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Chapter 4

Chapter 4. 15 March 2006. EventPro Strategies is looking for a part-time programmer (any language) who knows SQL. For more info, contact Ryan Taylor, ryan@eventprostrategies.com. Agenda. Chapter 4 – Math for Computer Graphics GLUT solids. Transformations.

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Chapter 4

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  1. Chapter 4 15 March 2006 EventPro Strategies is looking for a part-time programmer (any language) who knows SQL. For more info, contact Ryan Taylor, ryan@eventprostrategies.com

  2. Agenda • Chapter 4 – Math for Computer Graphics • GLUT solids

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

  4. Order of transformations 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(); • transformed vertex is NMLv

  5. 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

  6. 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.

  7. 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.

  8. 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

  9. VectorsVector and Vector Algebra 1 2 3 2 + 3 = 5 3 4 7

  10. Matrices • Rectangular array of numbers • Addition • QuickMath

  11. 1 3 1 Matrices • A vector in 3 space is a n x 1 matrix or column vector.

  12. Matrices • Multiplication 1 0 0 0 0 1 0 0 x 0 0 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

  13. Matrix multiplication • A is an n x m matrix with entries aij • B is an m x p matrix with entries bij • AB is an n x p matrix with entries cij m • cij = ais bsj s=1

  14. Matrix multiplication m • cij = ais bsj s=1 c11c12 c13 c14 c21c22 c23 c24 c31c32 c33 c34 c41c42 c43 c44 1 0 0 0 0 1 0 0 x 0 0 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 a b

  15. 2D Transformations • Translation: Pf = T + P xf = xo + dx yf = yo + dy • Rotation: Pf = R · P xf = xo * cos - yo *sin yf = xo * sin + yo *cos • Scale: Pf = S · P xf = sx * xo yf = sy * yo

  16. 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

  17. Homogeneous Coordinates 1 0 dx x 0 1 dy · y 0 0 1 1

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

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

  20. 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

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

  22. 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).

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

  24. 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.

  25. 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.

  26. glutSolidDodecahedron and glutWireDodecahedron • void glutSolidDodecahedron(void);

  27. glutSolidOctahedron and glutWireOctahedron . • void glutSolidOctahedron(void);

  28. glutSolidTetrahedron and glutWireTetrahedron • void glutSolidTetrahedron(void);

  29. glutSolidIcosahedron and glutWireIcosahedron • void glutSolidIcosahedron(void);

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

  31. Homework next week. • Study for Test on Chapters 1-4, 02/15/05

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