On the role of kinesthetic thinking in computational geometry

Computational geometry is a new (about 30 years) and rapidly growing branch of knowledge in computer science that deals with the analysis and design of algorithms for solving geometric problems. These problems typically arise in computer graphics, image processing, computer vision, robotics, manufacturing, knot theory, polymer physics and molecular biology. Since its inception many of the algorithms proposed for solving geometric problems, published in the literature, have been found to be incorrect. These incorrect algorithms rather than being ‘purely mathematical’ often contain a strong kinesthetic component. This paper explores the relationship between computational geometric thinking and kinesthetic thinking, the effect of the latter on the correctness and efficiency of the resulting algorithms, and their implications for education.

[1]  H. Gardner,et al.  Frames of Mind: The Theory of Multiple Intelligences , 1983 .

[2]  Donald D. Hoffman,et al.  Visual intelligence: How we create what we see , 1998 .

[3]  J. Hadamard,et al.  The Psychology of Invention in the Mathematical Field. , 1945 .

[4]  Josef Kittler,et al.  Pattern recognition theory and applications : proceedings of the NATO Advanced Study Institute held at St. Anne's College, Oxford, March 29-April 10, 1981 , 1982 .

[5]  J. O'Rourke.  Art gallery theorems and algorithms , 1987 .

[6]  Michael Samuels,et al.  Seeing With The Mind's Eye , 1975 .

[7]  Godfried T. Toussaint,et al.  A note on linear expected time algorithms for finding convex hulls , 2005, Computing.

[8]  Donis A. Dondis.  A primer of visual literacy , 1973 .

[9]  Ulrich Kortenkamp.  Foundations of dynamic geometry , 2000 .

[10]  G.T.E. Toussaint,et al.  Scanning the issue - Computational geometry (special issue intro.) , 1992, Proc. IEEE.

[11]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[12]  Godfried T. Toussaint,et al.  A Counterexample to a Diameter Algorithm for Convex Polygons , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[13]  I. Lakatos,et al.  Proofs and Refutations: Frontmatter , 1976 .

[14]  Dag Svanæs,et al.  Kinaesthetic thinking: The tacit dimension of interaction design , 1997 .

[15]  Louise Pelland.  Kinesthetic stimulation as a method for improved drawing-skill acquisition , 1980 .

[16]  Josef Kittler,et al.  Pattern Recognition Theory and Applications , 1987, NATO ASI Series.

[17]  Mark L. Johnson.  The Body in the Mind: The Bodily Basis of Meaning, Imagination, and Reason , 1989 .

[18]  Gary H. Meisters,et al.  POLYGONS HAVE EARS , 1975 .

[19]  William F. Eddy,et al.  A New Convex Hull Algorithm for Planar Sets , 1977, TOMS.

[20]  Godfried T. Toussaint,et al.  Efficient triangulation of simple polygons , 1991, The Visual Computer.

[21]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

[22]  Mark Rowlands,et al.  The body in mind , 1999 .

[23]  J. Pascual-Leone.  The Forms of Knowing in the Psychological Organism , 1976 .

[24]  Imre Lakatos,et al.  On the Uses of Rigorous Proof. (Book Reviews: Proofs and Refutations. The Logic of Mathematical Discovery) , 1977 .

[25]  N. J. Lennes.  Theorems on the Simple Finite Polygon and Polyhedron , 1911 .

[26]  Ray A. Jarvis,et al.  On the Identification of the Convex Hull of a Finite Set of Points in the Plane , 1973, Inf. Process. Lett..

[27]  A. Koestler.  The Act of Creation , 1964 .

[28]  Harold Abelson,et al.  Turtle geometry : the computer as a medium for exploring mathematics , 1983 .

[29]  D. J. Langridge,et al.  A Computational View of Perception , 1973, Perception.

[30]  S. G. Hoggar.  Mathematics for computer graphics , 1993, Cambridge tracts in theoretical computer science.