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A dynamic Complex Transformation generating FRACTALS

A dynamic Complex Transformation generating FRACTALS. Generation of Julia’s “rabbit”. Generation of the set of Mendelbrot. Review 1 : Complex Numbers set. The complex number z = a + i b is represented in the coordinates plane by a point M(a,b) or vector (a,b).

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A dynamic Complex Transformation generating FRACTALS

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  1. A dynamic Complex Transformation generating FRACTALS Fractals & Complex Numbers

  2. Generation of Julia’s “rabbit” Fractals & Complex Numbers

  3. Generation of the set of Mendelbrot Fractals & Complex Numbers

  4. Review 1 :Complex Numbers set The complex number z = a + i bis represented in the coordinates plane by a point M(a,b) or vector (a,b) • In polar coordinates z = r(cosj + i sin j) orr. e i j • r is the module of z : • r = |z| = • j is the argument : arg(z) = j Fractals & Complex Numbers

  5. The omplex numberz = a + i bis represented in the coordinates plane by the point M(a,b)where a and b are eal numbers and i an imaginary square root of (-1) Review 1.a :Complex Numbers set Fractals & Complex Numbers

  6. In polar coordinates z = r(cosj + i sin j) or z = r. e i j Review 1.b :Complex Numbers set • r is the module of z : • r = |z| = • j is the argument : arg(z) = j Fractals & Complex Numbers

  7. (1) Addition :if z = a + ib and z’ = a’ + ib’then z + z’ = (a + a’) + i (b + b’) Review 2Operations in (2) Multiplication :if z = r. e i j and z’ = r’. e i j’then z.z’ = r.r’.e i (j+j’) Fractals & Complex Numbers

  8. Construction of the Sumz = a + ib z’ = a’ + ib’ ================= z + z’ = (a + a’) + i (b + b’) Review 2.aOperations in The image of the sum is the sum of thevectors associated with the vectors representingz and z’ Fractals & Complex Numbers

  9. Construction of the productz = r. e i j z’ = r’. e i j’ ================= z.z’ = r. r’.e i (j + j’) Review 2.bOperations in The module of the product is the product of the modules The argument of the product is the Sum of the arguments Fractals & Complex Numbers

  10. Construction of the squarez = r. e i j z2 = r2. e i 2j Transformation in The module of the square is the square of the module. The argument of the square is the double of the argument. Fractals & Complex Numbers

  11. Construction of z2z = r. e i j z2 = r2. e i 2j Transformation (1.1) in 1st method : Square the module OM in OM1 Rotate the point M1 in M’ Fractals & Complex Numbers

  12. Construction of z2z = r. e i j z2 = r2. e i 2j Transformation (1.2) in 2nd method : Rotate the point M in M2 Square the module of OM2 in OM’ Fractals & Complex Numbers

  13. Transformation (1.3) in (Demo / Cabri / Fig.2) Fractals & Complex Numbers

  14. Construction of z2 + cz = r. e i j z2 + c = r2. e i 2j + c c is a complex constant represented by the point C Transformation (2.1) in 1st Method : Square the module of OM in OM1 Rotate the point M1(z1) in M’ Add the vector Fractals & Complex Numbers

  15. Construction of z2 + cz = r. e i j z2 + c = r2. e i 2j + c c is a complex constant represented by the point C Transformation (2.2) in 2nd Method : Rotate the point M(z) in M1 Square the module of OM1 in OM’ Add the vector Fractals & Complex Numbers

  16. Transformation (2.3) in (Demo / Cabri / Fig.3) Fractals & Complex Numbers

  17. Construction of “Julia’s rabbit” in by iterating the transformation Choose a point C of affix c in the Complex plane. Choose a point M0(z0) in the Complex plane. Build the image M1(z1) of M0(z0) by the above transformation in the coordinates plane. Build the image M2(z2) of M1(z1) by the above transformation in the coordinates plane. Fractals & Complex Numbers

  18. Construction of “Julia’s rabbit” in by iterating the transformation Continue to apply the transformation to each new point and mark them in the plane, until you get a sequence of 10 points or more … M0(z0) , M1(z1) , M2(z2) , M3(z3) ,…, M10(z10) ,… This set of points is called the orbit (轨道) of M0(z0) If the points get off the screen, we mark them in blue. if they stay inside the Unit circle we mark them in red Fractals & Complex Numbers

  19. Fractals & Complex Numbers

  20. Construction of Mendelbrot in by iterating the transformation Choose a point C of affix c in the Complex plane. Start from M0(z0) = O in the Complex plane. Build the image M1(z1 = c) of M0(z0) by the above transformation in the coordinates plane. Build the image M2(z2 = c2 + c) of M1(z1= c) by the transformation in the coordinates plane. Fractals & Complex Numbers

  21. Construction of Mendelbrot in by iterating the transformation Continue to apply the transformation to each new point and mark them in the plane, until you get a sequence of 10 points or more … O, M1(z1= c) , M2(z2= c2 + c) , M3(z3) ,…, M10(z10) ,… This set of points is called the orbit (轨道) of C If the points get off the screen, we mark C inred. if they stay inside the Unit circle we mark C in black. Fractals & Complex Numbers

  22. Fractals & Complex Numbers

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