cse 423 computer graphics | mushfiqur rouf [email protected] rouf/cse423 cse423:
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CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE423: Computer Graphics
Fractals
Mushfiqur [email protected]
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Fractal
• Arbitrarily fine structure
• Irregularity
• Self-similarity
• Fractional dimension
• Recursive Definition
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Dimension
• Number of independent parameters needed to describe a point
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Fractional Dimension
• how completely a fractal appears to fill space
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Self Similarity
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Examples
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Koch Snowflake
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Koch Snowflake: Construction
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Von Koch CurveGeneric Koch
• Specify an angle, not just 45º
• Dimension =
• An Example, for 85º • How would a 90º curve
look like?
cos12ln
4ln
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Quadratic Koch
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Quadratic Koch
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Gosper Curve
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Space Filling Curves
• Constructed using line segments only
• But each has dimension = 2
• Contain the entire 2-dimensional unit square – (or the 3-dimensional unit cube, or n-dimensional
hypercube)
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Hilbert CurveDimension = 2
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Hilbert Curve 3D
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Peano Curve
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Dragon Curve
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Dragon Curve
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Dragon Curve
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Dragon Curve: Self Similarity
22ln
2ln
Dimension
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Dragon Curve: Construction
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Sierpinski Triangle
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Sierpinski Triangle: Construction or Destruction
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Sierpinski Triangle: Self-Similarity
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Sierpinski Tetrahedron:3D Extension to Sierpinski
Triangle
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Sierpinski Carpet
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Menger Sponge
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Menger Sponge: Construction
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T-Box Curve
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T-Box Curve
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Hexaflake
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Apllonian Gasket
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Random Fractal: Brownian Tree
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Random Fractal: Brownian Tree
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Random Fractal: Mountain
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Mandelbrot Set
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
Mandelbrot Set: Self Similarity
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423
CSE 423 Computer Graphics | MUSHFIQUR ROUF [email protected]://groups.google.com/group/cse423spring07/ http://faculty.bu.ac.bd/~rouf/cse423