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To step from n to n + 1, generate a vector v uniformly distributed on the n-sphere S n, embed the n × n matrix in the next larger size with last column (0, ..., 0, 1), and rotate the larger matrix so the last column becomes v. As usual, we have special alternatives for the 3 × 3 case. Each of these methods begins with three independent random ...
One of the main motivations for using matrices to represent linear transformations is that transformations can then be easily composed and inverted. Composition is accomplished by matrix multiplication. Row and column vectors are operated upon by matrices, rows on the left and columns on the right. Since text reads from left to right, column ...
In the 2-dimensional case, if the density exists, each iso-density locus (the set of x 1,x 2 pairs all giving a particular value of ()) is an ellipse or a union of ellipses (hence the name elliptical distribution).
In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle .
Sequence of probes around the convex hull of a polygon to determine its diameter using Rotating Caliper method. In computational geometry, the method of rotating calipers is an algorithm design technique that can be used to solve optimization problems including finding the width or diameter of a set of points.
Let k be a unit vector defining a rotation axis, and let v be any vector to rotate about k by angle θ (right hand rule, anticlockwise in the figure), producing the rotated vector . Using the dot and cross products, the vector v can be decomposed into components parallel and perpendicular to the axis k,
The gyration radius is useful in estimating the stiffness of a column. If the principal moments of the two-dimensional gyration tensor are not equal, the column will tend to buckle around the axis with the smaller principal moment. For example, a column with an elliptical cross-section will tend to buckle in the direction of the smaller semiaxis.
When increases from zero, i.e., assumes positive values, the line evolves into an ellipse that is being traced out in the counterclockwise direction (looking in the direction of the propagating wave); this then corresponds to left-handed elliptical polarization; the semi-major axis is now oriented at an angle .