Fractals/Mathematics for computer graphic

Mathematics for computer graphic

• Numbers
  • by base
    • binary number
    • decimal
  • sequences
  • Period
  • Continued fraction
• Function, map, iterated function
  • angle doubling map
  • rotating map
  • Complex quadratic map
  • rational map
  • Derivative
• computations
  • Numerical methods, precision, error
    • Finding roots of equation
      • Newton method
      • Durand-Kerner method
    • Finding function from sequence , curve fitting, model fitting
  • Symbolic methods
    • Kneading sequences
• Group theory
  • Binary adding group
  • Basilica group
  • Kleinian group
• Vector field
  • Polynomial vector field in one complex variable
    • Line integral convolution ( LIC) global visualisation of vector field
  • From discrete dynamical systems to continuous dynamical systems
• dynamical system
  • discrete map
  • difference equation
  • differential equation
• Computational topology
  • topology of point cloud data ( PCD)
    • from PCD to simplicial complexes
  • surface topology: Normal curves and surfaces, Paths in the plane
  • Topological graph theory, Computation of Contour Trees and Topology of Level Sets
  • distance fields: SDF
  • noise
  • Morse theory relates critical points of a function on a manifold (geometry) to the topology of the manifold
• Computational Geometry has immediate applications in many other research areas, including computer graphics, mesh generation, geographic information systems, VLSI design, and robotics
  • Specific problems
    • convex hulls
    • Voronoi diagrams
    • line segment intersection
    • polygon triangulation
    • geometric range searching
    • low-dimensional linear programming
    • geometric shortest paths
    • visibility
  • parts
    • Discrete geometry ( it includes Digital geometry = computer graphics and image analysis, Geometry processing = mesh processing )
    • computational conformal geometry
    • Discrete Differential Geometry
    • Hyperbolic geometry
  • software
    • cgal The Computational Geometry Algorithms Library

       "It can be argued that the mathematics behind these images is even prettier than the pictures themselves." Robert L. Devaney

      "We choose to do mathematics, not because it is easy, but because it is hard." user "Haskell Curry "
      
      
       You don't need to be a mathematician to appreciate the impressive beauty of fractals, and many times intuition and curiosity are two of the more important ingredients that drive mathematical discovery. Víctor José García Garrido

      "whereas mathemathical idea is a timless thing, few things are more ephemeral then computer hardware and software" Tristan Needham in Visual Complex Analysis