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The graph of an involution (on the real numbers) is symmetric across the line y = x. This is due to the fact that the inverse of any general function will be its reflection over the line y = x. This can be seen by "swapping" x with y. If, in particular, the function is an involution, then its graph is its own reflection.
This reflection operation turns the gradient of any line into its reciprocal. [ 1 ] Assuming that f {\displaystyle f} has an inverse in a neighbourhood of x {\displaystyle x} and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at x {\displaystyle x} and have a derivative given by the above formula.
We only consider stretches along the x-axis and y-axis. A stretch along the x-axis has the form x' = kx; y' = y for some positive constant k. (Note that if k > 1, then this really is a "stretch"; if k < 1, it is technically a "compression", but we still call it a stretch. Also, if k = 1, then the transformation is an identity, i.e. it has no ...
In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation.It states that if an analytic function is defined on the upper half-plane, and has well-defined (non-singular) real values on the real axis, then it can be extended to the conjugate function on the lower half-plane.
Switching {X,Y} in a graph. A two-graph is equivalent to a switching class of graphs and also to a (signed) switching class of signed complete graphs.. Switching a set of vertices in a (simple) graph means reversing the adjacencies of each pair of vertices, one in the set and the other not in the set: thus the edge set is changed so that an adjacent pair becomes nonadjacent and a nonadjacent ...
A typical example of glide reflection in everyday life would be the track of footprints left in the sand by a person walking on a beach. Frieze group nr. 6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about a point on the line of reflection. It is isomorphic to a semi-direct product of Z and C 2.
A reflection through an axis. In mathematics, a reflection (also spelled reflexion) [1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.
It follows rather readily (see orthogonal matrix) that any orthogonal matrix can be decomposed into a product of 2 by 2 rotations, called Givens Rotations, and Householder reflections. This is appealing intuitively since multiplication of a vector by an orthogonal matrix preserves the length of that vector, and rotations and reflections exhaust ...