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Let P and Q be two sets, each containing N points in .We want to find the transformation from Q to P.For simplicity, we will consider the three-dimensional case (=).The sets P and Q can each be represented by N × 3 matrices with the first row containing the coordinates of the first point, the second row containing the coordinates of the second point, and so on, as shown in this matrix:
It can also be described as the shortest path distance in a rotation graph, a graph that has a vertex for each binary tree on a given left-to-right sequence of nodes and an edge for each rotation between two trees. [2] This rotation graph is exactly the graph of vertices and edges of an associahedron. [3]
A tree can be rebalanced using rotations. After a rotation, the side of the rotation increases its height by 1 whilst the side opposite the rotation decreases its height similarly. Therefore, one can strategically apply rotations to nodes whose left child and right child differ in height by more than 1.
A bitwise AND is a binary operation that takes two equal-length binary representations and performs the logical AND operation on each pair of the corresponding bits. Thus, if both bits in the compared position are 1, the bit in the resulting binary representation is 1 (1 × 1 = 1); otherwise, the result is 0 (1 × 0 = 0 and 0 × 0 = 0).
The two basic types are the arithmetic left shift and the arithmetic right shift. For binary numbers it is a bitwise operation that shifts all of the bits of its operand; every bit in the operand is simply moved a given number of bit positions, and the vacant bit-positions are filled in.
This motivates the following general definition: For a string s over an alphabet Σ, let shift(s) denote the set of circular shifts of s, and for a set L of strings, let shift(L) denote the set of all circular shifts of strings in L. If L is a cyclic code, then shift(L) ⊆ L; this is a necessary condition for L being a cyclic language.
In two dimensions this is a rotation through a right angle. As a further generalization, we can loosen the requirements of multilinearity and magnitude, and consider a general continuous function V d → V (where V is R n endowed with the Euclidean inner product and d ≥ 2), which is only required to satisfy the following two properties:
The binary degree, also known as the binary radian (or brad), is 1 / 256 turn. [21] The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte. Other measures of angle used in computing may be based on dividing one whole turn into 2 n equal parts for other values of n. [22]