1. Learning Objectives 3D Object Representations Polyhedron
Quadrics, SuperQuadrics
Spline, Bezier
Blobby
Constructive Solid Geometry

2. 3D Object Representations
Graphics scenes melibatkan berbagai jenis objek dan material surfaces
Trees, flowers, clouds, rocks, water, bricks, wood paneling, rubber, paper, marble, steel, glass, plastic, etc.
Bagaimana merepresentasikan objek 3D pada openGL?

3. 3D Object Representations
Untuk merepresentasikan objek 3D, ada beberapa teknik
Menggunakan polygon dan quadric untuk membuat objek seperti polyhedrons ataupun ellipsoids
Untuk membuat permukaan berkurva seperti pada sayap pesawat, gears, bodi mesin, etc, digunakan Spline surfaces
Constructive solid geometry untuk menyusun bentuk geometri dasar menjadi objek komplek
Untuk memodelkan pegunungan, awan, tumbuhan, atau air terjun digunakan procedural methods seperti fractals dan particle system

4. Predefined Objects
OpenGL sudah menyediakan fungsi menggambar beberapa objek dasar yang tinggal dipakai. Tak perlu membuat dari awal.
Objek-objek dari OpenGL ini dapat disusun untuk membuat bentuk yang kita inginkan
Beberapa yang sudah disediakan OpenGL antara lain:
Polyhedra
Polyhedron functions
Quadric Surfaces
Superquadrics

5. Object with Superquadrics

6. A. Polyhedron
A polyhedron is a connected mesh of simple planar polygons that encloses a finite amount of space.
Polyhedron adalah rangkaian jala polygon (polygon mesh) dengan kriteria sbb
Setiap edge dipakai oleh 2 faces
Sedikitnya 3 edge bertemu pada setiap vertex.
Faces tidak saling menembus, tetapi berhenti pada suatu edge.
Euler’s formula : If F, E, V represent the number of faces, vertices and edges of a polyhedron, then
V + F  E = 2.

7. 3D Object Representation
The data for polygonal meshes can be represented in two ways.
Method 1:
Vertex List
Normal List
Face List (Polygon List)
Method 2:
Edge List

8. Surface Normal n

9. Vertices and Faces - E.g. Cube2 5 5 4 3 1 1 1 5 6 6 7 7 2 2 3 3 4
Vertex Index
Face Index

10. Data representation using vertex, face and normal lists:
xyz axis

11. Data representation using vertex, face and edge lists:

12. Normal Vectors (OpenGL)

13. Regular Polyhedra (Platonic Solids)
Jika semua face pada polyhedron adalah identik dan berupa regular polygon, maka polyhedron tsb disebut platonic solid.
Hanya ada 5 jenis platonic solid

14. The Platonic Solids
Regular tetrahedron (or triangular pyramid) has 4 faces
Regular hexahedron (or cube) with 6 faces
Regular octahedron with 8 faces
Regular dodecahedron with 12 faces
Regular icosahedron with 20 faces

15. Menggambar polyhedron
Ada 2 cara
Method1 : Fitting the surface with a polygon mesh. Membungkus permukaan objek polyhedron dengan susunan jala polygon. Proses ini disebut juga dengan surface tesselation
Method 2 : Memakai fungsi yang disediakan library GLUT

16. Method-1 Polygon Mesh
In fitting polygons to a surface, we are not limited to using GL_POLYGON
We can also use
GL_TRIANGLES
GL_TRIANGLE_STRIP
GL_TRIANGLE_FAN
GL_QUADS
GL_QUAD_STRIP

17. Polygon Mesh
Polygon mesh ini juga bisa dipakai untuk memodelkan permukaan objek lainnya

18. Method 2- OpenGL Polyhedron Functions
5 functions produce wire frames which can be easily used
Ex: glutWireX(), where X is one of the names Cube, Tetrahedron, Octahedron, Dodecahedron, or Icosahedron (with the first letter capitalized).
5 functions produce polyhedra facets as shaded fill areas - the characteristics of these are determined by material and lighting properties.
Ex: glutSolidX(), where X is as above.

19. GLUT Library of Polyhedron Functions
Example: prog8OGLGLUTPolyhedra.cpp

20. glutWireTetrahedron() and glutWireCube(1.0)
4 faces
6 faces

21. glutWireOctahedron() and glutWireDodecahedron()
8 faces
12 faces

22. And, glutWireIcosahedron()
20 faces

23. B.Quadrics Objek yang didefinisikan sebagai persamaan quadratics
Sphere
Ellipsoid
Torus
General form

24. Quadric surfaces Double cones Ellipsoids Hyperboloids of one sheet
Hyperboloids of two sheets

25. Quadric surfaces
Elliptic paraboloids
Hyperbolic paraboloids

26. Superquadrics
the squaring operations are replaced by arbitrary powers.
Superellipses

27. Superquadrics
Superellipsoids

28. GLUT Quadric Functions – for Solids, Substitute Solid for Wire
glutWireSphere(radius, slices, stacks);
glutWireCone(base, height, slices, stacks);
glutWireTorus(innerRadius, outerRadius, nsides, rings);
and the following is provided also!
glutWireTeapot(size);

29. GLUT Quadric Functions
QuadricSurfs.cpp

30. GLU Quadric-Surface Functions
void gluSphere (GLUquadricObj *qobj, GLdouble radius,GLint slices, GLint stacks);
void gluCylinder (GLUquadricObj *qobj, GLdouble baseRadius,GLdouble topRadius, GLdouble height,GLint slices, GLint stacks);
void gluDisk (GLUquadricObj *qobj, GLdouble innerRadius,GLdouble outerRadius, GLint slices, GLint rings);

31. GLU Quadric-Surface Functions
void gluPartialDisk (GLUquadricObj *qobj, GLdouble innerRadius,GLdouble outerRadius, GLint slices, GLint rings,GLdouble startAngle, GLdouble sweepAngle);

32. GLU Quadric-Surface Functions
Quadric.c

33. WHY IS THE TEAPOT POPULAR?
Pada zaman dahulu belum ada library packages untuk 3D modelling. Pemodelan objek 3D dilakukan dengan tangan, menggambar kurva dan titik2nya dicatat secara manual.
Computer graphics researcher Martin Newell, ketika hendak mencari barang untuk dibuat model matematika tak sengaja menemukan teapot
Teapot adalah model yang ideal untuk eksperimen 3D modelling, karena
Mudah dikenal
Topologi yang komplek
Mempunyai proyeksi bayangan pada dirinya sendiri
Melibatkan topik hidden surface
Memiliki permukaan cekung dan cembung, juga saddle points (curved up and down)
Doesn't take much storage space

34. The Utah Teapot The real teapot---
The teapot was donated to the Boston Computer Museum but now resides in the Ephemera collection of the Computer History Museum where it's catalogued as "Teapot used for Computer Graphics rendering" catalogue number X

35. Many Versions of Teapots
From Steve Baker’s History of the Teapot site:

36. Teapot!
From Steve Baker’s History of the Teapot site:

37. wireframe

38. Lighting & shading

39. Texture mapped

40. Multiple Teapots of Various Materials
teapots.c

41. C. Spline Representations
Splines are used to design curves and surfaces based on a set of user-defined points
Control points
Himpunan titik koordinat yang mengontrol bentuk kurva
Interpolation
Semua control points tersambung satu sama lain pada garis kurva
Approximate
Semua atau beberapa control points terletak di luar garis kurva

42. Spline Representations
Interpolated
Approximate

43. Bezier Spline Curves
Developed by French engineer Pierre Bézier for use in the design of Renault automobile bodies
Easy to implement
Widely used in CAD systems, graphics, drawing and painting packages

44. Bezier Curve Equations
Diketahui sejumlah n +1 control points,
nilai k antara 0 sampai n
Persamaan garis Bezier akan membentuk titik-titik garis kurva sesuai control point yang didefinisikan

45. Bezier Curve Equations
Degree 1 – Linear Curve
Degree 2
Degree 3
Degree n

46. Bezier Spline Curves
A common use for Bezier curves is in font definition

47. Bezier Spline Curves
If we specify the first and the last control point as the same point, we can generate a closed Bezier curve

48. Bezier Surfaces
Two sets of Bezier curves can be used to design an object surface
with pj,k specifying the location of (m+1) by (n+1) control points

49. Bezier Surfaces
u and v parameters

50. Bezier Surfaces
An example Bezier surface

51. OpenGL Approximation Spline Functions
Bezier splines and B-splines can be displayed using OpenGL functions
The core library contains Bezier functions, and GLU has B-spline functions
Bezier functions are often hardware implemented

52. OpenGL Bezier-Spline Curve Functions
We specify parameters and activate the routines for Bezier-curve display with
glMap1*(GL_MAP1_VERTEX_3, uMin, uMax, stride, nPts, *ctrlPts);
glEnable(GL_MAP1_VERTEX_3);
and deactivate with
glDisable(GL_MAP1_VERTEX_3);
uMin and uMax are typically 0 and 1.0
stride=3 for 3D
nPts is the number of control points
ctrlPts is the array of control points

53. OpenGL Bezier-Spline Curve Functions
After setting parameters, we need to evaluate positions along the spline path and display the resulting curve. To calculate coordinate positions we use
glEvalCoord1*(uValue);
where uValue is assigned some value in the interval from uMin to uMax

54. Example OpenGL Code prog8OGLBezierCurve.cpp
GLfloat ctrlPts [4][3] = { {-40.0, 40.0, 0.0}, {-10.0, 200.0, 0.0},
{10.0, , 0.0}, {40.0, 40.0, 0.0} };
glMap1f (GL_MAP1_VERTEX_3, 0.0, 1.0, 3, 4, *ctrlPts);
glEnable (GL_MAP1_VERTEX_3);
GLint k;
glColor3f (0.0, 0.0, 1.0); // Set line color to blue.
glBegin (GL_LINE_STRIP); // Generate Bezier "curve".
for (k = 0; k <= 50; k++)
glEvalCoord1f (GLfloat (k) / 50.0);
glEnd ( );
glColor3f (1.0, 0.0, 0.0); // Set point color to red.
glPointSize (5.0); // Set point size to 5.0.
glBegin (GL_POINTS); // Plot control points.
for (k = 0; k < 4; k++)
glVertex3fv (&ctrlPts [k][0]);
prog8OGLBezierCurve.cpp

55. Example OpenGL Code
prog8OGLBezierCurve.cpp

56. OpenGL Bezier-Spline Surface Functions
We specify parameters and activate the routines for Bezier surface display with
glMap2*(GL_MAP2_VERTEX_3, uMin, uMax, uStride, nuPts, vMin, vMax, vStride, nvPts, *ctrlPts);
glEnable(GL_MAP2_VERTEX_3);
and deactivate with
glDisable(GL_MAP2_VERTEX_3);
uMin, uMax, vMin and vMax are typically 0 and 1.0
stride=3 for 3D
nuPts and nvPts are the size of the array
ctrlPts is the double subscripted array of control points

57. Example OpenGL Code bezsurf.c GLfloat ctrlpoints[4][4][3] = {
{{-1.5, -1.5, 4.0}, {-0.5, -1.5, 2.0},
{0.5, -1.5, -1.0}, {1.5, -1.5, 2.0}},
{{-1.5, -0.5, 1.0}, {-0.5, -0.5, 3.0},
{0.5, -0.5, 0.0}, {1.5, -0.5, -1.0}},
{{-1.5, 0.5, 4.0}, {-0.5, 0.5, 0.0},
{0.5, 0.5, 3.0}, {1.5, 0.5, 4.0}},
{{-1.5, 1.5, -2.0}, {-0.5, 1.5, -2.0},
{0.5, 1.5, 0.0}, {1.5, 1.5, -1.0}} };
glMap2f(GL_MAP2_VERTEX_3, 0, 1, 3, 4, 0, 1, 12, 4, &ctrlpoints[0][0][0]);
glEnable(GL_MAP2_VERTEX_3);
for (j = 0; j <= 8; j++) {
glBegin(GL_LINE_STRIP);
for (i = 0; i <= 30; i++)
glEvalCoord2f((GLfloat)i/30.0, (GLfloat)j/8.0);
glEnd();
glEvalCoord2f((GLfloat)j/8.0, (GLfloat)i/30.0); }
bezsurf.c

58. Example OpenGL Code
bezsurf.c

59. Bézier Surfaces: Example
Utah Teapot modeled by 32 Bézier Patches

60. D. Blobby Objects
Memodelkan objek yang dapat berubah bentuk tapi volumenya tetap
Contoh
Water drops
Molecules
Force fields

61. Blobby Objects
A collection of density functions
Equi-density surfaces

62. Metaballs (Blinn Blobbies)

63. E.Constructive Solid Geometry
Primitives
Transformed
Combined
Bermula dari objek geometri primitive, ditransformasikan dan
dikombinasikan membentuk objek yang kompleks