View
35
Download
0
Category
Preview:
DESCRIPTION
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). - PowerPoint PPT Presentation
Citation preview
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”
• 北京景山学校 纪光老师April 2010
Fractals & Complex Numbers 2
Generation of the set of Mendelbrot
• 北京景山学校 纪光老师April 2010
Fractals & Complex Numbers 3
Review 1 :Complex Numbers set
• 北京景山学校 纪光老师April 2010
4Fractals & Complex Numbers
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
• 北京景山学校 纪光老师April 2010
5Fractals & Complex Numbers
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
• 北京景山学校 纪光老师April 2010
6Fractals & Complex Numbers
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
• 北京景山学校 纪光老师April 2010
7Fractals & Complex Numbers
(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
• 北京景山学校 纪光老师April 2010
8Fractals & Complex Numbers
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
• 北京景山学校 纪光老师April 2010
9Fractals & Complex Numbers
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
• 北京景山学校 纪光老师April 2010
10Fractals & Complex Numbers
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
• 北京景山学校 纪光老师April 2010
11Fractals & Complex Numbers
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
Transformation (1.2) in
• 北京景山学校 纪光老师April 2010
12Fractals & Complex Numbers
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
Transformation (1.3) in
• 北京景山学校 纪光老师April 2010
13Fractals & Complex Numbers
£ z a z2
(Demo / Cabri / Fig.2)
Transformation (2.1) in
• 北京景山学校 纪光老师April 2010
14Fractals & Complex Numbers
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
Transformation (2.2) in
• 北京景山学校 纪光老师April 2010
15Fractals & Complex Numbers
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
Transformation (2.3) in
• 北京景山学校 纪光老师April 2010
16Fractals & Complex Numbers
£ z a z2
(Demo / Cabri / Fig.3)
Construction of “Julia’s rabbit” in
by iterating the transformation
• 北京景山学校 纪光老师April 2010
17Fractals & Complex Numbers
£
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
by iterating the transformation
• 北京景山学校 纪光老师April 2010
18Fractals & Complex Numbers
£
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) ,…
• 北京景山学校 纪光老师April 2010
19Fractals & Complex Numbers
£
Construction of Mendelbrot in
by iterating the transformation
• 北京景山学校 纪光老师April 2010
20Fractals & Complex Numbers
£
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
• 北京景山学校 纪光老师April 2010
21Fractals & Complex Numbers
£
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) ,…
• 北京景山学校 纪光老师April 2010
22Fractals & Complex Numbers
£
Recommended