Fractal and computational geometry (7ΕΠ12)
Instructor : Vasileios Drakopoulos
Course typeElective
Semester7
TermFall Semester
ECTS5
Teaching hours3
Laboratory hours
Description
Fractal sets and their geometry: Similarity, dimensions, dynamic system, iterated function system, complex analytic dynamics, Julia sets and the Mandelbrot set, computational methods for their construction and graphical representation in two- and three-dimensions. Design and analysis of geometric data processing algorithms: Geometric spaces and algebraic point representation, lines, curves, planes, surfaces, etc. geometric duality, space subdivisons and surface arrangements, the Zone theorem and its applications, Davenport – Schinzel sequences, applications, convex hull of points and algorithms for finding it, Voronoi diagrams and Delaunay triangulations, ways of computing them, proximity problem solutions, point and arrangement triangulations, applications, range searching techniques: subdivision trees, techniques based on random samples such as ε – net and ε – approximation, parametric searching, applications in robotics, computer vision, graphic and artificial design.
Course objectives
  • Recognising a self-similar object, i.e. an object that has the same shape as one or more of its parts, and its scale invariance.
  • Understanding the resolution independence of fractals, familiarising with the repetitive processes for generating fractals as well as recognising their basic characteristics and attributes.
  • Applying several methods and techniques for the design and construction of various fractals.
  • Recognising the use of a range of geometric data structures and computation methods.
  • The comparison, distinction and choice of appropriate geometric data structures and computation methods based on the criteria of functionality, performance in time and space and material requirements.
  • Implementing basic technical data structure design and computing methods to solve geometric problems.
  • Evaluating the importance of geometric calculations as applied to intricate, complicated and complex systems.
Textbooks/Bibliography
  • de Berg Mark, Cheong Otfried, van Kreveld Marc and Overmars Mark, Computational Geometry: Algorithms and applications, 3rd ed., Springer-Verlag, Berlin and Heidelberg, 2008
  • Peitgen Heinz-Otto, Jürgens Hartmut and Saupe Dietmar, Fractals for the classroom, Part One and Part Two, Springer-Verlag, New York, 1992
  • Preparata F.P. and Shamos M. I., Computational Geometry: An introduction, Springer-Verlag, New York, 1985
  • O'Rourke J., Computational Geometry in C, 2nd ed., Cambridge University Press, New York, 1998.
Assessment method
Course material