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Geometry
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
This chapter begins by discussing affine geometry, which can be defined, very roughly, as Euclidean geometry without any mention of measurement. Thus, affine theorems concern such things as incidence and parallelism. An affine space is defined in terms of an action of a vector space V on a set X, pursuant to which a vector v acts on a point A of X by sending it to another element v(A) of X, and two points A and B of X define a unique vector AB→, which acts on A by sending it to B. It might be helpful for the reader to think of AB→ as an arrow starting at point A and ending at point B. This vector then acts on an arbitrary point C of X by placing the starting point of the arrow at C and letting the end point of the arrow be v(C).
Digital geometry for image analysis and processing
Published in João Manuel, R. S. Tavares, R. M. Natal Jorge, Computational Modelling of Objects Represented in Images, 2018
As already mentioned, digital geometry is a modern discipline that sets up itself as such in relation to its nowadays applications. However, it has its roots in a number of classical mathematical disciplines, such as number theory (since C.F Gauss), geometry of numbers (since H. Minkowski), graph theory (since L. Euler), and combinatorial topology (since the middle of the 19th century). At present, research in digital geometry resorts to the above and some other mathematical disciplines. A more complete (although not exhaustive) list is given next.Number theory, geometry of numbersClassical Euclidean geometry, analytical geometry, affine geometry, projective geometryAlgebraic geometryVector spaces, metric spacesCombinatorial geometry, discrete geometry, tilings and patternsComputational geometryGeneral topology, combinatorial topologyGraph theoryLinear programming, integer programming, Dio-phantine equations, polyhedral combinatorics, lattice polytopesMathematical morphologyDiscrete dynamical systems, fractal theoryCombinatorics on wordsApproximation theory, Diophantine approximations, continued fractionsProbability theory and mathematical statisticsDesign and analysis of algorithms, complexity theory
Topological perception on attention to product shape
Published in International Journal of Design Creativity and Innovation, 2020
The shape of a product can be simplified to its basic geometry from a global viewpoint. In the product iteration process, shapes would be altered for competing with other brands, while the shapes of some products remain invariant, especially the identity of brand properties. There is a way to clarify the relationship between the invariants and their levels of structural stability. According to German mathematician Klein’s Erlangen program, any geometry is dealing with invariant properties under a certain transitive group action. Using this principle, he built a hierarchy of geometries, stratified in ascending order of stability: Euclidean geometry, affine geometry, projective geometry, and topology, which provides the highest stability (Kisil, 2012). Various studies on perceptual organization have demonstrated that the psychological reality (i.e. psychological representation of knowledge) of Klein’s development reveals a functional hierarchy of shape perception. Perceptual organization is the grouping of parts or regions of the field with one another, and the differentiation of the figure from the ground (Rock, 1986). This functional hierarchy is remarkably consistent with the stratification of geometries. For instance, the reaction times for perceiving differences in geometry are closely related to the geometrical stratification related to form stability (Chen, 2005). Specifically, the highest stability of topology provides the shortest reaction times among the gradations of geometry. In other words, people perceive topological features before any other geometrical variation. Therefore, we addressed customer attention to topological differences in the shape of products to foster innovative processes. Under normal circumstances, customers integrate information from stimuli with previous knowledge (Kruglanski & Gigerenzer, 2011). Especially for consumer products, consumers may be familiar with product shapes by recalling related memories. When a product with a novel shape is presented to consumers as a marketing stimulus, they integrate information from that stimulus with the previously known shape, and greater differences are likely to elicit more consumer attention.