v2009.01.01 - Convex Optimization

5.5. INVARIANCE 385
x 2 x 2
x 3 x 1 x 3
x 1
x 4 x 4
(a) (b) (c)
Figure 104: (a) Four points in quadrature in two dimensions about
their geometric center. (b) Complete EDM graph of diamond vertices.
(c) Quadrature rotation of a Euclidean body in R 2 first requires a shroud
that is the smallest Cartesian square containing it.
whereas reflection of any point through a hyperplane containing the origin
{ ∣ ∣∣∣∣ [ ] T
cos θ
∂H = x∈ R 2 x = 0}
(887)
sin θ
is 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 (888)
Rotation matrix Q is characterized by identical diagonal entries and by
antidiagonal entries equal but opposite in sign, whereas reflection matrix R
is characterized in the reverse sense.
Assign the diamond vertices { x l ∈ R 2 , l=1... 4 } to columns of a matrix
X = [x 1 x 2 x 3 x 4 ] ∈ R 2×4 (68)
Our scheme to prevent reflection enforces a rotation matrix characteristic
upon the coordinates of adjacent points themselves: First shift the geometric
center of X to the origin; for geometric centering matrix V ∈ S 4 (5.5.1.0.1),
define
Y ∆ = XV ∈ R 2×4 (889)