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Examples of inversion of circles A to J with respect to the red circle at O. Circles A to F, which pass through O, map to straight lines. Circles G to J, which do not, map to other circles. The reference circle and line L map to themselves. Circles intersect their inverses, if any, on the reference circle.
An inversion may be denoted by the pair of places (2, 4) or the pair of elements (5, 2). The inversions of this permutation using element-based notation are: (3, 1), (3, 2), (5, 1), (5, 2), and (5,4). In computer science and discrete mathematics, an inversion in a sequence is a pair of elements that are out of their natural order.
Applying this transformation twice on a 4-vector gives a transformation of the same form. The new symmetry of 'inversion' is given by the 3-tensor . This symmetry becomes Poincaré symmetry if we set = When = the second condition requires that is an orthogonal matrix. This transformation is 1-1 meaning that each point is mapped to a unique ...
Inversion in a point, or point reflection, a kind of isometric (distance-preserving) transformation in a Euclidean space; Inversion transformation, a conformal transformation (one which preserves angles of intersection) Method of inversion, the image of a harmonic function in a sphere (or plane); see Method of image charges
The point O in this construction is called the center of inversion, the circle the circle of inversion, and k the radius of inversion. An inversion applied twice is the identity transformation, so the inverse of an inverse curve with respect to the same circle is the original curve.
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Pyramidal inversion in nitrogen and amines is known as nitrogen inversion. [8] It is a rapid oscillation of the nitrogen atom and substituents, the nitrogen "moving" through the plane formed by the substituents (although the substituents also move - in the other direction); [ 9 ] the molecule passing through a planar transition state . [ 10 ]
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