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The line through segment AD and the line through segment B 1 B are skew lines because they are not in the same plane. In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron.
Two intersecting lines. In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line.Distinguishing these cases and finding the intersection have uses, for example, in computer graphics, motion planning, and collision detection.
The widely accepted interpretation of, e.g. the Poggendorff and Hering illusions as manifestation of expansion of acute angles at line intersections, is an example of successful implementation of a "bottom-up," physiological explanation of a geometrical–optical illusion. Ponzo illusion in a purely schematic form and, below, with perspective clues
One of the explanations for the Ponzo illusion is the "perspective hypothesis", which says that the perspective feature in the figure is produced by the converging lines ordinarily associated with distance; the two oblique lines appear to converge toward the horizon or a vanishing point. We interpret the upper line as though it were further ...
As predicted by catastrophe theory, singularities are generic, and stable under perturbation. This explains how the bright lines and surfaces are stable under perturbation. The caustics one sees at the bottom of a swimming pool, for example, have a distinctive texture and only has a few types of singular points, even though the surface of the ...
An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields and reaches its most sophisticated and advanced forms in quantum field theory.
For example, the first Napoleon point is the point of concurrency of the three lines each from a vertex to the centroid of the equilateral triangle drawn on the exterior of the opposite side from the vertex. A generalization of this notion is the Jacobi point. The de Longchamps point is the point of concurrence of several lines with the Euler line.
An example with continuous symmetry is given by a 3d analogue of the previous example, from rotating the graph around an axis through the top of the hill, or equivalently given by the graph (,) = (+). This is essentially the graph of the Mexican hat potential. This has a continuous symmetry given by rotation about the axis through the top of ...