DPD angles

7.2. DPD angles#

\begin{array}{r} \begin{array}{rcl} θ_{12} & = & acos (\frac{2 σ_{3} (- m_{1}^{2} - m_{3}^{2} + σ_{2}) - (m_{0}^{2} - m_{3}^{2} - σ_{3}) (m_{1}^{2} - m_{2}^{2} + σ_{3})}{\sqrt{λ (m_{0}^{2}, m_{3}^{2}, σ_{3})} \sqrt{λ (σ_{3}, m_{1}^{2}, m_{2}^{2})}}) \\ θ_{23} & = & acos (\frac{2 σ_{1} (- m_{1}^{2} - m_{2}^{2} + σ_{3}) - (m_{0}^{2} - m_{1}^{2} - σ_{1}) (m_{2}^{2} - m_{3}^{2} + σ_{1})}{\sqrt{λ (m_{0}^{2}, m_{1}^{2}, σ_{1})} \sqrt{λ (σ_{1}, m_{2}^{2}, m_{3}^{2})}}) \\ θ_{31} & = & acos (\frac{2 σ_{2} (- m_{2}^{2} - m_{3}^{2} + σ_{1}) - (m_{0}^{2} - m_{2}^{2} - σ_{2}) (- m_{1}^{2} + m_{3}^{2} + σ_{2})}{\sqrt{λ (m_{0}^{2}, m_{2}^{2}, σ_{2})} \sqrt{λ (σ_{2}, m_{3}^{2}, m_{1}^{2})}}) \end{array} \end{array}

Equation (A2):

for i in [1, 2, 3]:
    _, θii = formulate_theta_hat_angle(i, i)
    assert θii == 0

Equation (A3):

\begin{array}{r} \begin{array}{rcl} {\hat{θ}}_{3 (1)} & = & acos (\frac{- 2 m_{0}^{2} (- m_{1}^{2} - m_{3}^{2} + σ_{2}) + (m_{0}^{2} + m_{1}^{2} - σ_{1}) (m_{0}^{2} + m_{3}^{2} - σ_{3})}{\sqrt{λ (m_{0}^{2}, m_{1}^{2}, σ_{1})} \sqrt{λ (m_{0}^{2}, σ_{3}, m_{3}^{2})}}) \\ {\hat{θ}}_{1 (2)} & = & acos (\frac{- 2 m_{0}^{2} (- m_{1}^{2} - m_{2}^{2} + σ_{3}) + (m_{0}^{2} + m_{1}^{2} - σ_{1}) (m_{0}^{2} + m_{2}^{2} - σ_{2})}{\sqrt{λ (m_{0}^{2}, m_{2}^{2}, σ_{2})} \sqrt{λ (m_{0}^{2}, σ_{1}, m_{1}^{2})}}) \\ {\hat{θ}}_{2 (3)} & = & acos (\frac{- 2 m_{0}^{2} (- m_{2}^{2} - m_{3}^{2} + σ_{1}) + (m_{0}^{2} + m_{2}^{2} - σ_{2}) (m_{0}^{2} + m_{3}^{2} - σ_{3})}{\sqrt{λ (m_{0}^{2}, m_{3}^{2}, σ_{3})} \sqrt{λ (m_{0}^{2}, σ_{2}, m_{2}^{2})}}) \end{array} \end{array}

Equations (A4-5):

assert θ31hat == -θ13hat
assert θ12hat == -θ21hat
assert θ23hat == -θ32hat

Equations (A6):

for i in [1, 2, 3]:
    for k in [1, 2, 3]:
        _, ζi_k0 = formulate_zeta_angle(i, k, 0)
        _, ζi_ki = formulate_zeta_angle(i, k, i)
        _, ζi_kk = formulate_zeta_angle(i, k, k)
        assert ζi_ki == ζi_k0
        assert ζi_kk == 0

Equations (A7):

\begin{array}{r} \begin{array}{rcl} ζ_{1 (3)}^{1} & = & acos (\frac{2 m_{1}^{2} (- m_{0}^{2} - m_{2}^{2} + σ_{2}) + (m_{0}^{2} + m_{1}^{2} - σ_{1}) (- m_{1}^{2} - m_{2}^{2} + σ_{3})}{\sqrt{λ (m_{0}^{2}, m_{1}^{2}, σ_{1})} \sqrt{λ (σ_{3}, m_{1}^{2}, m_{2}^{2})}}) \\ ζ_{2 (1)}^{1} & = & acos (\frac{2 m_{1}^{2} (- m_{0}^{2} - m_{3}^{2} + σ_{3}) + (m_{0}^{2} + m_{1}^{2} - σ_{1}) (- m_{1}^{2} - m_{3}^{2} + σ_{2})}{\sqrt{λ (m_{0}^{2}, m_{1}^{2}, σ_{1})} \sqrt{λ (σ_{2}, m_{1}^{2}, m_{3}^{2})}}) \\ ζ_{2 (1)}^{2} & = & acos (\frac{2 m_{2}^{2} (- m_{0}^{2} - m_{3}^{2} + σ_{3}) + (m_{0}^{2} + m_{2}^{2} - σ_{2}) (- m_{2}^{2} - m_{3}^{2} + σ_{1})}{\sqrt{λ (m_{0}^{2}, m_{2}^{2}, σ_{2})} \sqrt{λ (σ_{1}, m_{2}^{2}, m_{3}^{2})}}) \\ ζ_{3 (2)}^{2} & = & acos (\frac{2 m_{2}^{2} (- m_{0}^{2} - m_{1}^{2} + σ_{1}) + (m_{0}^{2} + m_{2}^{2} - σ_{2}) (- m_{1}^{2} - m_{2}^{2} + σ_{3})}{\sqrt{λ (m_{0}^{2}, m_{2}^{2}, σ_{2})} \sqrt{λ (σ_{3}, m_{2}^{2}, m_{1}^{2})}}) \\ ζ_{3 (2)}^{3} & = & acos (\frac{2 m_{3}^{2} (- m_{0}^{2} - m_{1}^{2} + σ_{1}) + (m_{0}^{2} + m_{3}^{2} - σ_{3}) (- m_{1}^{2} - m_{3}^{2} + σ_{2})}{\sqrt{λ (m_{0}^{2}, m_{3}^{2}, σ_{3})} \sqrt{λ (σ_{2}, m_{3}^{2}, m_{1}^{2})}}) \\ ζ_{1 (3)}^{3} & = & acos (\frac{2 m_{3}^{2} (- m_{0}^{2} - m_{2}^{2} + σ_{2}) + (m_{0}^{2} + m_{3}^{2} - σ_{3}) (- m_{2}^{2} - m_{3}^{2} + σ_{1})}{\sqrt{λ (m_{0}^{2}, m_{3}^{2}, σ_{3})} \sqrt{λ (σ_{1}, m_{3}^{2}, m_{2}^{2})}}) \end{array} \end{array}

Equations (A8):

assert ζ1_12 == -ζ1_21
assert ζ2_23 == -ζ2_32
assert ζ3_31 == -ζ3_13

Equations (A10):

\begin{array}{r} \begin{array}{rcl} ζ_{2 (3)}^{1} & = & acos (\frac{2 m_{1}^{2} (m_{2}^{2} + m_{3}^{2} - σ_{1}) + (- m_{1}^{2} - m_{2}^{2} + σ_{3}) (- m_{1}^{2} - m_{3}^{2} + σ_{2})}{\sqrt{λ (σ_{2}, m_{3}^{2}, m_{1}^{2})} \sqrt{λ (σ_{3}, m_{1}^{2}, m_{2}^{2})}}) \\ ζ_{3 (1)}^{2} & = & acos (\frac{2 m_{2}^{2} (m_{1}^{2} + m_{3}^{2} - σ_{2}) + (- m_{1}^{2} - m_{2}^{2} + σ_{3}) (- m_{2}^{2} - m_{3}^{2} + σ_{1})}{\sqrt{λ (σ_{1}, m_{2}^{2}, m_{3}^{2})} \sqrt{λ (σ_{3}, m_{1}^{2}, m_{2}^{2})}}) \\ ζ_{1 (2)}^{3} & = & acos (\frac{2 m_{3}^{2} (m_{1}^{2} + m_{2}^{2} - σ_{3}) + (- m_{1}^{2} - m_{3}^{2} + σ_{2}) (- m_{2}^{2} - m_{3}^{2} + σ_{1})}{\sqrt{λ (σ_{1}, m_{2}^{2}, m_{3}^{2})} \sqrt{λ (σ_{2}, m_{3}^{2}, m_{1}^{2})}}) \end{array} \end{array}

Equations (A11):

assert ζ1_32 == -ζ1_23
assert ζ2_13 == -ζ2_31
assert ζ3_21 == -ζ3_12