} ,. . . , P = {E1, . . . , Em}problem finding concurrent edges (we draw lines NOT polygons)

find interior?store vertices in a given order (clock-wise, counter clock-wise)draw edge only from a vertex with a smaller index to one with a greater index value

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Object Modeling Recap

Solid Modeling

Solid Modeling

Boundary representations

polygonsexplicit: store vertices in the order they appear: V = {V1,V2, . . . ,Vn}duplicate coordinates, double drawing of edges, problem finding concurrent edgespointers to a list of vertex indices: VP1 = {1, 2, 4}, . . . , P = {VP1, . . . ,VPk}double drawing of edges, problem finding concurrent edges

pointers to a list of edges: E1 = {V1,V2,P1} ,. . . , P = {E1, . . . , Em}problem finding concurrent edges (we draw lines NOT polygons)

find interior?store vertices in a given order (clock-wise, counter clock-wise)draw edge only from a vertex with a smaller index to one with a greater index value

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Object Modeling Recap

Solid Modeling

Solid Modeling

Boundary representations

polygonsexplicit: store vertices in the order they appear: V = {V1,V2, . . . ,Vn}duplicate coordinates, double drawing of edges, problem finding concurrent edgespointers to a list of vertex indices: VP1 = {1, 2, 4}, . . . , P = {VP1, . . . ,VPk}double drawing of edges, problem finding concurrent edgespointers to a list of edges: E1 = {V1,V2,P1} ,. . . , P = {E1, . . . , Em}

problem finding concurrent edges (we draw lines NOT polygons)

find interior?store vertices in a given order (clock-wise, counter clock-wise)draw edge only from a vertex with a smaller index to one with a greater index value

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Object Modeling Recap

Solid Modeling

Solid Modeling

Boundary representations

polygonsexplicit: store vertices in the order they appear: V = {V1,V2, . . . ,Vn}duplicate coordinates, double drawing of edges, problem finding concurrent edgespointers to a list of vertex indices: VP1 = {1, 2, 4}, . . . , P = {VP1, . . . ,VPk}double drawing of edges, problem finding concurrent edgespointers to a list of edges: E1 = {V1,V2,P1} ,. . . , P = {E1, . . . , Em}problem finding concurrent edges (we draw lines NOT polygons)

find interior?store vertices in a given order (clock-wise, counter clock-wise)draw edge only from a vertex with a smaller index to one with a greater index value

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Object Modeling Recap

Solid Modeling

Solid Modeling

Boundary representations

polygonsexplicit: store vertices in the order they appear: V = {V1,V2, . . . ,Vn}duplicate coordinates, double drawing of edges, problem finding concurrent edgespointers to a list of vertex indices: VP1 = {1, 2, 4}, . . . , P = {VP1, . . . ,VPk}double drawing of edges, problem finding concurrent edgespointers to a list of edges: E1 = {V1,V2,P1} ,. . . , P = {E1, . . . , Em}problem finding concurrent edges (we draw lines NOT polygons)

find interior?store vertices in a given order (clock-wise, counter clock-wise)draw edge only from a vertex with a smaller index to one with a greater index value

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Object Modeling Recap

Solid Modeling

Solid Modeling

Boundary representations

polyhedrons: solid bounded by a set of polygons ⊕ set of constraintsw/o holesrepresentation:

vertices stored in a certain orderpolygonal faces stored through indices to a list of verticesedges stored through pairs of indices pointing to the vertex list

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Object Modeling Recap

Solid Modeling

Solid Modeling

Primitive instantiation

define a set of 3D solids relevant for the scene

primitives may be parameterized, i.e., family with members varying in a smallnumber of characteristics

represent complex objects through hierarchies

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Object Modeling Recap

Solid Modeling

Solid Modeling

Representation through sweeping (movement)

translational volume ⇒ 2D/3D object from a linear trajectory

rotational volume ⇒ 2D/3D object from rotation around an axis

general cylinder ⇒ 2D object

in general:objects can change shape and orientationtrajectory can arbitrary

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Object Modeling Recap

Solid Modeling

Solid Modeling

Voxels

– partition space into an uniform grid volume whose cells are called voxels ,i.e,volumetric pixels:

occupancy

color

density

temperature

. . .

Figure: Voxel partitioned space:http://www.cs.princeton.edu/courses/archive/spr03/cs426/#Coursework

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Object Modeling Recap

Solid Modeling

Solid Modeling

Voxels

advantages:simple, intuitive, unambiguoussame complexity for all objectsnatural acquisition for some applications, e.g., MRI, CATtrivial boolean operations (

⋃,⋂

, \)

disadvantages:approximativenot affine invariantlarge storage requirements, i.e., O(n3)expensive display

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Object Modeling Recap

Solid Modeling

Solid Modeling

Voxels

advantages:simple, intuitive, unambiguoussame complexity for all objectsnatural acquisition for some applications, e.g., MRI, CATtrivial boolean operations (

⋃,⋂

, \)

disadvantages:approximativenot affine invariantlarge storage requirements, i.e., O(n3)expensive display

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Object Modeling Recap

Solid Modeling

Solid Modeling

Quadtrees & octrees

– refine resolution of voxels hierarchically

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Object Modeling Recap

Solid Modeling

Solid Modeling

Quadtrees & octrees

Figure: Quadtree based surface http://www.infinitecode.com/?view_post=23

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Object Modeling Recap

Solid Modeling

Solid Modeling

Octrees

advantages:combinations very simplefast renderingspatial search possible

disadvantages:inexact representationlow image qualityrestricted transformationshigh memory costproblems in finding the neighbor of a node

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Object Modeling Recap

Solid Modeling

Solid Modeling

Octrees

advantages:combinations very simplefast renderingspatial search possible

disadvantages:inexact representationlow image qualityrestricted transformationshigh memory costproblems in finding the neighbor of a node

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Object Modeling Recap

Solid Modeling

Solid Modeling

Binary space partitioning

partition a convex region by a hyperplane

single operation → two child nodes added as leaf nodes:left child: interiorright child: exterior

point classification (in or out) can be easily done

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Object Modeling Recap

Solid Modeling

Solid Modeling

Constructive solid geometry

– simple primitives ⊕ boolean operations⇒tree with internal (operations) and terminal(primitives) nodes

can be created through CAD/CAM

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Object Modeling Recap

Solid Modeling

Solid Modeling

Multiple representations

– good for describing objects that changeover time → particle systems:

examples:cloudssmokefirefireworkswater(fall). . .

a particle:position, speed & accelerationlife & fading factorcolor, material & texture. . .

Figure: Smoke effect: Call of Duty 2

Figure: Explosion effect: Star Trek II

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Object Modeling Recap

Lecture Recap

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