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02mathematics 1


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02mathematics 1

  1. 1. Mathematics for Computer Graphics 고려대학교 컴퓨터 그래픽스 연구실
  2. 2. Contents <ul><li>Complex Numbers </li></ul><ul><ul><li>Pure imaginary number </li></ul></ul><ul><ul><li>Complex conjugate </li></ul></ul><ul><ul><li>Representation with polar coordinates </li></ul></ul><ul><li>Quaternions </li></ul><ul><ul><li>Definition </li></ul></ul><ul><ul><li>Addition, scalar multiplication, multiplication, division, and inverse </li></ul></ul><ul><ul><li>3D Rotations </li></ul></ul><ul><li>Nonparametric Representations </li></ul><ul><li>Parametric Representations </li></ul>
  3. 3. Complex Numbers <ul><li>Real Part + Imaginary Part: </li></ul><ul><li>Addition and Subtraction </li></ul><ul><li>Scalar Multiplication </li></ul><ul><li>Multiplication </li></ul>y x z imaginary axis real axis
  4. 4. Pure Imaginary Number & Complex Conjugate <ul><li>Imaginary Unit: </li></ul><ul><li>Complex Conjugate </li></ul><ul><ul><li>Modulus or absolute value </li></ul></ul><ul><li>Division </li></ul>
  5. 5. Representation with Polar Coordinates <ul><li>Euler’s Formula </li></ul><ul><li>Complex Multiplication and Division </li></ul><ul><li>n th Roots </li></ul>r θ z =( x , y ) imaginary axis real axis
  6. 6. Quaternions <ul><li>One Real Part + Three Imaginary Part </li></ul><ul><li>Properties: </li></ul><ul><li>Addition and Scalar Multiplication </li></ul>
  7. 7. Ordered-Pair Notation <ul><li>Scalar ‘ s ’ + Vector “ v = ( a , b , c ) ” </li></ul><ul><li>Addition: </li></ul><ul><li>Multiplication </li></ul><ul><li>Magnitude </li></ul><ul><li>Inverse </li></ul>
  8. 8. 3D Rotation <ul><li>For a 3D Point ( α , β , γ ) </li></ul><ul><ul><li>A unit quaternion its conjugate </li></ul></ul><ul><ul><li> Rotating ( α , β , γ ) by angle 2 θ about the axis parallel to ( a , b , c ) </li></ul></ul><ul><li>For </li></ul><ul><li>R q is a 3D Rotation about ( a , b , c ) by 2 θ </li></ul>
  9. 9. Nonparametric Representations <ul><li>Definition </li></ul><ul><ul><li>Object descriptions directly in terms of the coordinates of the reference frame </li></ul></ul><ul><ul><li>Ex. implicit expression: </li></ul></ul><ul><ul><li>Ex. explicit Expression: </li></ul></ul><ul><li>Advantages </li></ul><ul><ul><li>Useful in describing objects </li></ul></ul><ul><li>Disadvantages </li></ul><ul><ul><li>Change the independent variable as derivatives </li></ul></ul><ul><li>Implicit vs. Explicit Equations </li></ul>
  10. 10. Parametric Representations <ul><li>3D Curves with Parameter u </li></ul><ul><ul><li>Ex. </li></ul></ul><ul><li>3D Surfaces with Parameter u , v </li></ul><ul><ul><li>Ex. </li></ul></ul>