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An involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points. [6]: 24 Any projectivity that interchanges two points is an involution. The three pairs of opposite sides of a complete quadrangle meet any line (not through a vertex) in three pairs of an involution.
In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or π radians), while in three-dimensional Euclidean space a point reflection is an improper rotation which preserves distances but reverses orientation. A point reflection is an involution: applying it twice is the identity transformation.
The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. [2]By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, [3] but it doesn ...
Point Q is the reflection of point P through the line AB. In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.
If A represents a linear involution, then x→A(x−b)+b is an affine involution. One can check that any affine involution in fact has this form. Geometrically this means that any affine involution can be obtained by taking oblique reflections against any number from 0 through n hyperplanes going through a point b.
Many difficult problems in geometry become much more tractable when an inversion is applied. ... (i.e. an involution). ... If point R is the inverse of point P then ...
Social Security recipients saw their monthly check grow by 2.5% in 2025 thanks to a cost-of-living adjustment (COLA) designed to offset the dollar-diminishing power of inflation. Beneficiaries get ...
To verify the conditions that make this a projective plane, observe that every two rows have exactly one common column in which 1s appear (every pair of distinct points are on exactly one common line) and that every two columns have exactly one common row in which 1s appear (every pair of distinct lines meet at exactly one point). Among many ...