1. Modeling

2. Geometric Models  2D and 3D objects  Triangles, quadrilaterals, polygons  Spheres, cones, boxes  Surface characteristics  Color  Texture  Composite objects  Other objects and their relationships to each other 2

3. The basic idea  Describe objects using surfaces, which are polygons  Triangles, quadrilaterals, etc.  Important thing is that they are flat and convex  Convex : A line segment connecting any two points on the polygon is contained in the polygon.  Provide points in counterclock-wise order from the visible side 3 P Q P convex concave

4. Outline 4  OpenGL Elementary Rendering (codes)  Representation  Appendix: Vector Operations *

5. OpenGL Elementary Rendering  Geometric Primitives  Building Models  Managing OpenGL State 5

6. OpenGL Geometric Primitives  All geometric primitives are specified by vertices GL_QUAD_STRIP GL_POLYGON GL_TRIANGLE_STRIP GL_TRIANGLE_FAN GL_POINTS GL_LINES GL_LINE_LOOPGL_LINE_STRIP GL_TRIANGLES GL_QUADS 6

7. Simple Example void drawRhombus( GLfloat color[] ) { glBegin( GL_QUADS ); glColor3fv( color ); glVertex2f( 0.0, 0.0 ); glVertex2f( 1.0, 0.0 ); glVertex2f( 1.5, 1.118 ); glVertex2f( 0.5, 1.118 ); glEnd(); } 7

8. OpenGL Command Formats glVertex3fv( v ) Number of components 2 - (x,y) 3 - (x,y,z) 4 - (x,y,z,w) Data Type b - byte ub - unsigned byte s - short us - unsigned short i - int ui - unsigned int f - float d - double Vector omit “ v ” for scalar form glVertex2f( x, y ) 8

9. Specifying Geometric Primitives  Primitives are specified using glBegin( primType ); glEnd();  primType determines how vertices are combined GLfloat red, greed, blue; Glfloat coords[3]; glBegin( primType ); for ( i = 0; i < nVerts; ++i ) { glColor3f( red, green, blue ); glColor3f( red, green, blue ); glVertex3fv( coords ); glVertex3fv( coords );}glEnd(); 9

10. Shapes Tutorial 10

11. OpenGL Color Models *  RGBA or Color Index color index mode Display 1 2 4 8 16  RedGreenBlue 0 1 2 3 24 25 26 12321974  RGBA mode 11

12. Building models  Consider a mesh  There are 8 nodes and 12 edges  5 interior polygons  6 interior (shared) edges  Each vertex has a location v i = (x i y i z i ) v1v1 v2v2 v7v7 v6v6 v8v8 v5v5 v4v4 v3v3 e1e1 e8e8 e3e3 e2e2 e 11 e6e6 e7e7 e 10 e5e5 e4e4 e9e9 e 12

13. Simple Representation  Each polygon is defined by the geometric locations of its vertices, which leads to OpenGL code such as  Inefficient and unstructured glBegin(GL_POLYGON); glVertex3f(x1, y1, z1); glVertex3f(x7, y7, z7); glVertex3f(x8, y8, z8); glVertex3f(x6, y6, z6); glEnd();

14. Vertex Lists  Put the geometry in an array  Use pointers from the vertices into this array  Introduce a polygon list x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 x 4 y 4 z 4 x 5 y 5 z 5. x 6 y 6 z 6 x 7 y 7 z 7 x 8 y 8 z 8 P1 P2 P3 P4 P5 v1v7v6v1v7v6 v8v5v6v8v5v6 topology geometry

15. Example: Modeling a Cube  Define global arrays for vertices and colors GLfloat vertices[][3] = {{-1.0,-1.0,-1.0},{1.0,-1.0,-1.0}, {1.0,1.0,-1.0}, {-1.0,1.0,-1.0}, {-1.0,-1.0,1.0}, {1.0,-1.0,1.0}, {1.0,1.0,1.0}, {-1.0,1.0,1.0}}; GLfloat colors[][3] = {{0.0,0.0,0.0},{1.0,0.0,0.0}, {1.0,1.0,0.0}, {0.0,1.0,0.0}, {0.0,0.0,1.0}, {1.0,0.0,1.0}, {1.0,1.0,1.0}, {0.0,1.0,1.0}};

16.  Draw a quadrilateral from a list of indices into the array vertices and use color corresponding to first index void polygon(int a, int b, int c, int d) { glBegin(GL_POLYGON); glColor3fv(colors[a]); glVertex3fv(vertices[a]); glVertex3fv(vertices[b]); glVertex3fv(vertices[c]); glVertex3fv(vertices[d]); glEnd(); }

17.  Draw cube from faces  Note that vertices are ordered so that we obtain correct outward facing normals void colorcube( ) { polygon(0,3,2,1); polygon(2,3,7,6); polygon(0,4,7,3); polygon(1,2,6,5); polygon(4,5,6,7); polygon(0,1,5,4); } 0 56 2 4 7 1 3

18.  Efficiency  The problem is that we must do many function calls to draw the cube  6 glBegin, 6 glEnd  6 glColor  24 glVertex  More if we use texture and lighting

19. *  OpenGL provides vertex arrays to store array data  Six types of arrays supported  Vertices  Colors  Color indices  Normals  Texture coordinates  Edge flags  Here we will need only colors and vertices

20. *  Initialization  Using the same color and vertex data, first we enable glEnableClientState(GL_COLOR_ARRAY); glEnableClientState(GL_VERTEX_ARRAY);  Identify location of arrays glVertexPointer(3, GL_FLOAT, 0, vertices); glColorPointer(3, GL_FLOAT, 0, colors); 3d arraysstored as floats data contiguous data array

21. *  Form an array of face indices  Each successive four indices describe a face of the cube GLubyte cubeIndices[24] = {0,3,2,1,2,3,7,6 0,4,7,3,1,2,6,5,4,5,6,7,0,1,5,4};

22. *  Drawing the cube  Method 1:  Method 2: for(i=0; i<6; i++) glDrawElements(GL_POLYGON, 4, GL_UNSIGNED_BYTE, &cubeIndices[4*i]); format of index data start of index data what to draw number of indices glDrawElements(GL_QUADS, 24, GL_UNSIGNED_BYTE, cubeIndices); Draws cube with 1 function call!!

23. OpenGL’s State Machine  All rendering attributes are encapsulated in the OpenGL State  rendering styles  shading  lighting  texture mapping 23

24. Manipulating OpenGL State  Appearance is controlled by current state for each ( primitive to render ) { update OpenGL state render primitive }  Manipulating vertex attributes is most common way to manipulate state glColor*() / glIndex*() glNormal*()glTexCoord*() 24

25. Controlling current state  Setting State glPointSize( size ); glShadeModel( GL_SMOOTH );  Enabling Features glEnable( GL_LIGHTING ); glDisable( GL_TEXTURE_2D ); 25

26. Outline 26  OpenGL Elementary Rendering (codes)  Representation  Appendix: Vector Operations

27. Objectives  Introduce dimension and basis  Introduce coordinate systems for representing vectors spaces and frames for representing affine spaces  Discuss change of frames and bases  Introduce homogeneous coordinates

28. Vectors  Objects with length and direction  Examples include  Force  Velocity  Directed line segments  We will think of a vector as a displacement from one point to another  We use bold typeface to indicate vectors v 28

29. Vectors as Displacements 29  As a displacement, a vector has length and direction, but no inherent location  You can drive one mile East in Zhuhai  To represent vectors, we need firstly learn…

30. Linear Independence  A set of vectors v 1, v 2, …, v n is linearly independent if  1 v 1 +  2 v 2 +..  n v n =0 iff  1 =  2 =…=0  If a set of vectors is linearly independent, we cannot represent one in terms of the others  If a set of vectors is linearly dependent, as least one can be written in terms of the others

31. Dimension  In a vector space, the maximum number of linearly independent vectors is fixed and is called the dimension of the space  In an n-dimensional space, any set of n linearly independent vectors form a basis for the space  Given a basis v 1, v 2,…., v n, any vector v can be written as v=  1 v 1 +  2 v 2 +….+  n v n where the {  i } are unique

32. Coordinate Systems  Consider a basis v 1, v 2,…., v n  A vector is written v=  1 v 1 +  2 v 2 +….+  n v n  The list of scalars {  1,  2, ….  n } is the representation of v with respect to the given basis  Scalar: here an ordinary real (e.g., float) number.  Scalars alone have no geometric properties  We can write the representation as a row or column array of scalars a=[  1  2 ….  n ] T =

33. Examples  v=2v 1 +3v 2 -4v 3  a=[2 3 –4] T  Note that this representation is with respect to a particular basis  v = 5 v 1 -2v 2 in a plane

34. Coordinate Systems  Which is correct?  Both are because vectors have no fixed location v v

35. Frames  A coordinate system is insufficient to represent points  If we work in an affine space we can add a single point, the origin, to the basis vectors to form a frame P0P0 v1v1 v2v2 v3v3

36. Representation in a Frame  Frame determined by (P 0, v 1, v 2, v 3 )  Within this frame, every vector can be written as v=  1 v 1 +  2 v 2 +….+  n v n  Every point can be written as P = P 0 +  1 v 1 +  2 v 2 +….+  n v n

37. Confusing Points and Vectors  Consider the point and the vector P = P 0 +  1 v 1 +  2 v 2 +….+  n v n v=  1 v 1 +  2 v 2 +….+  n v n  They appear to have the similar representations p=[  1  2  3 ] T v=[  1  2  3 ] T which confuses the point with the vector v p v Vector can be placed anywhere point: fixed

38. A Single Representation  If we define 0P = 0 and 1P =P then we can write v=  1 v 1 +  2 v 2 +  3 v 3 = [v 1 v 2 v 3 P 0 ] [  1  2  3 0 ] T P = P 0 +  1 v 1 +  2 v 2 +  3 v 3 = [v 1 v 2 v 3 P 0 ] [  1  2  3 1 ] T Thus we obtain the four-dimensional homogeneous coordinate representation v = [  1  2  3 0 ] T p = [  1  2  3 1 ] T

39. Homogeneous Coordinates  The homogeneous coordinates form for a three dimensional point [x y z] is given as p =[x’ y’ z’ w] T =[wx wy wz w] T We return to a three dimensional point (for w  0 ) by x  x’ /w y  y’/w z  z’/w  If w=0, the representation is that of a vector  For w=1, the representation of a point is [x y z 1]  Note that homogeneous coordinates replaces points in three dimensions by lines through the origin in four dimensions

40.  Homogeneous coordinates are key to all computer graphics systems  All standard transformations (rotation, translation, scaling) can be implemented with matrix multiplications using 4 x 4 matrices  Hardware pipeline works with 4 dimensional representations  For orthographic viewing, we can maintain w=0 for vectors and w=1 for points  For perspective we need a perspective division

41. Change of Coordinate Systems  Consider two representations of a same vector with respect to two different bases. The representations are a=[  1  2  3 ] T b=[  1  2  3 ] T where v=  1 v 1 +  2 v 2 +  3 v 3 = [v 1 v 2 v 3 ] [  1  2  3 ] T =  1 u 1 +  2 u 2 +  3 u 3 = [u 1 u 2 u 3 ] [  1  2  3 ] T

42. Representing 2 nd basis in terms of 1 st  Each of the basis vectors, u 1,u 2, u 3, are vectors that can be represented in terms of the first basis u 1 =  11 v 1 +  12 v 2 +  13 v 3 u 2 =  21 v 1 +  22 v 2 +  23 v 3 u 3 =  31 v 1 +  32 v 2 +  33 v 3 v [u 1 u 2 u 3 ] = [v 1 v 2 v 3 ]M T

43. Simple example 43

44. Matrix Form The coefficients define a 3 x 3 matrix and the representations can be related by a=M T b M =

45. Change of Frames  We can apply a similar process in homogeneous coordinates to the representations of both points and vectors  Any point or vector can be represented in either frame  We can represent Q 0, u 1, u 2, u 3 in terms of P 0, v 1, v 2, v 3 Consider two frames: (P 0, v 1, v 2, v 3 ) (Q 0, u 1, u 2, u 3 ) P0P0 v1v1 v2v2 v3v3 Q0Q0 u1u1 u2u2 u3u3

46. Representing 2 nd Frame in the 1 st u 1 =  11 v 1 +  12 v 2 +  13 v 3 u 2 =  21 v 1 +  22 v 2 +  23 v 3 u 3 =  31 v 1 +  32 v 2 +  33 v 3 Q 0 =  41 v 1 +  42 v 2 +  43 v 3 +  44 P 0 Extending what we did with change of bases defining a 4 x 4 matrix M =

47. Working with Representations  Within the two frames any point or vector has a representation of the same form a=[  1  2  3  4 ] T in the first frame b=[  1  2  3  4 ] T in the second frame where  4   4  for points and  4   4  for vectors and The matrix M is 4 x 4 and specifies an affine transformation in homogeneous coordinates a=M T b

48. Representing geometry elements  Points  Location in space  Operations allowed between points and vectors  Point-point subtraction yields a vector  Equivalent to point-vector addition P=v+QP=v+Q v=P-Qv=P-Q 48

49. Lines  Consider all points of the form  P(  )=P 0 +  d  Set of all points that pass through P 0 in the direction of the vector d 49

50. Rays and Line Segments  If  >= 0, then P(  ) is the ray leaving P 0 in the direction d  If we use two points to define v, then P(  ) = Q+  v = Q +  (R-Q) =  R + (1-  )Q  For 0<=  <=1 we get all the points on the line segment joining R and Q 50

51. Planes  A plane can be defined by a point and two vectors or by three points P( ,  )=R+  u+  v P( ,  )=R+  (Q-R)+  (P-R) u v R P R Q 51

52. Barycentric Coordinates 52  We can reorder the terms of the equation: with:  This coordinate system is called barycentric coordinates  For any point P inside the triangle (a, b, c):  Point on an edge: one coefficient is 0  Vertex: two coefficients are 0, remaining one is 1

53. Outline 53  OpenGL Elementary Rendering  Representation  Appendix: Vector Operations *

54. Vector Operations 54  Addition and multiplication by a scalar:  a + b  s*a  Examples:

55. Vector Addition 55  Head to tail: At the end of a, place the start of b.  Parallelogram rule:  Components: [a 1 + b 1, a 2 + b 2 ]

56. Vector Subtraction 56  To draw a - b, go forward along a and then backward along b.  Components: [a 1 - b 1, a 2 - b 2 ]

57. The Dot Product 57  Also called the inner product of two vectors  Produces a number, not a vector  In 2D:  In 3D:  Symmetry:  Linearity:  Homogeneity:

58. Length of a Vector 58  The length (or norm) of a vector a is:  The dot product of a vector with itself yields the length squared:

59. Unit Vectors and Normalization 59  A unit vector is a vector whose length equals 1  Example: (1, 0, 0), (0, 1, 0), (0, 0, 1)  Divide any nonzero vector v by its length |v|. The result is a unit vector in the same direction as v.  This is called normalizing a vector:  Any vector can be written as its magnitude times its direction:

60. The Angle Between Vectors 60  The angle between vectors can be computed by:  Using normalized vectors:

61. The Sign of ab 61  Imagine the cosine function:  Two vectors a and b are perpendicular, orthogonal, or normal if ab= 0

62. The end 62  Questions and answers