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CSCE 552 Spring 2011

CSCE 552 Spring 2011. Math. By Jijun Tang. Layered. Initialization/Shutdown. Resource Acquisition Is Initialization A useful rule to minimalize mismatch errors in the initialization and shutdown steps

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CSCE 552 Spring 2011

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  1. CSCE 552 Spring 2011 Math By Jijun Tang

  2. Layered

  3. Initialization/Shutdown • Resource Acquisition Is Initialization • A useful rule to minimalize mismatch errors in the initialization and shutdown steps • Means that creating an object acquires and initializes all the necessary resources, and destroying it destroys and shuts down all those resources • Optimizations • Fast shutdown • Warm reboot

  4. Main Game Loop • Structure • Hard-coded loops • Multiple game loops: for each major game state • Consider steps as tasks to be iterated through • Coupling • Can decouple the rendering step from simulation and update steps • Results in higher frame rate, smoother animation, and greater responsiveness • Implementation is tricky and can be error-prone

  5. Math

  6. Applied Trigonometry • Trigonometric functions • Defined using right triangle h y a x

  7. Applied Trigonometry • Angles measured in radians • Full circle contains 2p radians

  8. Applied Trigonometry • Sine and cosine used to decompose a point into horizontal and vertical components y r r sin a a x r cos a

  9. Trigonometry

  10. Trigonometric identities

  11. Inverse trigonometric functions • Return angle for which sin, cos, or tan function produces a particular value • If sin a = z, then a = sin-1z • If cos a = z, then a = cos-1z • If tan a = z, then a = tan-1z

  12. arcs

  13. Applied Trigonometry • Law of sines • Law of cosines Reduces to Pythagorean theorem wheng = 90 degrees a c b b g a

  14. Vectors and Scalars • Scalars represent quantities that can be described fully using one value • Mass • Time • Distance • Vectors describe a magnitude and direction together using multiple values

  15. Examples of vectors • Difference between two points • Magnitude is the distance between the points • Direction points from one point to the other • Velocity of a projectile • Magnitude is the speed of the projectile • Direction is the direction in which it’s traveling • A force is applied along a direction

  16. Visualize Vectors • The length represents the magnitude • The arrowhead indicates the direction • Multiplying a vector by a scalar changes the arrow’s length 2V V –V

  17. Vectors Add and Subtraction • Two vectors V and W are added by placing the beginning of W at the end of V • Subtraction reverses the second vector V W V + W W –W V V – W V

  18. High-Dimension Vectors • An n-dimensional vector V is represented by n components • In three dimensions, the components are named x, y, and z • Individual components are expressed using the name as a subscript:

  19. Add/Subtract Componentwise

  20. Vector Magnitude • The magnitude of an n-dimensional vector V is given by • In three dimensions, this is

  21. Vector Normalization • A vector having a magnitude of 1 is called a unit vector • Any vector V can be resized to unit length by dividing it by its magnitude: • This process is called normalization

  22. Matrices • A matrix is a rectangular array of numbers arranged as rows and columns • A matrix having n rows and m columns is an nm matrix • At the right, M is a2  3 matrix • If n = m, the matrix is a square matrix

  23. Matrix Representation • The entry of a matrix M in the i-th row and j-th column is denoted Mij • For example,

  24. Matrix Transpose The transpose of a matrix M is denoted MT and has its rows and columns exchanged:

  25. Vectors and Matrices • An n-dimensional vector V can be thought of as an n 1 column matrix: • Or a 1 n row matrix:

  26. Vectors and Matrices • Product of two matrices A and B • Number of columns of A must equal number of rows of B • Entries of the product are given by • If A is a nm matrix, and B is an mp matrix, then AB is an np matrix

  27. Vectors and Matrices • Example matrix product

  28. Vectors and Matrices • Matrices are used to transform vectors from one coordinate system to another • In three dimensions, the product of a matrix and a column vector looks like:

  29. Identity Matrix In For any nn matrix M, the product with the identity matrix is M itself • InM = M • MIn = M

  30. Invertible • An nn matrix M is invertible if there exists another matrix G such that • The inverse of M is denoted M-1

  31. Properties of Inverse • Not every matrix has an inverse • A noninvertible matrix is called singular • Whether a matrix is invertible can be determined by calculating a scalar quantity called the determinant

  32. Determinant • The determinant of a square matrix M is denoted det M or |M| • A matrix is invertible if its determinant is not zero • For a 2  2 matrix,

  33. Determinant • The determinant of a 3  3 matrix is

  34. Inverse Explicit formulas exist for matrix inverses • These are good for small matrices, but other methods are generally used for larger matrices • In computer graphics, we are usually dealing with 2  2, 3  3, and a special form of 4  4 matrices

  35. Inverse of 2x2 and 3x3 • The inverse of a 2  2 matrix M is • The inverse of a 3  3 matrix M is

  36. 4x4 • A special type of 4  4 matrix used in computer graphics looks like • R is a 3  3 rotation matrix, and T is a translation vector

  37. Inverse of 4x4

  38. General Matrix Inverse • (AB) -1 = B-1A-1 • A general nxn matrix can be inverted using methods such as • Gauss-Jordan elimination • Gaussian elimination • LU decomposition

  39. Gauss-Jordan Elimination Gaussian Elimination

  40. The Dot Product • The dot product is a product between two vectors that produces a scalar • The dot product between twon-dimensional vectors V and W is given by • In three dimensions,

  41. Dot Product Usage • The dot product can be used to project one vector onto another V a W

  42. Dot Product Properties • The dot product satisfies the formula • a is the angle between the two vectors • Dot product is always 0 between perpendicular vectors • If V and W are unit vectors, the dot product is 1 for parallel vectors pointing in the same direction, -1 for opposite

  43. Self Dot Product • The dot product of a vector with itself produces the squared magnitude • Often, the notation V 2 is used as shorthand for VV

  44. The Cross Product • The cross product is a product between two vectors the produces a vector • The cross product only applies in three dimensions • The cross product is perpendicular to both vectors being multiplied together • The cross product between two parallel vectors is the zero vector (0, 0, 0)

  45. Illustration • The cross product satisfies the trigonometric relationship • This is the area ofthe parallelogramformed byV and W V ||V|| sin a a W

  46. Area and Cross Product • The area A of a triangle with vertices P1, P2, and P3 is thus given by

  47. Rules • Cross products obey the right hand rule • If first vector points along right thumb, and second vector points along right fingers, • Then cross product points out of right palm • Reversing order of vectors negates the cross product: • Cross product is anticommutative

  48. Rules in Figure

  49. Formular • The cross product between V and W is • A helpful tool for remembering this formula is the pseudodeterminant (x, y, z) are unit vectors

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