10 Extension of the Monogenic Phasor Method to Extract Displacements and Their Derivatives from 3-D Fringe Patterns 69 10.4 Derivation of the 4D Hyper-Sphere One can generalize the equivalent of the Poincare sphere in the 3D complex space [2] by resorting to a 4D complex space. The 4D expression for a hyper-sphere of radius RDIsp in the complex space is given by the 4D complex vector, )I sp D 2 6 6 4 I1 I2 I3 I4 3 7 7 5 (10.21) where the modulus of )I sp is given by, )I sp DIsp D qI2 1 CI 2 2 CI 2 3 CI 2 4 DR2 (10.22) The components of the vector are, I1 DIspcos¥ cos™1 cos™2 (10.23) I2 DIspcos¥cos™1 sin™2 (10.24) I3 DIspcos¥sin™1 (10.25) I4 DIspsin¥ (10.26) Figure 10.8 illustrates the relationship between the vector Ip and the vector Isp in the complex plane. Equation (10.15) provides Ip. From Eq. (10.2), dp.x/ D qu2 1 .x1/ Cu2 2 .x2/ Cu2 3 .x3/ (10.27) Calling dp, the modulus of dp and the local phase , from Eq. (10.6) it follows, ¥D 2 dp p (10.28) From Fig. 10.8, Isp D Ip cos¥ (10.29) Then one has obtained all the elements of the complex vector )I sp. Fig. 10.8 Determination of I4 in the complex plane Isp Ip I4=Iqu φ i j 0
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