Computer Graphics! LECTURE 4 HIERARCHICAL MODELING INTRODUCTION TO 3D CONCEPTS. Doç. Dr. Mehmet Gokturk Gebze Institute of Technology!

Transcription

1 Computer Graphics! LECTURE 4 HIERARCHICAL MODELING INTRODUCTION TO 3D CONCEPTS Doç. Dr. Mehmet Gokturk Gebze Institute of Technology!

2 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();! M. Gokturk 2

3 Structure Concept! Editing advantages Structure attributes can be set within structure Applying transformations Labeling structures! M. Gokturk 3

4 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();! }! M. Gokturk 4

5 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! M. Gokturk 5

6 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 M. Gokturk 6

7 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! M. Gokturk 7

8 Symbol Hierarchies - Example! Logic circuit AND NOR OR AND One level hierarchical description of a circuit formed with logic gates M. Gokturk 8

9 Symbol Hierarchies Example 2! FACILITY workarea1 workarea2... TABLE CHAIR Cabinet TABLE Shelves Cabinet Desk... A two-level hierarchical description of a facility layout M. Gokturk 9

10 3D Concepts! 3D display methods! M. Gokturk 10

11 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! M. Gokturk 11

12 Three Dimensional Display Methods! Screen is a 2D flat surface 3D coordinates must be tranformed into 2D viewport coordinates in display plane PROJECTION! M. Gokturk 12

13 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 M. Gokturk 13

14 Parallel Projection! Project Points on the object surface along the paralle lines onto the display plane Used in CAD and technical drawing! M. Gokturk 14

15 Mapping to Pixel Coordinates! This process is often called the viewport transformation M. Gokturk 15

16 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. M. Gokturk 16

17 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! M. Gokturk 17

18 Depth Cueing! Distant objects are dimmed slightly Effect of haze, air and scattering Color can be changed slightly as well! M. Gokturk 18

19 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! M. Gokturk 19

20 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! M. Gokturk 20

21 Exploded and Cutaway Views! Used in CAD and technical drawing Relationships between parts of the objects are clearer Engine parts! M. Gokturk 21

22 3D and Stereoscopic Views! 2 Different viewing positions are displayed to each eye separately! HOLOVIDEO (holographic autostereo display) M. Gokturk 22

23 Caves and Fishbowls! M. Gokturk 23

24 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! M. Gokturk 24

25 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! M. Gokturk 25

26 V1 Polygon Tables! Vertex Table E1 E3 E6 Edge Table Surface Table! V2 E2 E4 V3 E5 V5 V4 V1 V2 V3 VERTEX TABLE x1,y1,z1 x2,y2,z2 x3,y3,z3 E1 E2 E3 EDGE TABLE V1,V2 V2,V3 V3,V1 POLYGON-SURFACE TABLE S1 S2 E1,E2,E3 E3,E4,E5,E6 V4 x4,y4,z4 E4 V3,V4 V5 x5,y5,z5 E5 V4,V5 E6 V5,V1 M. Gokturk 26

27 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! M. Gokturk 27

28 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 (x 1,y 1,z 1 ),(x 2,y 2,z 2 ),(x 3,y 3,z 3 )! M. Gokturk 28

29 Plane Equations! A=y 1 (z 2 -z 3 )+y 2 (z 3 -z 1 )+y 3 (z 1 -z 2 ) B=z 1 (x 2 -x 3 )+z 2 (x 3 -x 1 )+z 3 (x 1 -x 2 ) C=x 1 (y 2 -y 3 )+x 2 (y 3 -y 1 )+x 3 (y 1 -y 2 ) D= -x 1 (y 2 z 3 -y 3 z 2 )-x 2 (y 3 z 1 -y 1 z 3 )-x 3 (y 1 z 2 -y 2 z 1 ) 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.! M. Gokturk 29

30 Normal Vector! N=(A,B,C)! N M. Gokturk 30

31 Normal Vector! 3 vertex calculation! V 3 N V 1 V 2 N= (V 2 -V 1 )X(V 3 -V 2 ) N= (V 2 -V 1 )X(V 3 -V 1 ) N.P = - D M. Gokturk 31

32 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 M. Gokturk 32

33 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! M. Gokturk 33

34 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! M. Gokturk 34

35 Why triangular meshes?! Rasterizing triangles Interpolating parameters Post-triangle rendering Data structure 3 vertices Can approximate arbitrary shapes! M. Gokturk 35

36 Curved Lines and Surfaces! Quadric Surfaces Sphere Ellipsoid Torus Superquadrics (play with mode parameters) Superellipse Superellipsoid! M. Gokturk 36

37 Sphere! z P(x,y,z) φ y x θ M. Gokturk 37

38 Ellipsoid rz rx ry z P(x,y,z) φ y x θ M. Gokturk 38

39 Blobby Objects! Blobbiness Molecular shapes Gravitational, proximity conditions Field conditions Gaussian density functions! M. Gokturk 39

40 Blobby Objects (example-metaballs)! M. Gokturk 40

41 Blobby Objects - Example! M. Gokturk 41

42 Polynomials! Good candidates for freeform shape representation! M. Gokturk 42

43 Splines! Original spline device used in technical drawing Spline surfaces Control points Interpolation (Bezier) Approximation (Cubic)! M. Gokturk 43

44 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! M. Gokturk 44

45 Splines: Approximation 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 M. Gokturk 45

46 Parametric Continuity Conditions! Non continuous C -1 Zero order C 0 First order C 1 Second order C 2 Nth order C n! nth DERIVATIVES ARE EQUAL FROM BOTH SIDES M. Gokturk 46

47 Spline Specifications! X(u) = a x u 3 + b x u 2 + c x u + d x, 0<= u <= 1! p0 p1 u=0 p u=1 constraints Basis matrix M. Gokturk 47

48 Hermite Interpolation! 4 Equations requires 4 known boundary conditions 2 endpoints with 2 coordinates and 2 derivatives! p k p k+1 M. Gokturk 48

49 Hermite Interpolation! p k p k+1 M. Gokturk 49

50 Hermite Interpolation! M H M. Gokturk 50

51 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):! M. Gokturk 51

52 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! M. Gokturk 52

53 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. M. Gokturk 53

54 Constructive Solid Geometry - CSG! Regular set of BOOLEAN operations - union, intersection, subtraction, and complement:! M. Gokturk 54

55 CSG Trees! To construct complex objects Perform boolean operations re-cursively! M. Gokturk 55

56 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! M. Gokturk 56

57 Octrees!? M. Gokturk 57

58 Octrees M. Gokturk 58

59 Octrees M. Gokturk 59

60 Octrees M. Gokturk 60

61 Octrees - numbering! Construct the tree hierarchically Question: why is boolean operatins among voxels is easy? M. Gokturk 61

62 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! M. Gokturk 62

63 Fractal Based Modeling! Self Similarity Function Randomized Fractals Simple Example Start with a simple line! M. Gokturk 63

64 Fractal based modeling! Insert a triangle in the midpoint! M. Gokturk 64

65 Fractal based modeling Do the same for new lines M. Gokturk 65

66 Fractal based modeling Continue until you rech max resolution point or until you exhaust! Endless posibilities! M. Gokturk 66

67 Fractal Based Modeling - Examples! x = y = Zero(12000); for i = 2 to do x[i] = y[i-1]*(1 + Sin(0.7*x[i-1]/Rad)) - 1.2*Sqrt(Abs(x[i-1])); y[i] = x[i-1]; enddo plot('equal', x, y, 'dots'); M. Gokturk 67

68 Fractal Based Modeling! Nature has a lot of fractal structures! Real picture of a califlower (Karnıbahar in Turkish) M. Gokturk 68

69 Other methods! Particle Systems Physically Based Modeling Data visualization modeling - These will be studied in future courses! M. Gokturk 69

70 Questions?! Homeworks? You must read the book! Get a good grade, not a passing grade.! M. Gokturk 70