A dynamic Complex Transformation generating FRACTALS

A dynamic Complex Transformation

generating FRACTALS

• 北京景山学校纪光老师April 2010

1Fractals & Complex Numbers

Generation of Julia’s “rabbit”


Fractals & Complex Numbers 2

Generation of the set of Mendelbrot


Fractals & Complex Numbers 3

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)

£

In polar coordinates z = r (cos j + i sin j) or r. e i j

• r is the module of z : r = |z| = • j is the argument :

arg(z) = j

OMu ruuu

≡ e

r; OM

u ruuu( ) 2π[ ]

OMu ruuu

= a2 +b2

Review 1.a :

Complex Numbers set



The omplex numberz = a + i b

is 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.b :

Complex Numbers set



In polar coordinates

z = r (cos j + i sin j)

or z = r. e i j

£

• r is the module of z :

r = |z| = OMu ruuu

= a2 +b2

• j is the argument :

arg(z) = j ≡ e

r; OM

u ruuu( ) 2π[ ]

Review 2Operations in



(1) Addition : if z = a + i b and z’ = a’ + i b’then z + z’ = (a + a’) + i (b + b’)

(2) Multiplication :if z = r. e i j and z’ = r’. e i j’

then z.z’ = r.r’.e i (j+j’)

£

Review 2.aOperations in



Construction of the Sum z = a + i b

z’ = a’ + i b’=================

z + z’ = (a + a’) + i (b + b’)

£

The image of the sumis the sum of the

vectors associated with the vectors representing

z and z’

Review 2.bOperations in



Construction of the product z = r. e i j

z’ = r’. e i j’

================= z.z’ = r. r’. e i (j + j’)

£

The module of the product is the product of the modules

The argument of the product is the Sum of the

arguments

Transformation in



Construction of the square z = r. e i j

z2 = r2. e i 2j

£

The module of the square is the square of the module.

The argument of the square is the double of the

argument.

z a z2

Transformation (1.1) in



Construction of z2

z = r. e i j

z2 = r2. e i 2j

£

1st method :

1. Square the module OM in OM1

2. Rotate the point M1 in M’

z a z2




Construction of z2

z = r. e i j

z2 = r2. e i 2j

£

2nd method :

1. Rotate the point M in M2

2. Square the module of OM2 in OM’

z a z2




£ z a z2

(Demo / Cabri / Fig.2)




Construction of z2 + c z = r. e i j

z2 + c = r2. e i 2j + cc is a complex constantrepresented by the point C

£

1st Method :

1. Square the module of OM in OM1

2. Rotate the point M1(z1) in M’

3. Add the vector

z a z2 +c

OCu ruu

to O ′Mu ruuu




Construction of z2 + c z = r. e i j

z2 + c = r2. e i 2j + cc is a complex constantrepresented by the point C

£

2nd Method :

1. Rotate the point M(z) in M1

2. Square the module of OM1 in OM’

3. Add the vector

z a z2 +c

OCu ruu

to O ′Mu ruuu




£ z a z2

(Demo / Cabri / Fig.3)

Construction of “Julia’s rabbit” in

by iterating the transformation



£

z a z2 + c1. Choose a point C of affix c in the Complex plane.

2. Choose a point M0(z0) in the Complex plane.

3. Build the image M1(z1) of M0(z0) by the above transformation in the coordinates plane.

4. Build the image M2(z2) of M1(z1) by the above transformation in the coordinates plane.

Construction of “Julia’s rabbit” in




£

z a z2 + c5. 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 …

6. If the points get off the screen, we mark them in blue.

This set of points is called the orbit ( 轨道 ) of M0(z0)

6. if they stay inside the Unit circle we mark them in red

M0(z0) , M1(z1) , M2(z2) , M3(z3) ,…, M10(z10) ,…



£

Construction of Mendelbrot in




£

z a z2 + c1. Choose a point C of affix c in the Complex plane.

2. Start from M0(z0) = O in the Complex plane.

3. Build the image M1(z1 = c) of M0(z0) by the above transformation in the coordinates plane.

4. Build the image M2(z2 = c2 + c) of M1(z1= c) by the transformation in the coordinates plane.

Construction of Mendelbrot inby iterating the transformation



£

z a z2 + c5. 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 …

6. If the points get off the screen, we mark C in red.

This set of points is called the orbit ( 轨道 ) of C

6. if they stay inside the Unit circle we mark C in black.

O, M1(z1= c) , M2(z2= c2 + c) , M3(z3) ,…, M10(z10) ,…



£

Documents

A dynamic Complex Transformation generating FRACTALS