v2010.10.26 - Convex Optimization

5.5. INVARIANCE 437x 2 x 2x 3 x 1 x 3x 1x 4 x 4(a) (b) (c)Figure 128: (a) Four points in quadrature in two dimensions about theirgeometric center. (b) Complete EDM graph of diamond-shaped vertices.(c) Quadrature rotation of a Euclidean body in R 2 first requires a shroudthat is the smallest Cartesian square containing it.whereas reflection of any point through a hyperplane containing the origin{ ∣ ∣∣∣∣ [ ] Tcos θ∂H = x∈ R 2 x = 0}(982)sin θis accomplished via multiplication with symmetric orthogonal matrix (B.5.2)[ ]sin(θ)R =2 − cos(θ) 2 −2 sin(θ) cos(θ)−2 sin(θ) cos(θ) cos(θ) 2 − sin(θ) 2 (983)Rotation matrix Q is characterized by identical diagonal entries and byantidiagonal entries equal but opposite in sign, whereas reflection matrix Ris characterized in the reverse sense.Assign the diamond vertices { x l ∈ R 2 , l=1... 4 } to columns of a matrixX = [x 1 x 2 x 3 x 4 ] ∈ R 2×4 (76)Our scheme to prevent reflection enforces a rotation matrix characteristicupon the coordinates of adjacent points themselves: First shift the geometriccenter of X to the origin; for geometric centering matrix V ∈ S 4 (5.5.1.0.1),defineY XV ∈ R 2×4 (984)