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In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3) , the group of all rotation matrices ...
Most rotation matrices fit this description, and for them it can be shown that (Q − I)(Q + I) −1 is a skew-symmetric matrix, A. Thus A T = − A ; and since the diagonal is necessarily zero, and since the upper triangle determines the lower one, A contains 1 / 2 n ( n − 1) independent numbers.
O'Rourke's approach uses a 3-dimensional rotating calipers technique, and is based on lemmas characterizing the minimum enclosing box: There must exist two neighbouring faces of the smallest-volume enclosing box which both contain an edge of the convex hull of the point set.
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
Curve fitting [1] [2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, [3] possibly subject to constraints. [ 4 ] [ 5 ] Curve fitting can involve either interpolation , [ 6 ] [ 7 ] where an exact fit to the data is required, or smoothing , [ 8 ] [ 9 ] in which a "smooth ...
Function calls and blocks of code, such as code contained within a loop, are often replaced by a one-line natural language sentence. Depending on the writer, pseudocode may therefore vary widely in style, from a near-exact imitation of a real programming language at one extreme, to a description approaching formatted prose at the other.
The rotating calipers technique for designing geometric algorithms may also be interpreted as a form of the plane sweep, in the projective dual of the input plane: a form of projective duality transforms the slope of a line in one plane into the x-coordinate of a point in the dual plane, so the progression through lines in sorted order by their ...
The figure shows the three transformation steps of an ordinary Procrustes fit for two configurations of landmarks. (a) Scaling of both configurations to the same size; (b) Transposition to the same position of the center of gravity; (c) Rotation to the orientation that provides the minimum sum of squared distances between corresponding landmarks.