Computer Graphics !

LECTURE 4 HIERARCHICAL MODELING

INTRODUCTION TO 3D CONCEPTS

Doç. Dr. Mehmet Gokturk Gebze Institute of Technology

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Structure Concept!● In many graphics applications, it is necessary to be able to modify

parts of the whole model/picture. ● Therefore hierachical structures are used. ● All labeled set of output primitives are called structure: glBegin(GL_POLYGON);

glVertex3f (0.25, 0.25, 0.0); glVertex3f (0.75, 0.25, 0.0); glVertex3f (0.75, 0.75, 0.0); glVertex3f (0.25, 0.75, 0.0); glEnd(); !

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Structure Concept!●  Editing advantages ●  Structure attributes can be set within structure ●  Applying transformations ●  Labeling structures!

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Example (openGL)!/* Draw the field lines */! r1 = 12;! r2 = 16;! for (j=0;j<360;j+=20) {! glPushMatrix();! glRotatef((double)j,0.0,1.0,0.0);! glBegin(GL_LINE_STRIP);! glColor3f(grey.r,grey.g,grey.b);! for (i=-140;i<140;i++) {! x = r1 + r1 * cos(i*DTOR);! y = r2 * sin(i*DTOR);! z = 0;! glVertex3f(x,y,z); ! } ! glEnd();! glPopMatrix(); ! }!

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Basic Modeling Concepts!

●  Engineering ●  Architecture ●  Design ●  Economic and Social

●  The creation and manipulation of a system representation is called modeling

●  Any single representation is called a model of the system (can be descriptive or graphical)

●  Graphical models are often called as geometric models!

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Symbols and Instances!

●  Component parts of the system are displayed as geometric structures, called symbols

●  Each occurrence of a symbol within a model is called an instance of that symbol!

Model of a logic circuit

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Symbol Hieracrchies!

●  Many models can be organized as a hierarchy of symbols ●  The basic “building blocks” for the model are defined as

simple geometric shapes appropriate to the type of model under consideration

●  The basic symols can be used to form composite objects, called modules

●  They can be grouped to form higher-level modules!

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Symbol Hierarchies - Example!

Logic circuit

AND NOR OR AND

One level hierarchical description of a circuit formed with logic gates

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Symbol Hierarchies – Example 2!

...

... TABLE CHAIR Cabinet TABLE Shelves Cabinet Desk

workarea2 workarea1

FACILITY

A two-level hierarchical description of a facility layout

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3D Concepts!

●  3D display methods!

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Three Dimensional Space!

●  The real space is three dimensional ●  Human eye expects a 3D picture ●  Do not confuse 3D with StereoVision ●  Positioning of objects are done in 3D space!

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Three Dimensional Display Methods!

●  Screen is a 2D flat surface ●  3D coordinates must be tranformed into 2D viewport

coordinates in display plane ●  PROJECTION!

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Camera View!

Reorient the entire scene so the the camera is located and aligned at the origin

We need to perform rotations and translations so that the camera is aligned with the coordinate frames

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Parallel Projection!

●  Project Points on the object surface along the paralle lines onto the display plane

●  Used in CAD and technical drawing!

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Mapping to Pixel Coordinates!

This process is often called the viewport transformation

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Perspective Transformation!

Artists (Donatello,Brunelleschi and Da Vinci) during renaissance discovered the importance of perspective for making images appear realistic. Perspective causes objects nearer to the viewer to appear larger than the same object would appear farther away.

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Perspective Projection!

●  Project points to display plane along converging paths ●  Objects farther from the viewing poisition appear smaller ●  Parallel lines are projected as non-parallel lines, converging

at some distant point ●  Vanishing point

●  Perspective projection is more realistic ●  Human eye perceives the world with perspective projection!

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Depth Cueing!

●  Distant objects are dimmed slightly ●  Effect of haze, air and scattering ●  Color can be changed slightly as well!

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Visible Line and Surface Identification!

●  Depth relationship in a wireframe display is enhanced

●  Highlight visible lines ●  Use dashed lines for nonvisible ●  Removing altogether sometimes is not good ●  Visible Surface: ●  Hidden surfaces are obscured ●  Various algorithms are present and still developing!

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Surface Rendering!

●  Setting surface intensity of objects according to the lighting condition ●  Polygon rendering and Surface rendering (OpenGL) using scan

line methods ●  Ray tracing pixel by pixel ●  Radiosity techniques by volume!

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Exploded and Cutaway Views!

●  Used in CAD and technical drawing ●  Relationships between parts of the objects are clearer ●  Engine parts!

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3D and Stereoscopic Views!

●  2 Different viewing positions are displayed to each eye separately!

HOLOVIDEO (holographic autostereo display)

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Caves and Fishbowls!

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3D Object Representations!

●  Objects in 3D world must be modeled and recorded to the computer memory in order to be processed

●  Representing complex objects is usually a challenge ●  Performance considerations!

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Polygon Surfaces!

●  Enclose object within the polygon surfaces ●  Not a unique model for the object ●  Simple and fast ●  Lots of polygons ●  Polygon Mesh ●  Easy to display in wireframe during design stage!

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Polygon Tables!

●  Vertex Table ●  Edge Table ●  Surface Table!

E1

E2 E4

E3

E5

E6

V1

V5 V3 V2

V4

VERTEX TABLE V1 x1,y1,z1 V2 x2,y2,z2 V3 x3,y3,z3 V4 x4,y4,z4 V5 x5,y5,z5

POLYGON-SURFACE TABLE

S1 E1,E2,E3 S2 E3,E4,E5,E6

EDGE TABLE E1 V1,V2 E2 V2,V3 E3 V3,V1 E4 V3,V4 E5 V4,V5 E6 V5,V1

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Wireframe representation!

● Represent as set of points and lines with limited topological information (a graph) ● Can be ambiguous - Non-unique ● Question: Are there other objects with this wireframe? ● Not used for internal representation ● Used often for fast display in interactive systems!

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Plane Equations!

●  Ax+By+Cz+D=0 where x,y,z is any point on a plane ●  Three noncollinear points can determine A,B,C,D ●  Select (x1,y1,z1),(x2,y2,z2),(x3,y3,z3)!

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Plane Equations!

A=y1(z2-z3)+y2(z3-z1)+y3(z1-z2) B=z1(x2-x3)+z2(x3-x1)+z3(x1-x2) C=x1(y2-y3)+x2(y3-y1)+x3(y1-y2) D= -x1(y2z3-y3z2)-x2(y3z1-y1z3)-x3(y1z2-y2z1) ●  As vertex values and other information are entered into the

polygon data structure, values for A, B,C and D are computed for each polygon and then can be stored with other polygon data.!

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Normal Vector!●  N=(A,B,C)!

N

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Normal Vector!3 vertex calculation!

N

V1 V2

V3

N= (V2-V1)X(V3-V2) N= (V2-V1)X(V3-V1) N.P = - D

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Inside - Outside!

●  If Ax + By + Cz + D < 0 point is inside ●  If Ax + By + Cz + D > 0 point is outside ●  If Ax + By + Cz + D = 0 point is on the plane!

N

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Polygon Meshes!

●  Construct the object from triangles or quadrilaterals by tiling ●  Most models are constructed this way ●  Easy to work with ●  Easy to render ●  Rates about 300 million triangles/sec available with

ordinary graphics cards for PC computers!

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Polygon Meshes -triangular meshes-!●  Free form/Boolean disadvantages: complexity of description &

algorithms ●  Representing a model as closed triangular mesh is EASY –

Operations & Description format very simple ●  Base representation for scanned data ●  Most popular web representation!

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Why triangular meshes?!

●  Rasterizing triangles ●  Interpolating parameters ●  Post-triangle rendering ●  Data structure ●  3 vertices ●  Can approximate arbitrary shapes!

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Curved Lines and Surfaces!

●  Quadric Surfaces ●  Sphere ●  Ellipsoid ●  Torus

●  Superquadrics (play with mode parameters) ●  Superellipse ●  Superellipsoid!

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Sphere!

P(x,y,z)

x

y

z

φ θ

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Ellipsoid

P(x,y,z)

x

y

z

φ θ

rx

ry

rz

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Blobby Objects!

●  Blobbiness ●  Molecular shapes ●  Gravitational, proximity conditions ●  Field conditions ●  Gaussian density functions!

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Blobby Objects (example-metaballs)!

http://www.cs.wpi.edu/~matt/courses/cs563/talks/metaballs.html

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Blobby Objects - Example!

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Polynomials!

●  Good candidates for freeform shape representation!

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Splines!

●  Original spline device used in technical drawing ●  Spline surfaces ●  Control points ●  Interpolation (Bezier) ●  Approximation (Cubic)!

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Splines: Interpolation!

● When polynomial sections are fitted so that the curve passes through each control point, the resulting curve is said to interpolate the set of control points.

● Interpolation curves are commonly used to digitize drawings or to specify animation paths!

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• When the polynomials are fitted to the general control point path without necessarily passing through any control point, the resulting curve is said to approximate the set of control points.

• Used as design tools to structure object surfaces

Splines: Approximation

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Parametric Continuity Conditions!

●  Non continuous C-1

●  Zero order C0

●  First order C1

●  Second order C2

●  Nth order Cn!

nth DERIVATIVES ARE EQUAL FROM BOTH SIDES

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Spline Specifications!

●  X(u) = axu3 + bxu2 + cxu + dx, 0<= u <= 1!

p0

p

p1 u=0 u=1

Basis matrix

constraints

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Hermite Interpolation!

●  4 Equations requires 4 known boundary conditions

●  2 endpoints with 2 coordinates and 2 derivatives!

pk pk+1

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Hermite Interpolation! pk pk+1

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Hermite Interpolation!

MH

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Hermite Interpolation!

●  We use the following formula to calculate points on the curve as u goes from 0 to 1 (for all x,y and z):!

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Other Types of Splines!

●  Lagrange (Interpolation Scheme) ●  Kochanek-Bartel (Interpolation Scheme) ●  Catmull-Rom (Interpolation Scheme) ●  Bézier (Approximation Scheme) ●  B-spline (Approximation Scheme) ●  Beta Splines (Interpolation Scheme) ●  Cardinal Splines (Interpolation Scheme) ●  Rational Splines (Approximation Scheme)

●  These will be studied in COMP304!

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Sweep Representations!

●  Useful for objects that possess translational, rotational, or other symmetries

●  Specify a 2D shape and a swepp that moves the shape through a reion of space!

Helix tube created by sweeping circle along helix

Horn created by adding scaling to the sweep.

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Constructive Solid Geometry - CSG!

●  Regular set of BOOLEAN operations - union, intersection, subtraction, and complement:!

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CSG Trees!

●  To construct complex objects

●  Perform boolean operations re-cursively!

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Octrees!

●  Used to subdivide space recursively ●  Hieracrhical tree structure representing solid objects ●  Each node corresponds to a space in 3D space ●  Convenient way to store large object information in 3D ●  EXAMPLE: Medical imaging data ●  2D version is called Quadtree!

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Octrees!

?

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Octrees

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Octrees

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Octrees

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Octrees - numbering!

● Construct the tree hierarchically ● Question: why is boolean operatins among voxels is easy?

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Quadtrees!

● Data structure for storage of 2D data by successive subdivision of regions into quadrants ● Leaf quadrants contain uniform data (full/empty) ● Non-uniform quadrants are subdivided further!

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Fractal Based Modeling!

●  Self Similarity Function ●  Randomized Fractals ●  Simple Example ●  Start with a simple line!

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Fractal based modeling!

●  Insert a triangle in the midpoint!

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Fractal based modeling

•  Do the same for new lines

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Fractal based modeling

• Continue until you rech max resolution point or until you exhaust !

• Endless posibilities !

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Fractal Based Modeling - Examples!

x = y = Zero(12000);

for i = 2 to 12000 do

x[i] = y[i-1]*(1 + Sin(0.7*x[i-1]/Rad)) - 1.2*Sqrt(Abs(x[i-1]));

y[i] = 0.21 - x[i-1];

enddo

plot('equal', x, y, 'dots');

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Fractal Based Modeling!

●  Nature has a lot of fractal structures!

Real picture of a califlower (Karnıbahar in Turkish)

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Other methods!

● Particle Systems ● Physically Based Modeling ● Data visualization modeling - These will be studied in future courses!

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Questions ?!

● Homeworks? ● You must read the book! ● Get a good grade, not a passing grade.!