1. Chapter 4 10 February 2004

2. Agenda Program 2 – Due 2/17 Chapter 4 – transformations GLUT solids

3. OpenGL Buffers Color –can be divided into front and back for double buffering Alpha Depth Stencil Accumulation

4. Double Buffering

5. Animating Using Double Buffering Request a double buffered color buffer glutInitDisplayMode (GLUT_RGB | GLUT_DOUBLE); Clear color buffer –glClear(GL_COLOR_BUFFER_BIT); Render Scene Request swap of front and back buffers –glutSwapBuffers(); Repeat steps 2-4 for animation.

6. Depth Buffering

7. 3D Coords --> Raster coords Transformations Clipping Viewport transformation.

8. Transformations Prior to rendering: view, locate and orient –eye / camera position –3D geometry Manage the matrices –including the matrix stack Combine (composite) transformations

9. Camera Analogy

10. Stages of Vertex Transformation

11. Transformations 45-degree counterclockwise rotation about the origin around the z-axis a translation down the x-axis

12. Order of transformations transformed vertex is NMLv glMatrixMode(GL_MODELVIEW); glLoadIdentity(); glMultMatrixf(N); /* apply transformation N */ glMultMatrixf(M); /* apply transformation M */ glMultMatrixf(L); /* apply transformation L */ glBegin(GL_POINTS); glVertex3f(v); /* draw transformed vertex v */ glEnd();

13. Translation void glTranslate{fd} (TYPE x, TYPE y, TYPE z); Multiplies the current matrix by a matrix that moves (translates) an object by the given x, y, and z values

14. Rotation void glRotate{fd}(TYPE angle, TYPE x, TYPE y, TYPE z); Multiplies the current matrix by a matrix that rotates an object in a counterclockwise direction about the ray from the origin through the point (x, y, z). The angle parameter specifies the angle of rotation in degrees.

15. Scale void glScale{fd} (TYPEx, TYPE y, TYPEz); Multiplies the current matrix by a matrix that stretches, shrinks, or reflects an object along the axes.

16. Vectors N tuple of real numbers (n = 2 for 2D, n = 3 for 3D) Directed line segment Example –Velocity vector (speed and direction) Operations –Addition –Multiplication by a scalar –Dot product

17. Vectors 1 2 3 2 + 3 = 5 3 4 7

18. Matrices Rectangular array of numbers A vector in 3 space is a n x 1 matrix or columnvector. Multiplication 1 0 0 0 0 1 0 0 x 0 0 0 0 1/k 1 Cos α 0 sin α 0 0 1 0 m -sin α 0 cos α n 0 0 0 1 131131

19. Matrix Multiplication A is an n x m matrix with entries a ij B is an m x p matrix with entries b ij AB is an n x p matrix with entries c ij m c ij =  a is b sj s=1

20. 2D Transformations Translation: Pf = T + P x f = x o + dx y f = y o + dy Rotation: Pf = R · P x f = x o * cos  - y o *sin  y f = x o * sin  + y o *cos  Scale: Pf = S · P x f = sx * x o y f = sy * y o

21. Homogeneous Coordinates Want to treat all transforms in a consistent way so they can be combined easily Developed in geometry (‘46 in cambridge) and applied to graphics Add a third coordinate to a point (x, y, w) (x1, y1, w1) is the same point as (x2, y2, w2) if one is a multiple of another Homogenize a point by dividing by w

22. Homogeneous Coordinates 1 0 dxx 0 1 dy ·y 0 0 11

23. Homogeneous Coordinates sx 0 0 x 0 sy 0 · y 0 0 1 1

24. Homogeneous Coordinates Cos  -sin  0 x sin  cos  0 · y 0 0 1 1

25. Combining 2D Transformations Rotate a house about the origin Rotate the house about one of its corners –translate so that a corner of the house is at the origin –rotate the house about the origin –translate so that the corner returns to its original position

26. GLUT Solids Sphere Cube Cone Torus Dodecahedron Octahedron Tetrahedron Icosahedron Teapot

27. glutSolidSphere and glutWireSphere void glutSolidSphere(GLdouble radius, GLint slices, GLint stacks); radius - The radius of the sphere. slices - The number of subdivisions around the Z axis (similar to lines of longitude). stacks - The number of subdivisions along the Z axis (similar to lines of latitude).

28. glutSolidCube and glutWireCube void glutSolidCube(GLdouble size); size – length of sides

29. glutSolidCone and glutWireCone void glutSolidCone(GLdouble base, GLdouble height, GLint slices, GLint stacks); base - The radius of the base of the cone. height - The height of the cone. slices - The number of subdivisions around the Z axis. stacks - The number of subdivisions along the Z axis.

30. glutSolidTorus and glutWireTorus void glutSolidTorus(GLdouble innerRadius,GLdouble outerRadius, GLint nsides, GLint rings); innerRadius - Inner radius of the torus. outerRadius - Outer radius of the torus. nsides - Number of sides for each radial section. rings - Number of radial divisions for the torus.

31. glutSolidDodecahedron and glutWireDodecahedron void glutSolidDodecahedron(void);

32. glutSolidOctahedron and glutWireOctahedron. void glutSolidOctahedron(void);

33. glutSolidTetrahedron and glutWireTetrahedron void glutSolidTetrahedron(void);

34. glutSolidIcosahedron and glutWireIcosahedron void glutSolidIcosahedron(void);

35. glutSolidTeapot and glutWireTeapot void glutSolidTeapot(GLdouble size); size - Relative size of the teapot.

36. Homework for next week. Program 2 due 2/17 Study for Test on Chapters 1-4, 2/19