cse 872 dr. charles b. owen advanced computer graphics1 polygon meshes and other modeling tools...

CSE 872 Dr. Charles B. OwenAdvanced Computer Graphics1

Polygon Meshes and other modeling tools• Object representation• Polygon meshes• Normals• Scene graphs• Level of detail• Space partitioning


How can we represent objects/scenes• Polygon meshes• Scene graphs• Parametric representations• Equations• Bunch o points


Why?• Rendering• Manipulate

– User– System


Point Clouds

• Unstructured set of 3D point samples– Why– What for?


Polygon/Triangle Soup

• Unstructured set up polygons/triangles– Just a list of vertices for each polygon/triangle– Why?– What for? class Triangle

{ public: Triangle() {m_material = NULL;}

CGrPoint3F m_vertices[3]; CGrPoint3F m_normals[3]; CGrPoint2F m_tvertices[3]; const void *m_material; };


What is tessellation?

• Representing a surface with flat approximating polygons– A sampled representation

• Geometric Aliasing– Error in the representation due to the sampling with

polygons


Polygon/Triangle mesh

• Common method for describing geometry

• Set of vertices– Coordinates– Normal– Texture coordinates– Etc.

• Set of surfaces– Pointers to vertices– Material properties

Common: Triangle mesh


class CTriangleMesh{public: CTriangleMesh(void); ~CTriangleMesh(void);

// This structure defines a triangle struct Triangle { int m[3]; const CMaterial *material; };

int VertexCnt() const {return (int)m_vertices.size();} int TriangleCnt() const {return (int)m_triangles.size();}

const std::vector<CGrPoint3F> &Vertices() const {return m_vertices;} const std::vector<CGrPoint3F> &Normals() const {return m_normals;} const std::vector<CGrPoint2F> &TVertices() const {return m_tvertices;}

const std::vector<Triangle> &Triangles() const {return m_triangles;} const CMaterials &Materials() const {return m_materials;}

private: CMaterials m_materials; // The materials collection

// The vertices, normals, and texture vertices. // These three all correspond with each other. std::vector<CGrPoint3F> m_vertices; std::vector<CGrPoint3F> m_normals; std::vector<CGrPoint2F> m_tvertices;

std::vector<Triangle> m_triangles;};


Polygon Mesh

• Retains shared-vertex relationships

Vertex a,b,c,d;

Polygon P1(a, b, c);Polygon P2(c, b, d);

b,c shared by two polygons

a

b

c

d


Computing a surface normal

• Assume the following polygon:

p1

p2

p3

p4

)()(

)()(

1412

1412

pppp

ppppN

N


The Method of Projected Areas

• Assume this 2D example line:

p1

p2

x

y

xy

xyN

,

,

Projection of lengthto one axis


Method of Projected Areas

• The projection of the area onto the x/y plane is the z part of the normal

zyx

zyx

nii

n

iniiz

nii

n

iniiy

nii

n

iniix

NNN

NNNN

yyxxN

xxzzN

zzyyN

,,

,,

))((

))((

))((

)1%(1

)1%(

)1%(1

)1%(

)1%(1

)1%(


Where does this come from?

zyx

zyx

nii

n

iniiz

nii

n

iniiy

nii

n

iniix

NNN

NNNN

yyxxN

xxzzN

zzyyN

,,

,,

))((

))((

))((

)1%(1

)1%(

)1%(1

)1%(

)1%(1

)1%(

x

y

Why would we want to use this?


Vertex normals

• Add normals for surfaces incident on vertex:


The Winged-Edge Data Structure

• Maintains– List of vertices– List of edges– List of faces


Winged-Edge Data Structure

Vertex Incident Edge

A a

B b

C e

D a

Face Incident Edge

1 a

2 c

3 e

4 b

Vertex table:

Face table:


Edge Table

a

c

e

b d

1 2

B

D

Edge Vertices Faces Left Traverse Right Traverse

Name Start End Left Right Pred Succ Prec Succ

c B D 1 2 b a e d


Voxels

• Grid of volumetric samples– CAT, MRI, etc.


Scene Graphs

PolygonPolygon PolygonPolygon…

CompositeComposite

TranslateTranslate

ColorColor


CompositeComposite

ColorColor

TranslateTranslateTranslateTranslate

CompositeComposite

BarbellEnds

BarbellBar

SeparatorSeparator SeparatorSeparator SeparatorSeparator


Scene Graphs


Collapsing a Scene Graph

• It’s often possible to apply transformations in a scene graph once to the underlying vertices, collapsing some nodes. Example:

TranslationTranslation

RotationRotation

PolygonPolygon

PolygonPolygonCollapse to


Cloning

• If part of a scene graph has two edges to it, it will be necessary to clone before collapsing


RotationRotation

PolygonPolygon


RotationRotation

PolygonPolygon


RotationRotation

Polygon’Polygon’


Integration of mesheswith scene graphs


CompositeComposite

TranslateTranslate

ColorColor

Polygon MeshPolygon Mesh

ColorColor

TranslateTranslateTranslateTranslate

CompositeComposite

BarbellEnds

BarbellBar

SeparatorSeparator SeparatorSeparator SeparatorSeparator

Other possibilities?


Bounding Boxes

• Bounding boxes describe the X,Y,Z extents (minimum and maximum values).– If we know everything below a certain point in the scene

graph is in a box, we can avoid traversal if the box is not on-screen


class CBBoxF{ public: class CBBoxF() {}; class CBBoxF(const CGrPoint3F &p) : m_min(p), m_max(p) {}

void Set(const CGrPoint3F &p) {m_min = p; m_max = p;} void Set(const CBBoxF &b) {m_min = b.m_min; m_max=b.m_max;} void Include(const CGrPoint3F &p); void Include(const CBBoxF &b) {Include(b.m_min); Include(b.m_max);}

const CGrPoint3F &Max() const {return m_max;} const CGrPoint3F &Min() const {return m_min;} const float MinForD(int d) const {return m_min[d];} double Extent(int d) const {return m_max[d] - m_min[d];}

private: CGrPoint3F m_min; CGrPoint3F m_max;};


Levels of Detail (lod)

• Alternative scene graphs with different resolutions– Varying tessellation– Which we use depends on how far away the camera is.– Significant performance enhancement


BSP Trees

• Binary Space Partitioning– Divides space into half-spaces– We can then use this method to

• limit how much scene graph we examine• examine the scene graph in certain orders


k-d Trees

• k is dimensions• Node

– Info– x, y– lt-link– ge-link


Levels

• Level 0 – Index X– Level 1 – Index Y

• Level 2 – Index X

• level = 0 if root, level(parent)+1 otherwise• For each level, index: level mod k

• 3D: Index X, Index Y, Index Z, Index X, etc…


Example

• Banja Luca (19, 45)• Derventa (40, 50)• Teslic (38, 38)• Tuzla (54, 40)• Sinj (4, 4)

Banja Luca (19, 45)Banja Luca (19, 45)

Derventa (40, 50)Derventa (40, 50)

Teslic (38, 38)Teslic (38, 38)

Tuzla (54, 40)Tuzla (54, 40)

Sinj (4, 4)Sinj (4, 4)

X

Y

X

Y

Insert: East Broko (21, 57)Search for exact?Search for nearest?Range searches?


k-d Trees

• Search for exact– Okay, but may be o(k)

• Search for nearest– In a moment

• Range search?


Range Search

• Consider each tree level to be region• Recursively search all regions that overlap search

range


Example Regions




Tuzla (54, 40)Tuzla (54, 40)

Sinj (4, 4)Sinj (4, 4)

(-,+) (-,+)

(-,19) (-,+) [19,+) (-,+)

[19,+) (-,50)

[38,+) (-,50)

East Broko (21, 57)East Broko (21, 57)

[19,+) [50,+)

Range: (10,20) (20, 45)


k-d Trees

• Fast and easy• Tend to be rather tall (unbalanced)• How could we extend to disk structure?• What if k=1?


Quadtrees

• 2-d only• Split data 4 ways• Node:

– info– x,y– nw,sw,ne,se


Example

• Banja Luca (19, 45)

• Derventa (40, 50)

• Teslic (38, 38)

• Tuzla (54, 40)

• Sinj (4, 4)




Tuzla (54, 40)Tuzla (54, 40)

Sinj (4, 4)Sinj (4, 4)

ne senw

se

Insert: East Broko (21, 57)Search for exact?Search for nearest?Range searches?


Octtrees

• 3-d• Split data 8 ways• Node:

– info– x,y,z– up nw,sw,ne,se, down nw,sw,ne,se

cse 872 dr. charles b. owen advanced computer graphics1 polygon meshes and other modeling tools...

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