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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 tim e, 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 usecalculus • 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 usecalculus • A matrix can: • Translate (move) a vertex • Rotate a vertex • Scale a vertex • Math libraries cover up these details • Learn it anyway!

  4. 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)

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

  6. 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…

  7. 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)

  8. 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)

  9. 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)

  10. 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)

  11. 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…

  12. A few Special Matrices(Used for Rotation)

  13. 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

  14. 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

  15. 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 =

  16. 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 =

  17. 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?

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

  19. 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: =

  20. 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: =

  21. 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: =

  22. 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: =

  23. 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?

  24. 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: =

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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…

  35. 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 ?

  36. 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 =

  37. 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 = Oldpoint Newpoint

  38. JEFF WAS HERE

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