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Geometry
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Changing the parallel postulate results in other geometries:(5; for hyperbolic geometry) Through a point not on a given straight line, infinitely many lines can be drawn that never meet the given line. For example, the surface of a hyperboloid is an example of hyperbolic geometry.(5; for elliptic geometry) Through a point not on a given straight line, no lines can be drawn that never meet the given line. For example, the surface of a sphere is an example of elliptic geometry.
Hyperbolic Prism, Poincaré Disk, and Foams
Published in Christos H. Skiadas, Charilaos Skiadas, Handbook of Applications of Chaos Theory, 2017
Alberto Tufaile, Adriana Pedrosa, Biscaia Tufaile
According to Needham [2], the Poincaré disk model is a model for hyperbolic geometry, in which a line is represented as an arc of a circle, the ends of which are perpendicular to the disk boundary. What is the definition of parallel rays in this disk? Two arcs that do not meet correspond to parallel rays. In that geometry, arcs that meet orthogonally correspond to perpendicular lines, and arcs that meet on the boundary are a pair of limit rays.
Design, analysis and manufacturing of lattice structures: an overview
Published in International Journal of Computer Integrated Manufacturing, 2018
TPMS are minimal surfaces, which are a subset of hyperbolic surfaces. A hyperbolic surface is one created utilising hyperbolic geometry. The main difference between a surface in normal Euclidean geometry and hyperbolic geometry is that in hyperbolic geometry, there are at least two distinct lines that pass through a given point and are parallel to a given line. Alternatively, Euclidean geometry has exactly one line through a given point in the same plane as a given line which is never intersected.