Resolving circular imports in physics.vector

[tjstienstra]

There are various circular dependencies in the sympy.physics.vector module. This was also pointed out in #26384, where a new circular import will be added.
The aim of this issue is to see whether it might be possible to reduce the number of circular.
As an overview, I've made a (mostly) complete list of all circular imports.

vector.py:

• No import from physics.vector at the start of the module.
• Vector.dot: from .dyadic import Dyadic, _check_dyadic both used for the same type check (Dyadic could be removed)
• Vector.outer: from .dyadic import Dyadic uses Dyadic(...)
• Vector._pretty: from sympy.printing.pretty.stringpict import prettyForm
• Vector.cross: from .dyadic import Dyadic, _check_dyadic both used for the same type check and uses Dyadic(0)
• Vector.diff: from sympy.physics.vector.frame import _check_frame
• Vector.express: from .functions import express used as forward call
• Vector.dt: from .functions import time_derivative used as forward call
• Vector.free_dynamicsymbols: from sympy.physics.mechanics.functions import find_dynamicsymbols

dyadic.py:

• No import from physics.vector at the start of the module.
• Dyadic.dot: from .vector import Vector, _check_vector used for type checking and creation of Vector(0)
• Dyadic.cross: from .vector import _check_vector used for type checking
• Dyadic.express: from .functions import express used as forward call
• Dyadic.dt: from .functions import time_derivative used as forward call

frame.py:

• from .vector import Vector, _check_vector at the top of the module. Utilized abundantly.
• ReferenceFrame._w_diff_dcm: from .functions import dynamicsymbols utilized to get the time symbol
• ReferenceFrame.orient_axis: from .functions import dynamicsymbols utilized to get the time symbol
• ReferenceFrame.orient_body_fixed: from .functions import dynamicsymbols utilized to get the time symbol
• ReferenceFrame.orient_space_fixed: from .functions import dynamicsymbols utilized to get the time symbol
• ReferenceFrame.orient_quaternion: from .functions import dynamicsymbols utilized to get the time symbol
• _check_frame: from .vector import VectorTypeError to raise an exception (this one could be moved to the top of the module)
• ReferenceFrame.partial_velocity: from .functions import partial_velocity (will be introduced in physics/vector: Improved speed of Point/Frame partial_velocity methods #26384)

point.py:

• Import various at the top of the module:
  • from .vector import Vector, _check_vector used for type checking and creation of Vector(0)
  • from .frame import _check_frame
• Point.partial_velocity: from .functions import partial_velocity (will be introduced in physics/vector: Improved speed of Point/Frame partial_velocity methods #26384)

functions.py:

• Imports various at the top of the module:
  • from .vector import Vector, _check_vector
  • from .frame import CoordinateSym, _check_frame
  • from .dyadic import Dyadic
• No circular imports
• cross uses Vector and Dyadic for type checking
• dot uses Vector and Dyadic for type checking
• express uses _check_frame, CoordinateSym, Dyadic, and Vector for type checking. Vector(...), Vector.simp, and Dyadic(0) are used.
• time_derivative uses _check_frame, Vector, and Dyadic for type checking. Vector(...) and Dyadic(0) are used.
• outer uses Vector for type checking
• kinematic_equations uses none
• get_motion_params uses _check_vector, _check_frame, and Vector(0)
• partial_velocity uses Vector(0)
• dynamicsymbols uses none (even no other functions from functions.py)

Some preliminary conclusions:

• Vector and Dyadic form the basis of the module on which everything depends. ReferenceFrame is next, followed by Point. functions should be a gathering place of overarching functions.
• Several of the above could be resolved if we would move _check_... to a separate module without them using any isinstance checks.
• We could/should move dynamicsymbols and find_dynamicsymbols also to a separate module.

[tjstienstra]

• Vector and Dyadic form the basis of the module on which everything depends. ReferenceFrame is next, followed by Point. functions should be a gathering place of overarching functions.

As function.py should be seen as a gathering place of overarching functions, I would also suggest having the type-specific implementation of express and time_derivative to Vector and Dyadic.

[moorepants]

Yes, we should just be able to make express and dt methods in Vector and Dyadic and they be the primary implementations. Then the express and time_derivative in functions.py are just wrappers around the methods.

[tjstienstra]

I just noticed that printing.py actually also has a several local imports. Multiple from functions.py because it requires dynamicsymbols._t.