1 Dr. Scott Schaefer Fractals and Iterated Affine Transformations.

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Presentation transcript:

1 Dr. Scott Schaefer Fractals and Iterated Affine Transformations

2/193 What are Fractals?

3/193 What are Fractals?

4/193 What are Fractals?

5/193 What are Fractals?

6/193 What are Fractals? Recursion made visible A self-similar shape created from a set of contractive transformations

7/193 Contractive Transformations A transformation F(X) is contractive if, for all compact sets Transformations on sets? Distance between sets?

8/193 Transformations on Sets Given a transformation F(x), In other words, F(X) means apply the transformation to each point in the set X

9/193 Hausdorff Distance

10/193 Hausdorff Distance

11/193 Hausdorff Distance

12/193 Hausdorff Distance

13/193 Hausdorff Distance

14/193 Hausdorff Distance

15/193 Hausdorff Distance

16/193 Hausdorff Distance

17/193 Hausdorff Distance

18/193 Hausdorff Distance

19/193 Contractive Transformations A transformation F(X) is contractive if, for all compact sets

20/193 Iterated Affine Transformations Special class of fractals where each transformation is an affine transformation

21/193 Rendering Fractals Given starting set X 0 Attractor is

22/193 Rendering Fractals Given starting set X 0 Attractor is

23/193 Rendering Fractals Given starting set X 0 Attractor is

24/193 Rendering Fractals Given starting set X 0 Attractor is

25/193 Rendering Fractals Given starting set X 0 Attractor is

26/193 Rendering Fractals

27/193 Rendering Fractals

28/193 Rendering Fractals

29/193 Rendering Fractals

30/193 Rendering Fractals

31/193 Rendering Fractals

32/193 Rendering Fractals

33/193 Rendering Fractals

34/193 Rendering Fractals

35/193 Rendering Fractals

36/193 Rendering Fractals

37/193 Rendering Fractals

38/193 Rendering Fractals

39/193 Rendering Fractals

40/193 Rendering Fractals

41/193 Rendering Fractals

42/193 Rendering Fractals

43/193 Rendering Fractals

44/193 Rendering Fractals

45/193 Rendering Fractals

46/193 Rendering Fractals

47/193 Rendering Fractals

48/193 Rendering Fractals

49/193 Rendering Fractals

50/193 Contractive Transformations A transformation F(X) is contractive if, for all compact sets A set of transformations has a unique attractor if all transformations are contractive  That attractor is independent of the starting shape!!!

51/193 Rendering Fractals

52/193 Rendering Fractals

53/193 Rendering Fractals

54/193 Rendering Fractals

55/193 Rendering Fractals

56/193 Rendering Fractals

57/193 Rendering Fractals

58/193 Rendering Fractals

59/193 Rendering Fractals

60/193 Rendering Fractals

61/193 Rendering Fractals

62/193 Rendering Fractals

63/193 Rendering Fractals

64/193 Rendering Fractals

65/193 Finding Transformations

66/193 Finding Transformations

67/193 Finding Transformations

68/193 Finding Transformations

69/193 Finding Transformations

70/193 Fractal Tennis Start with any point x For ( i=1; i<100; i++ ) x = M random *x For ( i=1; i<100000; i++ ) draw(x) x = M random *x

71/193 Fractal Tennis Start with any point x For ( i=1; i<100; i++ ) x = M random *x For ( i=1; i<100000; i++ ) draw(x) x = M random *x Gets a point on the fractal

72/193 Fractal Tennis Start with any point x For ( i=1; i<100; i++ ) x = M random *x For ( i=1; i<100000; i++ ) draw(x) x = M random *x Creates new points on the fractal

73/193 Fractal Tennis – Example 25,000 Points

74/193 Fractal Tennis – Example 50,000 Points

75/193 Fractal Tennis – Example 75,000 Points

76/193 Fractal Tennis – Example 100,000 Points

77/193 Fractal Tennis – Example 125,000 Points

78/193 Fractal Tennis – Example 150,000 Points

79/193 Fractal Tennis – Example 175,000 Points

80/193 Fractal Tennis – Example 200,000 Points

81/193 Modified Fractal Tennis

82/193 Modified Fractal Tennis 25,000 Points

83/193 Modified Fractal Tennis 50,000 Points

84/193 Modified Fractal Tennis 75,000 Points

85/193 Modified Fractal Tennis 100,000 Points

86/193 Modified Fractal Tennis 125,000 Points

87/193 Modified Fractal Tennis 150,000 Points

88/193 Modified Fractal Tennis 175,000 Points

89/193 Modified Fractal Tennis 200,000 Points

90/193 Modified Fractal Tennis Weight probabilities based on area 200,000 Points

91/193 Modified Fractal Tennis Weight probabilities based on area 25,000 Points

92/193 Modified Fractal Tennis Weight probabilities based on area 50,000 Points

93/193 Modified Fractal Tennis Weight probabilities based on area 75,000 Points

94/193 Modified Fractal Tennis Weight probabilities based on area 100,000 Points

95/193 Modified Fractal Tennis Weight probabilities based on area 125,000 Points

96/193 Modified Fractal Tennis Weight probabilities based on area 150,000 Points

97/193 Modified Fractal Tennis Weight probabilities based on area 175,000 Points

98/193 Modified Fractal Tennis Weight probabilities based on area 200,000 Points

99/193 Condensation Sets Given a condensation set C

100/193 Condensation Set Example

101/193 Condensation Sets – Example Condensation Set

102/193 Condensation Sets – Example

103/193 Condensation Sets – Example

104/193 Condensation Sets – Example

105/193 Condensation Sets – Example

106/193 Condensation Sets – Example

107/193 Condensation Sets – Example

108/193 Condensation Sets – Example

109/193 Condensation Sets – Example

110/193 Condensation Sets – Example

111/193 Condensation Sets – Example

112/193 Condensation Sets – Example

113/193 Condensation Sets – Example

114/193 Condensation Sets – Example

115/193 Condensation Sets – Example

116/193 Condensation Sets – Example

117/193 Condensation Sets – Example

118/193 Condensation Sets – Example

119/193 Condensation Sets – Example

120/193 Condensation Sets – Example

121/193 Condensation Sets – Example

122/193 Rendering Fractals

123/193 Rendering Fractals

124/193 Rendering Fractals

125/193 Rendering Fractals

126/193 Rendering Fractals

127/193 Rendering Fractals

128/193 Rendering Fractals

129/193 Rendering Fractals

130/193 Condensation Set Example

131/193 Condensation Set Example

132/193 Condensation Set Example

133/193 Condensation Set Example

134/193 Condensation Set Example

135/193 Condensation Set Example

136/193 Condensation Set Example

137/193 Condensation Set Example

138/193 Condensation Set Example

139/193 Condensation Set Example

140/193 Condensation Set Example

141/193 Fractal Dimension Fractal dimension =

142/193 Fractal Dimension Fractal dimension =

143/193 Fractal Dimension Fractal dimension =

144/193 Fractal Dimension Fractal dimension =

145/193 Fractal Dimension Fractal dimension =

146/193 Fractal Dimension Fractal dimension =

147/193 Fractal Dimension Fractal dimension =

148/193 Fractal Dimension Fractal dimension =

149/193 Fractal Dimension Fractal dimension =

150/193 Fractal Dimension Fractal dimension =

151/193 Fractal Dimension Fractal dimension =

152/193 Fractal Dimension Fractal dimension =

153/193 Fractal Dimension Fractal dimension =

154/193 Fractal Dimension Fractal dimension =

155/193 Fractal Dimension Fractal dimension =

156/193 Fractal Dimension Fractal dimension =

157/193 Fractal Dimension Fractal dimension =

158/193 Fractal Dimension Fractal dimension =

159/193 Fractal Dimension Fractal dimension =

160/193 Fractal Dimension Fractal dimension =

161/193 Fractal Dimension Fractal dimension =

162/193 Fractal Dimension Fractal dimension =

163/193 Fractal Dimension Fractal dimension =

164/193 Fractal Dimension Fractal dimension =

165/193 Fractal Dimension Fractal dimension =

166/193 Fractal Dimension Fractal dimension =

167/193 Fractal Dimension Fractal dimension =

168/193 Fractal Dimension Fractal dimension =

169/193 Fractal Dimension Fractal dimension =

170/193 Fractal Dimension Fractal dimension =

171/193 Fractal Dimension Fractal dimension =

172/193 Fractal Dimension Fractal dimension =

173/193 Fractal Dimension Fractal dimension =

174/193 Fractal Dimension Fractal dimension =

175/193 Fractal Dimension Fractal dimension =

176/193 Fractal Dimension Fractal dimension =

177/193 Fractal Dimension Fractal dimension =

178/193 Fractal Dimension Fractal dimension =

179/193 Weird Fractal Properties Fractal curves can have infinite length!!!

180/193 Weird Fractal Properties Fractal curves can have infinite length!!!

181/193 Weird Fractal Properties Fractal curves can have infinite length!!!

182/193 Weird Fractal Properties Fractal curves can have infinite length!!!

183/193 Weird Fractal Properties Fractal curves can have infinite length!!!

184/193 Weird Fractal Properties Fractal curves can have infinite length!!!

185/193 Weird Fractal Properties Fractal curves can have infinite length!!!

186/193 Weird Fractal Properties Fractal curves can have infinite length!!! But only enclose finite area?

187/193 Weird Fractal Properties Fractal curves can have infinite length!!! But only enclose finite area?

188/193 Weird Fractal Properties Fractal curves can have infinite length!!! But only enclose finite area?

189/193 Weird Fractal Properties Fractal curves can have infinite length!!! But only enclose finite area?

190/193 Weird Fractal Properties Fractal curves can have infinite length!!! But only enclose finite area?

191/193 Weird Fractal Properties Fractal curves can have infinite length!!! But only enclose finite area?

192/193 Weird Fractal Properties Fractal curves can have infinite length!!! But only enclose finite area?

193/193 Weird Fractal Properties