idtx.py¶
Equation definitions and derivations using SymPy.
- class gme.core.idtx.IdtxMixin¶
Define indicatrix and figuratrix equations.
- define_idtx_fgtx_eqns() → None¶
Define indicatrix and figuratrix equations.
- pz_cosbeta_varphi_eqn¶
\(p_{z}^{4} = \dfrac{\cos^{4}{\left(\beta \right)}} {\varphi^{4} \left(1 - \cos^{2}{\left(\beta \right)}\right)^{3}}\)
- Type
- cosbeta_pz_varphi_solns¶
\(\left[ -\frac{ \left( -6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2} - 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} + 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2\right)^{\frac{2}{3}} + 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2} + 2 \sqrt[3]{2} \right) }{6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2}} ,\dots \right]\)
- Type
- fgtx_cosbeta_pz_varphi_eqn¶
\(\cos^{2}{\left(\beta \right)} = - \frac{- 6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2} - 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} + 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2\right)^{\frac{2}{3}} + 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2} + 2 \sqrt[3]{2}}{6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2}}\)
- Type
- fgtx_tanbeta_pz_varphi_eqn¶
\(\tan{\left(\beta \right)} = \sqrt{\frac{- 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} + 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2\right)^{\frac{2}{3}} + 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2} + 2 \sqrt[3]{2}}{6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2} + 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} - 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2\right)^{\frac{2}{3}} - 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2} - 2 \sqrt[3]{2}}}\)
- Type
- fgtx_px_pz_varphi_eqn¶
\(p_{x} = - p_{z} \sqrt{\frac{- 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} + 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2\right)^{\frac{2}{3}} + 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2} + 2 \sqrt[3]{2}}{6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2} + 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} - 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2\right)^{\frac{2}{3}} - 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2} - 2 \sqrt[3]{2}}}\)
- Type
- idtx_rdotx_pz_varphi_eqn¶
\({r}^x = \dfrac{\sqrt{6} \left(- 81 \cdot 2^{\frac{2}{3}} \varphi^{12} p_{z}^{12} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} - 378 \varphi^{12} p_{z}^{12} - 9 \cdot 2^{\frac{2}{3}} \sqrt{3} \varphi^{10} p_{z}^{10} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} - 42 \sqrt{3} \varphi^{10} p_{z}^{10} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} + 45\sqrt[3]{2}\varphi^{8}p_{z}^{8} \left(27 \varphi^{8} p_{z}^{8} + 3\sqrt{3}\varphi^{6}p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} + 96 \cdot 2^{\frac{2}{3}} \varphi^{8} p_{z}^{8} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 306 \varphi^{8} p_{z}^{8} + \sqrt[3]{2} \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4}\left(27\varphi^{8}p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} -18\varphi^{4}p_{z}^{4} + 2\right)^{\frac{2}{3}} + 2 \cdot 2^{\frac{2}{3}} \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 6 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 20 \sqrt[3]{2} \varphi^{4} p_{z}^{4} \left(27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} - 26 \cdot 2^{\frac{2}{3}} \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} - 64 \varphi^{4} p_{z}^{4} + 2 \sqrt[3]{2}\left(27\varphi^{8}p_{z}^{8}+3\sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} + 2 \cdot 2^{\frac{2}{3}} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 4\right)}{72 \varphi^{4} p_{z}^{5} \left(\frac{\sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2}}{6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} - 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} -2\sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} - 2 \sqrt[3]{2}}\right)^{\frac{3}{2}} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} \left(- 54 \cdot 2^{\frac{2}{3}} \varphi^{12} p_{z}^{12} - 6 \cdot 2^{\frac{2}{3}} \sqrt{3} \varphi^{10} p_{z}^{10} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} + 6 \varphi^{8} p_{z}^{8} \left(27 \varphi^{8} p_{z}^{8} + 3\sqrt{3}\varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} + 33 \sqrt[3]{2} \varphi^{8} p_{z}^{8} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 78 \cdot 2^{\frac{2}{3}} \varphi^{8} p_{z}^{8} + \sqrt[3]{2} \sqrt{3} \varphi^{6} p_{z}^{6}\sqrt{27\varphi^{4} p_{z}^{4} - 4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 2 \cdot 2^{\frac{2}{3}} \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 12 \varphi^{4} p_{z}^{4} \left(27 \varphi^{8} p_{z}^{8} + 3\sqrt{3}\varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} - 18 \sqrt[3]{2} \varphi^{4} p_{z}^{4} \sqrt[3]{27\varphi^{8}p_{z}^{8} + 3\sqrt{3}\varphi^{6}p_{z}^{6} \sqrt{27\varphi^{4}p_{z}^{4} - 4} - 18\varphi^{4}p_{z}^{4} + 2} - 24 \cdot 2^{\frac{2}{3}} \varphi^{4} p_{z}^{4} + 2 \left(27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} + 2 \sqrt[3]{2} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 2\cdot 2^{\frac{2}{3}}\right)}\)
- Type
- idtx_rdotz_pz_varphi_eqn¶
\({r}^z = \dfrac{\sqrt{6} \sqrt{\frac{- 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} + 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} + 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 2 \sqrt[3]{2}}{6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} - 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} - 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} - 2 \sqrt[3]{2}}} \left(27 \cdot 2^{\frac{2}{3}} \varphi^{8} p_{z}^{8} + 3 \cdot 2^{\frac{2}{3}} \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} - 18 \cdot 2^{\frac{2}{3}} \varphi^{4} p_{z}^{4} + 2 \left(27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} + 2 \sqrt[3]{2} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 2 \cdot 2^{\frac{2}{3}}\right)} {72 \varphi^{4} p_{z}^{5} \left(\frac{\sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2}} {6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} - 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} - 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} - 2 \sqrt[3]{2}}\right)^{\frac{3}{2}} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} \left(- 6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} - 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} + 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} + 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 2 \sqrt[3]{2}\right)}\)
- Type
Code¶
"""
Equation definitions and derivations using :mod:`SymPy <sympy>`.
---------------------------------------------------------------------
Requires Python packages/modules:
- :mod:`SymPy <sympy>`
- `GME`_
.. _GMPLib: https://github.com/geomorphysics/GMPLib
.. _GME: https://github.com/geomorphysics/GME
.. _Matrix:
https://docs.sympy.org/latest/modules/matrices/immutablematrices.html
.. _Equality: https://docs.sympy.org/latest/modules/core.html
#sympy.core.relational.Equality
---------------------------------------------------------------------
"""
# Disable these pylint errors because it doesn't understand SymPy syntax
# - notably minus signs in equations flag an error
# pylint: disable=invalid-unary-operand-type, not-callable
# Library
import warnings
import logging
# from typing import Dict, Type, Optional # , Tuple, Eq, List
# SymPy
from sympy import (
Eq,
Rational,
factor,
N,
re,
im,
sqrt,
solve,
sin,
cos,
tan,
Abs,
)
# GME
from gme.core.symbols import (
p,
rx,
rz,
px,
pz,
beta,
eta,
rvec,
varphi,
varphi_r,
)
warnings.filterwarnings("ignore")
__all__ = ["IdtxMixin"]
class IdtxMixin:
"""Define indicatrix and figuratrix equations."""
# Prerequisites
eta_: float
beta_type: str
pz_p_beta_eqn: Eq
p_varphi_beta_eqn: Eq
gstar_varphi_pxpz_eqn: Eq
# Definitions
pz_cosbeta_varphi_eqn: Eq
cosbeta_pz_varphi_solns: Eq
cosbeta_pz_varphi_soln: Eq
fgtx_cosbeta_pz_varphi_eqn: Eq
fgtx_tanbeta_pz_varphi_eqn: Eq
fgtx_px_pz_varphi_eqn: Eq
idtx_rdotx_pz_varphi_eqn: Eq
idtx_rdotz_pz_varphi_eqn: Eq
def define_idtx_fgtx_eqns(self) -> None:
r"""
Define indicatrix and figuratrix equations.
Attributes:
pz_cosbeta_varphi_eqn (`Equality`_):
:math:`p_{z}^{4} = \dfrac{\cos^{4}{\left(\beta \right)}}
{\varphi^{4} \left(1
- \cos^{2}{\left(\beta \right)}\right)^{3}}`
cosbeta_pz_varphi_solns (list):
:math:`\left[
-\frac{ \left( -6 \varphi^{4} p_{z}^{4} \sqrt[3]{27
\varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4}
+ \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12}
p_{z}^{12}} + 2} - 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4}
+ 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8}
- 18 \varphi^{4} p_{z}^{4}
+ \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12}
p_{z}^{12}} + 2\right)^{\frac{2}{3}}
+ 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4}
p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16}
- 108 \varphi^{12} p_{z}^{12}} + 2} + 2 \sqrt[3]{2} \right)
}{6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8}
- 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16}
p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2}}
,\dots \right]`
fgtx_cosbeta_pz_varphi_eqn (`Equality`_):
:math:`\cos^{2}{\left(\beta \right)}
= - \frac{- 6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8}
p_{z}^{8} - 18 \varphi^{4} p_{z}^{4}
+ \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12}
p_{z}^{12}} + 2} - 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4}
+ 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8}
- 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16}
- 108 \varphi^{12} p_{z}^{12}} + 2\right)^{\frac{2}{3}}
+ 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8}
- 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16}
p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2}
+ 2 \sqrt[3]{2}}{6 \varphi^{4} p_{z}^{4} \sqrt[3]{27
\varphi^{8} p_{z}^{8} - 18 \varphi^{4} p_{z}^{4}
+ \sqrt{729 \varphi^{16} p_{z}^{16} - 108 \varphi^{12}
p_{z}^{12}} + 2}}`
fgtx_tanbeta_pz_varphi_eqn (`Equality`_):
:math:`\tan{\left(\beta \right)}
= \sqrt{\frac{- 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4}
+ 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8}
- 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16}
p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}}
+ 2\right)^{\frac{2}{3}}
+ 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4}
p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16}
- 108 \varphi^{12} p_{z}^{12}} + 2} + 2 \sqrt[3]{2}}{6
\varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8}
- 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16}
p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2}
+ 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} - 2^{\frac{2}{3}}
\left(27 \varphi^{8} p_{z}^{8}
- 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16}
p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}}
+ 2\right)^{\frac{2}{3}}
- 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4}
p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16}
- 108 \varphi^{12} p_{z}^{12}} + 2} - 2 \sqrt[3]{2}}}`
fgtx_px_pz_varphi_eqn (`Equality`_):
:math:`p_{x}
= - p_{z} \sqrt{\frac{- 12 \sqrt[3]{2} \varphi^{4}
p_{z}^{4} + 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8}
- 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16}
p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}}
+ 2\right)^{\frac{2}{3}}
+ 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4}
p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16}
- 108 \varphi^{12} p_{z}^{12}} + 2} + 2 \sqrt[3]{2}}{6
\varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8}
- 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16}
p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}} + 2}
+ 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} - 2^{\frac{2}{3}}
\left(27 \varphi^{8} p_{z}^{8}
- 18 \varphi^{4} p_{z}^{4} + \sqrt{729 \varphi^{16}
p_{z}^{16} - 108 \varphi^{12} p_{z}^{12}}
+ 2\right)^{\frac{2}{3}}
- 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} - 18 \varphi^{4}
p_{z}^{4} + \sqrt{729 \varphi^{16} p_{z}^{16}
- 108 \varphi^{12} p_{z}^{12}} + 2} - 2 \sqrt[3]{2}}}`
idtx_rdotx_pz_varphi_eqn (`Equality`_):
:math:`{r}^x = \dfrac{\sqrt{6} \left(- 81
\cdot 2^{\frac{2}{3}} \varphi^{12} p_{z}^{12} \sqrt[3]{27
\varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4}
p_{z}^{4} + 2} - 378 \varphi^{12} p_{z}^{12} - 9 \cdot
2^{\frac{2}{3}} \sqrt{3} \varphi^{10} p_{z}^{10}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} \sqrt[3]{27 \varphi^{8}
p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27
\varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2}
- 42 \sqrt{3} \varphi^{10} p_{z}^{10} \sqrt{27
\varphi^{4} p_{z}^{4} - 4} + 45\sqrt[3]{2}\varphi^{8}p_{z}^{8}
\left(27 \varphi^{8} p_{z}^{8} + 3\sqrt{3}\varphi^{6}p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4}
p_{z}^{4} + 2\right)^{\frac{2}{3}} + 96 \cdot 2^{\frac{2}{3}}
\varphi^{8} p_{z}^{8} \sqrt[3]{27 \varphi^{8} p_{z}^{8}
+ 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4}
p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2}
+ 306 \varphi^{8} p_{z}^{8} + \sqrt[3]{2}
\sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4}\left(27\varphi^{8}p_{z}^{8}
+ 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4}
p_{z}^{4} - 4} -18\varphi^{4}p_{z}^{4} + 2\right)^{\frac{2}{3}}
+ 2 \cdot 2^{\frac{2}{3}} \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} \sqrt[3]{27 \varphi^{8}
p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27
\varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2}
+ 6 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4}
p_{z}^{4} - 4} - 20 \sqrt[3]{2} \varphi^{4} p_{z}^{4} \left(27
\varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4}
+ 2\right)^{\frac{2}{3}} - 26 \cdot 2^{\frac{2}{3}}
\varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8}
+ 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4}
p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} - 64 \varphi^{4}
p_{z}^{4} + 2 \sqrt[3]{2}\left(27\varphi^{8}p_{z}^{8}+3\sqrt{3}
\varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18
\varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}}
+ 2 \cdot 2^{\frac{2}{3}} \sqrt[3]{27 \varphi^{8} p_{z}^{8} +
3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4}
- 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 4\right)}{72 \varphi^{4}
p_{z}^{5} \left(\frac{\sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3
\sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4}
- 4} - 18 \varphi^{4} p_{z}^{4} + 2}}{6 \varphi^{4} p_{z}^{4}
\sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6}
p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4}
p_{z}^{4} + 2} + 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} -
2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3}
\varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} -
18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} -2\sqrt[3]{27
\varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4}
- 18 \varphi^{4} p_{z}^{4} + 2}
- 2 \sqrt[3]{2}}\right)^{\frac{3}{2}} \sqrt[3]{27 \varphi^{8}
p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4}
+ 2} \left(- 54 \cdot 2^{\frac{2}{3}} \varphi^{12} p_{z}^{12}
- 6 \cdot 2^{\frac{2}{3}} \sqrt{3} \varphi^{10} p_{z}^{10}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} + 6 \varphi^{8} p_{z}^{8}
\left(27 \varphi^{8} p_{z}^{8} + 3\sqrt{3}\varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4}
+ 2\right)^{\frac{2}{3}} + 33 \sqrt[3]{2} \varphi^{8} p_{z}^{8}
\sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6}
p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4}
p_{z}^{4} + 2} + 78 \cdot 2^{\frac{2}{3}} \varphi^{8} p_{z}^{8}
+ \sqrt[3]{2} \sqrt{3} \varphi^{6} p_{z}^{6}\sqrt{27\varphi^{4}
p_{z}^{4} - 4} \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3}
\varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} -
18 \varphi^{4} p_{z}^{4} + 2}
+ 2 \cdot 2^{\frac{2}{3}} \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 12 \varphi^{4} p_{z}^{4}
\left(27 \varphi^{8} p_{z}^{8} + 3\sqrt{3}\varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4}
+ 2\right)^{\frac{2}{3}} - 18 \sqrt[3]{2} \varphi^{4} p_{z}^{4}
\sqrt[3]{27\varphi^{8}p_{z}^{8} + 3\sqrt{3}\varphi^{6}p_{z}^{6}
\sqrt{27\varphi^{4}p_{z}^{4} - 4} - 18\varphi^{4}p_{z}^{4} + 2}
- 24 \cdot 2^{\frac{2}{3}} \varphi^{4} p_{z}^{4}
+ 2 \left(27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6}
p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4}
p_{z}^{4} + 2\right)^{\frac{2}{3}} + 2 \sqrt[3]{2}
\sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6}
p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} -
18 \varphi^{4} p_{z}^{4} + 2} + 2\cdot 2^{\frac{2}{3}}\right)}`
idtx_rdotz_pz_varphi_eqn (`Equality`_):
:math:`{r}^z = \dfrac{\sqrt{6} \sqrt{\frac{- 12 \sqrt[3]{2}
\varphi^{4} p_{z}^{4} + 2^{\frac{2}{3}} \left(27 \varphi^{8}
p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27
\varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4}
+ 2\right)^{\frac{2}{3}} + 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8}
+ 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4}
p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} + 2
\sqrt[3]{2}}{6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8}
p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27
\varphi^{4} p_{z}^{4} - 4} - 18
\varphi^{4} p_{z}^{4} + 2} + 12 \sqrt[3]{2} \varphi^{4}
p_{z}^{4} - 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8} +
3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4}
p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} +
2\right)^{\frac{2}{3}} - 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8}
+ 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27
\varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2} - 2
\sqrt[3]{2}}} \left(27 \cdot 2^{\frac{2}{3}} \varphi^{8}
p_{z}^{8} + 3 \cdot 2^{\frac{2}{3}}
\sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4}
- 4} - 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4} \sqrt[3]{27
\varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4}
p_{z}^{4} + 2} - 18 \cdot 2^{\frac{2}{3}} \varphi^{4}
p_{z}^{4} + 2 \left(27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3}
\varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4} -
18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}} +
2 \sqrt[3]{2} \sqrt[3]{27
\varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4}
p_{z}^{4} + 2} + 2 \cdot 2^{\frac{2}{3}}\right)}
{72 \varphi^{4} p_{z}^{5} \left(\frac{\sqrt[3]{27 \varphi^{8}
p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27
\varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2}}
{6 \varphi^{4} p_{z}^{4} \sqrt[3]{27 \varphi^{8} p_{z}^{8}
+ 3 \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} -
18 \varphi^{4} p_{z}^{4} + 2} + 12 \sqrt[3]{2}
\varphi^{4} p_{z}^{4} - 2^{\frac{2}{3}} \left(27
\varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} -
18 \varphi^{4} p_{z}^{4} + 2\right)^{\frac{2}{3}}
- 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8} + 3 \sqrt{3}
\varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4} p_{z}^{4} - 4}
- 18 \varphi^{4} p_{z}^{4} + 2} -
2 \sqrt[3]{2}}\right)^{\frac{3}{2}} \sqrt[3]{27 \varphi^{8}
p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4}
p_{z}^{4} + 2} \left(- 6 \varphi^{4} p_{z}^{4} \sqrt[3]{27
\varphi^{8} p_{z}^{8} + 3 \sqrt{3} \varphi^{6} p_{z}^{6}
\sqrt{27 \varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4}
p_{z}^{4} + 2} - 12 \sqrt[3]{2} \varphi^{4} p_{z}^{4}
+ 2^{\frac{2}{3}} \left(27 \varphi^{8} p_{z}^{8}
+ 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27
\varphi^{4} p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4}
+ 2\right)^{\frac{2}{3}} + 2 \sqrt[3]{27 \varphi^{8} p_{z}^{8}
+ 3 \sqrt{3} \varphi^{6} p_{z}^{6} \sqrt{27 \varphi^{4}
p_{z}^{4} - 4} - 18 \varphi^{4} p_{z}^{4} + 2}
+ 2 \sqrt[3]{2}\right)}`
"""
logging.info("gme.core.idtx.define_idtx_fgtx_eqns")
# if self.eta_ == 2:
# pz_tanbeta_varphi_eqn = ( self.pz_p_beta_eqn
# .subs({p:self.p_varphi_beta_eqn.rhs})
# .subs({varphi_r(rvec):varphi})
# .subs({cos(beta):sqrt(1/(1+tan(beta)**2))})
# .subs({Abs(tan(beta)):tan(beta)})
# )
# tanbeta_pz_varphi_solns = solve( pz_tanbeta_varphi_eqn, tan(beta) )
# tanbeta_pz_varphi_eqn = Eq(tan(beta),
# ([soln for soln in tanbeta_pz_varphi_solns
# if Abs(im(N(soln.subs({varphi:1,pz:1}))))<1e-20][0]) )
# self.fgtx_tanbeta_pz_varphi_eqn = tanbeta_pz_varphi_eqn
# self.fgtx_cosbeta_pz_varphi_eqn
# = Eq(cos(beta)**2, 1/(1+tanbeta_pz_varphi_eqn.rhs**2))
# else:
eta_sub = {eta: self.eta_}
pz_cosbeta_varphi_tmp_eqn = (
self.pz_p_beta_eqn.subs({p: self.p_varphi_beta_eqn.rhs})
.subs({varphi_r(rvec): varphi})
.subs(eta_sub)
.subs({Abs(tan(beta)): Abs(sin(beta)) / Abs(cos(beta))})
.subs({Abs(cos(beta)): cos(beta), Abs(sin(beta)): sin(beta)})
.subs({sin(beta): sqrt(1 - cos(beta) ** 2)})
)
pz_cosbeta_varphi_eqn = Eq(
pz_cosbeta_varphi_tmp_eqn.lhs ** self.eta_, # __dbldenom
pz_cosbeta_varphi_tmp_eqn.rhs ** self.eta_,
) # __dbldenom
# New
# pz_cosbeta_varphi_eqn \
# = (self.pz_varphi_beta_eqn
# .subs({Abs(sin(beta)**self.eta_):sin(beta)**self.eta_})
# .subs({varphi_r(rvec):varphi})
# .subs({sin(beta):sqrt(1-cos(beta)**2)}))
# cosbeta_pz_varphi_soln
# = (solve( pz_cosbeta_varphi_eqn, cos(beta)**2 ))[0]
self.pz_cosbeta_varphi_eqn = pz_cosbeta_varphi_eqn
self.cosbeta_pz_varphi_solns = None
self.cosbeta_pz_varphi_soln = None
self.fgtx_cosbeta_pz_varphi_eqn = None
self.fgtx_tanbeta_pz_varphi_eqn = None
self.fgtx_px_pz_varphi_eqn = None
self.idtx_rdotx_pz_varphi_eqn = None
self.idtx_rdotz_pz_varphi_eqn = None
self.cosbeta_pz_varphi_solns = solve(
self.pz_cosbeta_varphi_eqn, cos(beta)
)
# self.cosbetasqrd_pz_varphi_solns \
# = solve(self.pz_cosbeta_varphi_eqn, cos(beta)**2)
if (
self.eta_ == Rational(1, 4) or self.eta_ == Rational(3, 2)
) and self.beta_type == "tan":
print(
"Cannot compute all indicatrix equations for "
+ rf"$\tan\beta$ model and $\eta={self.eta_}$"
)
return
def find_cosbeta_root(sub):
# logging.info([
# soln.subs(sub) for soln in self.cosbeta_pz_varphi_solns
# ])
rtn = [
soln
for soln in self.cosbeta_pz_varphi_solns
if Abs(im(N(soln.subs(sub)))) < 1e-20
and (re(N(soln.subs(sub)))) >= 0
]
# logging.info(rtn)
return rtn
self.cosbeta_pz_varphi_soln = find_cosbeta_root({varphi: 1, pz: -0.01})
if self.cosbeta_pz_varphi_soln == []:
self.cosbeta_pz_varphi_soln = find_cosbeta_root(
{varphi: 10, pz: -0.5}
)
self.fgtx_cosbeta_pz_varphi_eqn = Eq(
cos(beta), self.cosbeta_pz_varphi_soln[0]
)
self.fgtx_tanbeta_pz_varphi_eqn = Eq(
tan(beta), (1 / (self.fgtx_cosbeta_pz_varphi_eqn.rhs) - 1)
)
self.fgtx_px_pz_varphi_eqn = factor(
Eq(px, -pz * self.fgtx_tanbeta_pz_varphi_eqn.rhs)
)
g_xx = self.gstar_varphi_pxpz_eqn.rhs[0, 0]
g_zx = self.gstar_varphi_pxpz_eqn.rhs[1, 0]
g_xz = self.gstar_varphi_pxpz_eqn.rhs[0, 1]
g_zz = self.gstar_varphi_pxpz_eqn.rhs[1, 1]
self.idtx_rdotx_pz_varphi_eqn = factor(
Eq(
rx,
(g_xx * px + g_xz * pz).subs(
{px: self.fgtx_px_pz_varphi_eqn.rhs, varphi_r(rvec): varphi}
),
)
)
self.idtx_rdotz_pz_varphi_eqn = factor(
factor(
Eq(
rz,
(g_zx * px + g_zz * pz).subs(
{
px: self.fgtx_px_pz_varphi_eqn.rhs,
varphi_r(rvec): varphi,
}
),
)
)
)
#
