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The Math Lecture

The Math Lecture. ( Part I - Matrices). Introduction. For 2D games, we use a lot of trigonometry For 3D games, we use a lot of linear algebra Most of the time, we don’t have to use calculus A matrix can: Translate (move) a vertex Rotate a vertex Scale a vertex

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The Math Lecture

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  1. The Math Lecture (Part I - Matrices)

  2. Introduction • For 2D games, we use a lot of trigonometry • For 3D games, we use a lot of linear algebra • Most of the time, we don’t have to use calculus • A matrix can: • Translate (move) a vertex • Rotate a vertex • Scale a vertex • Math libraries cover up these details • Learn it anyway!

  3. Introduction • For 2D games, we use a lot of trigonometry • For 3D games, we use a lot of linear algebra • Most of the time, we don’t have to use calculus • A matrix can: • Translate (move) a vertex • Rotate a vertex • Scale a vertex • Math libraries cover up these details • Learn it anyway!

  4. What is the matrix? • Control • Position, scaling, and rotationof all your 3D objects in OpenGL

  5. What is a matrix? • m x n array of scalars M =

  6. A Frame in Matrix Form • Where P0 is the origin of the frame M: v2 v1 P0 v3

  7. Matrix Operations • Scalar multiplication, addition, matrix-matrix multiplication… • In this class, we really only care about: • matrix-matrix multiplications with square (m x m) matrices • matrix-vector or matrix-point multiplication

  8. Matrix-Point (or vector) Multiplication In GLM you can use the overloaded operator * for this

  9. Matrix-Matrix Multiplication Does A*B = B*A? NO What does the identity do? ? ? In GLM you can use the overloaded operator * for this

  10. Examples ? ?

  11. Activity: 4x4 Matrix Multiplication • (5 min) In CG, we mostly use 4x4 matrices • Multiply the following: • Use GLM!

  12. Properties of Matrices • Associative • A(BC) = (AB)C • NOT commutative (usually) • AB = BA • However, • IA = AI = A In GLM the * operator adheres to these rules

  13. Matrix Algebra • Assume AB=C • Assume A and C are known. • B is unknown • What is B? • Lets do some algebra!! • A-1AB=A-1C • A-1A = Identity • Then B=A-1C

  14. MAtrices • The foundation of all geometric operations (translation, rotation, scaling, skewing…) • Have multiple rows and columns (usually 3x3 or 4x4) • Below is an identity matrix • We can multiply a matrix • with another matrix, and get a matrix • with a vector of “appropriate” dimension (later)

  15. Transposing a Matrix • Have a transpose (denoted ) where • In other words, make the rows columns! • glm::transpose Transpose

  16. Transposing a Matrix • Have a transpose (denoted ) where • In other words, make the rows columns! • Mirror image along the diagonal Transpose The Matrix has you…

  17. Matrices • There’s also an inverse matrix (denoted ) where • Not all matrices are invertible • Can check by getting the determinant of the matrix (looking for non-zero) • glm::inverse

  18. Matrices • There’s also an inverse matrix (denoted ) where • Not all matrices are invertible • Can check by getting the determinant of the matrix (looking for non-zero)

  19. Matrices • There’s also an inverse matrix (denoted ) where • Not all matrices are invertible • Can check by getting the determinant of the matrix (looking for non-zero)

  20. Matrices • There’s also an inverse matrix (denoted ) where • Not all matrices are invertible • Can check by getting the determinant of the matrix (looking for non-zero)

  21. Matrices • There’s also an inverse matrix (denoted ) where • Not all matrices are invertible • Can check by getting the determinant of the matrix (looking for non-zero) Follow the white rabbit…

  22. A few Special Matrices(Used for Rotation)

  23. Matrix Multiplication • Multiplying a point by a matrix gives us a new point! • Let’s say that we want to rotate the point around the z axis by 90 degrees • Points are always stored with a 1 for the 4th element = Oldpoint Newpoint

  24. Axis-Angle Rotations in GLM glm::rotate( angle, x, y, z); Where angle is the angle and (x,y,z) is a vector you want to rotate around (normalize it first!) Example: Rotate 30 deg around the vector (1,1,1) glm::vec3 v = normalize(vec3(1,1,1)); glm::rotate(30, v.x, v.y, v.z)

  25. Euler Angles More intuitive: represent rotations by 3 angles, one for each axis glm::rotate(anglex,1,0,0) * glm::rotate(angley,0,1,0) * glm::rotate(anglez,0,0,1); Think: if we have a torus unstranformed at the origin, what will the torus look like if you have anglex=90, angley=90, and anglez=90 Transformed

  26. Gimbal Lock Rotate around these Initial orientation (x=blue, y=green, z = red) How can we gain altitude here?

  27. Quaternions • Quaternions represent 3D rotations in 4D using imaginary numbers – a 4-tuple Q = (w,x,y,z) • IF YOU EVER RUN INTO GIMBAL LOCK: • 1) Convert to Quaternion from euler, matrix, or angle-axis • 2) Do rotations in quaternion form • 3) Convert to angle-axis or matrix • Quaternions are easy to use in GLM • http://glm.g-truc.net/api-0.9.0/a00184.html

  28. Matrix Multiplication • Multiplying a point by a matrix gives us a new point! • Let’s say that we want to rotate the point around the z axis by 90 degrees • Points are always stored with a 1 for the 4th element = -1 0

  29. Matrix Multiplication • Multiplying a point by a matrix gives us a new point! • Let’s say that we want to rotate the point around the z axis by 90 degrees • Points are always stored with a 1 for the 4th element =

  30. Matrix Multiplication • Multiplying a point by a matrix gives us a new point! • Let’s say that we want to rotate the point around the z axis by 90 degrees • Points are always stored with a 1 for the 4th element =

  31. Matrix Multiplication • Multiplying a point by a matrix gives us a new point! • Let’s say that we want to rotate the point around the z axis by 90 degrees • Points are always stored with a 1 for the 4th element = What IS real?

  32. Translation • It’s a piece of cake, because the 4th column is the translation!

  33. Translation • It’s a piece of cake, because the 4th column is the translation! • Imagine we want to move the point by (2, 1, -3). Then: =

  34. Translation • It’s a piece of cake, because the 4th column is the translation! • Imagine we want to move the point by (2, 1, -3). Then: =

  35. Translation • It’s a piece of cake, because the 4th column is the translation! • Imagine we want to move the point by (2, 1, -3). Then: =

  36. Translation • It’s a piece of cake, because the 4th column is the translation! • Imagine we want to move the point by (2, 1, -3). Then: =

  37. Translation • It’s a piece of cake, because the 4th column is the translation! • Imagine we want to move the point by (2, 1, -3). Then: = Why, oh why, didn’t I take the blue pill?

  38. Translation • It’s a piece of cake, because the 4th column is the translation! • Imagine we want to move the point by (2, 1, -3). Then: =

  39. SCALING • It’s a piece of cake too, because it’s the diagonal! • We can scale along just one axis, or more than one! • Imagine we want to scale the point by the values x, y and z. Then: = Trace program: running

  40. Matrix Multiplication • What if you want to rotate a point and then translate it? • Need a rotation matrix • Need a translation matrix • Returns a 4x4 matrix

  41. Matrix Multiplication • What if you want to rotate a point and then translate it? • Need a rotation matrix • Need a translation matrix • Returns a 4x4 matrix

  42. Matrix Multiplication • What if you want to rotate a point and then translate it? • Need a rotation matrix • Need a translation matrix • Returns a 4x4 matrix

  43. Matrix Multiplication • What if you want to rotate a point and then translate it? • Need a rotation matrix • Need a translation matrix • Returns a 4x4 matrix

  44. Matrix Multiplication • What if you want to rotate a point and then translate it? • Need a rotation matrix • Need a translation matrix • Returns a 4x4 matrix

  45. Matrix Multiplication • What if you want to rotate a point and then translate it? • Need a rotation matrix • Need a translation matrix • Returns a 4x4 matrix

  46. Matrix Multiplication • What if you want to rotate a point and then translate it? • Need a rotation matrix • Need a translation matrix • Returns a 4x4 matrix

  47. Matrix Multiplication • What if you want to rotate a point and then translate it? • Need a rotation matrix • Need a translation matrix • Returns a 4x4 matrix

  48. Matrix Multiplication • What if you want to rotate a point and then translate it? • Need a rotation matrix • Need a translation matrix • Returns a 4x4 matrix There is no spoon…

  49. Matrix Multiplication • What if you want to rotate a point and then translate it? • Need a rotation matrix • Need a translation matrix • Returns a 4x4 matrix ?

  50. Matrix Multiplication • What if you want to rotate a point and then translate it? • Need a rotation matrix • Need a translation matrix • Returns a 4x4 matrix =

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