AndroidRemoteController/python/py/Lib/site-packages/sympy/physics/mechanics/tests/test_pathway.py

"""Tests for the ``sympy.physics.mechanics.pathway.py`` module."""

import pytest

from sympy import (
    Rational,
    Symbol,
    cos,
    pi,
    sin,
    sqrt,
)
from sympy.physics.mechanics import (
    Force,
    LinearPathway,
    ObstacleSetPathway,
    PathwayBase,
    Point,
    ReferenceFrame,
    WrappingCylinder,
    WrappingGeometryBase,
    WrappingPathway,
    WrappingSphere,
    dynamicsymbols,
)
from sympy.simplify.simplify import simplify


def _simplify_loads(loads):
    return [
        load.__class__(load.location, load.vector.simplify())
        for load in loads
    ]


class TestLinearPathway:

    def test_is_pathway_base_subclass(self):
        assert issubclass(LinearPathway, PathwayBase)

    @staticmethod
    @pytest.mark.parametrize(
        'args, kwargs',
        [
            ((Point('pA'), Point('pB')), {}),
        ]
    )
    def test_valid_constructor(args, kwargs):
        pointA, pointB = args
        instance = LinearPathway(*args, **kwargs)
        assert isinstance(instance, LinearPathway)
        assert hasattr(instance, 'attachments')
        assert len(instance.attachments) == 2
        assert instance.attachments[0] is pointA
        assert instance.attachments[1] is pointB
        assert isinstance(instance.attachments[0], Point)
        assert instance.attachments[0].name == 'pA'
        assert isinstance(instance.attachments[1], Point)
        assert instance.attachments[1].name == 'pB'

    @staticmethod
    @pytest.mark.parametrize(
        'attachments',
        [
            (Point('pA'), ),
            (Point('pA'), Point('pB'), Point('pZ')),
        ]
    )
    def test_invalid_attachments_incorrect_number(attachments):
        with pytest.raises(ValueError):
            _ = LinearPathway(*attachments)

    @staticmethod
    @pytest.mark.parametrize(
        'attachments',
        [
            (None, Point('pB')),
            (Point('pA'), None),
        ]
    )
    def test_invalid_attachments_not_point(attachments):
        with pytest.raises(TypeError):
            _ = LinearPathway(*attachments)

    @pytest.fixture(autouse=True)
    def _linear_pathway_fixture(self):
        self.N = ReferenceFrame('N')
        self.pA = Point('pA')
        self.pB = Point('pB')
        self.pathway = LinearPathway(self.pA, self.pB)
        self.q1 = dynamicsymbols('q1')
        self.q2 = dynamicsymbols('q2')
        self.q3 = dynamicsymbols('q3')
        self.q1d = dynamicsymbols('q1', 1)
        self.q2d = dynamicsymbols('q2', 1)
        self.q3d = dynamicsymbols('q3', 1)
        self.F = Symbol('F')

    def test_properties_are_immutable(self):
        instance = LinearPathway(self.pA, self.pB)
        with pytest.raises(AttributeError):
            instance.attachments = None
        with pytest.raises(TypeError):
            instance.attachments[0] = None
        with pytest.raises(TypeError):
            instance.attachments[1] = None

    def test_repr(self):
        pathway = LinearPathway(self.pA, self.pB)
        expected = 'LinearPathway(pA, pB)'
        assert repr(pathway) == expected

    def test_static_pathway_length(self):
        self.pB.set_pos(self.pA, 2*self.N.x)
        assert self.pathway.length == 2

    def test_static_pathway_extension_velocity(self):
        self.pB.set_pos(self.pA, 2*self.N.x)
        assert self.pathway.extension_velocity == 0

    def test_static_pathway_to_loads(self):
        self.pB.set_pos(self.pA, 2*self.N.x)
        expected = [
            (self.pA, - self.F*self.N.x),
            (self.pB, self.F*self.N.x),
        ]
        assert self.pathway.to_loads(self.F) == expected

    def test_2D_pathway_length(self):
        self.pB.set_pos(self.pA, 2*self.q1*self.N.x)
        expected = 2*sqrt(self.q1**2)
        assert self.pathway.length == expected

    def test_2D_pathway_extension_velocity(self):
        self.pB.set_pos(self.pA, 2*self.q1*self.N.x)
        expected = 2*sqrt(self.q1**2)*self.q1d/self.q1
        assert self.pathway.extension_velocity == expected

    def test_2D_pathway_to_loads(self):
        self.pB.set_pos(self.pA, 2*self.q1*self.N.x)
        expected = [
            (self.pA, - self.F*(self.q1 / sqrt(self.q1**2))*self.N.x),
            (self.pB, self.F*(self.q1 / sqrt(self.q1**2))*self.N.x),
        ]
        assert self.pathway.to_loads(self.F) == expected

    def test_3D_pathway_length(self):
        self.pB.set_pos(
            self.pA,
            self.q1*self.N.x - self.q2*self.N.y + 2*self.q3*self.N.z,
        )
        expected = sqrt(self.q1**2 + self.q2**2 + 4*self.q3**2)
        assert simplify(self.pathway.length - expected) == 0

    def test_3D_pathway_extension_velocity(self):
        self.pB.set_pos(
            self.pA,
            self.q1*self.N.x - self.q2*self.N.y + 2*self.q3*self.N.z,
        )
        length = sqrt(self.q1**2 + self.q2**2 + 4*self.q3**2)
        expected = (
            self.q1*self.q1d/length
            + self.q2*self.q2d/length
            + 4*self.q3*self.q3d/length
        )
        assert simplify(self.pathway.extension_velocity - expected) == 0

    def test_3D_pathway_to_loads(self):
        self.pB.set_pos(
            self.pA,
            self.q1*self.N.x - self.q2*self.N.y + 2*self.q3*self.N.z,
        )
        length = sqrt(self.q1**2 + self.q2**2 + 4*self.q3**2)
        pO_force = (
            - self.F*self.q1*self.N.x/length
            + self.F*self.q2*self.N.y/length
            - 2*self.F*self.q3*self.N.z/length
        )
        pI_force = (
            self.F*self.q1*self.N.x/length
            - self.F*self.q2*self.N.y/length
            + 2*self.F*self.q3*self.N.z/length
        )
        expected = [
            (self.pA, pO_force),
            (self.pB, pI_force),
        ]
        assert self.pathway.to_loads(self.F) == expected


class TestObstacleSetPathway:

    def test_is_pathway_base_subclass(self):
        assert issubclass(ObstacleSetPathway, PathwayBase)

    @staticmethod
    @pytest.mark.parametrize(
        'num_attachments, attachments',
        [
            (3, [Point(name) for name in ('pO', 'pA', 'pI')]),
            (4, [Point(name) for name in ('pO', 'pA', 'pB', 'pI')]),
            (5, [Point(name) for name in ('pO', 'pA', 'pB', 'pC', 'pI')]),
            (6, [Point(name) for name in ('pO', 'pA', 'pB', 'pC', 'pD', 'pI')]),
        ]
    )
    def test_valid_constructor(num_attachments, attachments):
        instance = ObstacleSetPathway(*attachments)
        assert isinstance(instance, ObstacleSetPathway)
        assert hasattr(instance, 'attachments')
        assert len(instance.attachments) == num_attachments
        for attachment in instance.attachments:
            assert isinstance(attachment, Point)

    @staticmethod
    @pytest.mark.parametrize(
        'attachments',
        [[Point('pO')], [Point('pO'), Point('pI')]],
    )
    def test_invalid_constructor_attachments_incorrect_number(attachments):
        with pytest.raises(ValueError):
            _ = ObstacleSetPathway(*attachments)

    @staticmethod
    @pytest.mark.parametrize(
        'attachments',
        [
            (None, Point('pA'), Point('pI')),
            (Point('pO'), None, Point('pI')),
            (Point('pO'), Point('pA'), None),
        ]
    )
    def test_invalid_constructor_attachments_not_point(attachments):
        with pytest.raises(TypeError):
            _ = WrappingPathway(*attachments)  # type: ignore

    def test_properties_are_immutable(self):
        pathway = ObstacleSetPathway(Point('pO'), Point('pA'), Point('pI'))
        with pytest.raises(AttributeError):
            pathway.attachments = None  # type: ignore
        with pytest.raises(TypeError):
            pathway.attachments[0] = None  # type: ignore
        with pytest.raises(TypeError):
            pathway.attachments[1] = None  # type: ignore
        with pytest.raises(TypeError):
            pathway.attachments[-1] = None  # type: ignore

    @staticmethod
    @pytest.mark.parametrize(
        'attachments, expected',
        [
            (
                [Point(name) for name in ('pO', 'pA', 'pI')],
                'ObstacleSetPathway(pO, pA, pI)'
            ),
            (
                [Point(name) for name in ('pO', 'pA', 'pB', 'pI')],
                'ObstacleSetPathway(pO, pA, pB, pI)'
            ),
            (
                [Point(name) for name in ('pO', 'pA', 'pB', 'pC', 'pI')],
                'ObstacleSetPathway(pO, pA, pB, pC, pI)'
            ),
        ]
    )
    def test_repr(attachments, expected):
        pathway = ObstacleSetPathway(*attachments)
        assert repr(pathway) == expected

    @pytest.fixture(autouse=True)
    def _obstacle_set_pathway_fixture(self):
        self.N = ReferenceFrame('N')
        self.pO = Point('pO')
        self.pI = Point('pI')
        self.pA = Point('pA')
        self.pB = Point('pB')
        self.q = dynamicsymbols('q')
        self.qd = dynamicsymbols('q', 1)
        self.F = Symbol('F')

    def test_static_pathway_length(self):
        self.pA.set_pos(self.pO, self.N.x)
        self.pB.set_pos(self.pO, self.N.y)
        self.pI.set_pos(self.pO, self.N.z)
        pathway = ObstacleSetPathway(self.pO, self.pA, self.pB, self.pI)
        assert pathway.length == 1 + 2 * sqrt(2)

    def test_static_pathway_extension_velocity(self):
        self.pA.set_pos(self.pO, self.N.x)
        self.pB.set_pos(self.pO, self.N.y)
        self.pI.set_pos(self.pO, self.N.z)
        pathway = ObstacleSetPathway(self.pO, self.pA, self.pB, self.pI)
        assert pathway.extension_velocity == 0

    def test_static_pathway_to_loads(self):
        self.pA.set_pos(self.pO, self.N.x)
        self.pB.set_pos(self.pO, self.N.y)
        self.pI.set_pos(self.pO, self.N.z)
        pathway = ObstacleSetPathway(self.pO, self.pA, self.pB, self.pI)
        expected = [
            Force(self.pO, -self.F * self.N.x),
            Force(self.pA, self.F * self.N.x),
            Force(self.pA, self.F * sqrt(2) / 2 * (self.N.x - self.N.y)),
            Force(self.pB, self.F * sqrt(2) / 2 * (self.N.y - self.N.x)),
            Force(self.pB, self.F * sqrt(2) / 2 * (self.N.y - self.N.z)),
            Force(self.pI, self.F * sqrt(2) / 2 * (self.N.z - self.N.y)),
        ]
        assert pathway.to_loads(self.F) == expected

    def test_2D_pathway_length(self):
        self.pA.set_pos(self.pO, -(self.N.x + self.N.y))
        self.pB.set_pos(
            self.pO, cos(self.q) * self.N.x - (sin(self.q) + 1) * self.N.y
        )
        self.pI.set_pos(
            self.pO, sin(self.q) * self.N.x + (cos(self.q) - 1) * self.N.y
        )
        pathway = ObstacleSetPathway(self.pO, self.pA, self.pB, self.pI)
        expected = 2 * sqrt(2) + sqrt(2 + 2*cos(self.q))
        assert (pathway.length - expected).simplify() == 0

    def test_2D_pathway_extension_velocity(self):
        self.pA.set_pos(self.pO, -(self.N.x + self.N.y))
        self.pB.set_pos(
            self.pO, cos(self.q) * self.N.x - (sin(self.q) + 1) * self.N.y
        )
        self.pI.set_pos(
            self.pO, sin(self.q) * self.N.x + (cos(self.q) - 1) * self.N.y
        )
        pathway = ObstacleSetPathway(self.pO, self.pA, self.pB, self.pI)
        expected = - (sqrt(2) * sin(self.q) * self.qd) / (2 * sqrt(cos(self.q) + 1))
        assert (pathway.extension_velocity - expected).simplify() == 0

    def test_2D_pathway_to_loads(self):
        self.pA.set_pos(self.pO, -(self.N.x + self.N.y))
        self.pB.set_pos(
            self.pO, cos(self.q) * self.N.x - (sin(self.q) + 1) * self.N.y
        )
        self.pI.set_pos(
            self.pO, sin(self.q) * self.N.x + (cos(self.q) - 1) * self.N.y
        )
        pathway = ObstacleSetPathway(self.pO, self.pA, self.pB, self.pI)
        pO_pA_force_vec = sqrt(2) / 2 * (self.N.x + self.N.y)
        pA_pB_force_vec = (
            - sqrt(2 * cos(self.q) + 2) / 2 * self.N.x
            + sqrt(2) * sin(self.q) / (2 * sqrt(cos(self.q) + 1)) * self.N.y
        )
        pB_pI_force_vec = cos(self.q + pi/4) * self.N.x - sin(self.q + pi/4) * self.N.y
        expected = [
            Force(self.pO, self.F * pO_pA_force_vec),
            Force(self.pA, -self.F * pO_pA_force_vec),
            Force(self.pA, self.F * pA_pB_force_vec),
            Force(self.pB, -self.F * pA_pB_force_vec),
            Force(self.pB, self.F * pB_pI_force_vec),
            Force(self.pI, -self.F * pB_pI_force_vec),
        ]
        assert _simplify_loads(pathway.to_loads(self.F)) == expected


class TestWrappingPathway:

    def test_is_pathway_base_subclass(self):
        assert issubclass(WrappingPathway, PathwayBase)

    @pytest.fixture(autouse=True)
    def _wrapping_pathway_fixture(self):
        self.pA = Point('pA')
        self.pB = Point('pB')
        self.r = Symbol('r', positive=True)
        self.pO = Point('pO')
        self.N = ReferenceFrame('N')
        self.ax = self.N.z
        self.sphere = WrappingSphere(self.r, self.pO)
        self.cylinder = WrappingCylinder(self.r, self.pO, self.ax)
        self.pathway = WrappingPathway(self.pA, self.pB, self.cylinder)
        self.F = Symbol('F')

    def test_valid_constructor(self):
        instance = WrappingPathway(self.pA, self.pB, self.cylinder)
        assert isinstance(instance, WrappingPathway)
        assert hasattr(instance, 'attachments')
        assert len(instance.attachments) == 2
        assert isinstance(instance.attachments[0], Point)
        assert instance.attachments[0] == self.pA
        assert isinstance(instance.attachments[1], Point)
        assert instance.attachments[1] == self.pB
        assert hasattr(instance, 'geometry')
        assert isinstance(instance.geometry, WrappingGeometryBase)
        assert instance.geometry == self.cylinder

    @pytest.mark.parametrize(
        'attachments',
        [
            (Point('pA'), ),
            (Point('pA'), Point('pB'), Point('pZ')),
        ]
    )
    def test_invalid_constructor_attachments_incorrect_number(self, attachments):
        with pytest.raises(TypeError):
            _ = WrappingPathway(*attachments, self.cylinder)

    @staticmethod
    @pytest.mark.parametrize(
        'attachments',
        [
            (None, Point('pB')),
            (Point('pA'), None),
        ]
    )
    def test_invalid_constructor_attachments_not_point(attachments):
        with pytest.raises(TypeError):
            _ = WrappingPathway(*attachments)

    def test_invalid_constructor_geometry_is_not_supplied(self):
        with pytest.raises(TypeError):
            _ = WrappingPathway(self.pA, self.pB)

    @pytest.mark.parametrize(
        'geometry',
        [
            Symbol('r'),
            dynamicsymbols('q'),
            ReferenceFrame('N'),
            ReferenceFrame('N').x,
        ]
    )
    def test_invalid_geometry_not_geometry(self, geometry):
        with pytest.raises(TypeError):
            _ = WrappingPathway(self.pA, self.pB, geometry)

    def test_attachments_property_is_immutable(self):
        with pytest.raises(TypeError):
            self.pathway.attachments[0] = self.pB
        with pytest.raises(TypeError):
            self.pathway.attachments[1] = self.pA

    def test_geometry_property_is_immutable(self):
        with pytest.raises(AttributeError):
            self.pathway.geometry = None

    def test_repr(self):
        expected = (
            f'WrappingPathway(pA, pB, '
            f'geometry={self.cylinder!r})'
        )
        assert repr(self.pathway) == expected

    @staticmethod
    def _expand_pos_to_vec(pos, frame):
        return sum(mag*unit for (mag, unit) in zip(pos, frame))

    @pytest.mark.parametrize(
        'pA_vec, pB_vec, factor',
        [
            ((1, 0, 0), (0, 1, 0), pi/2),
            ((0, 1, 0), (sqrt(2)/2, -sqrt(2)/2, 0), 3*pi/4),
            ((1, 0, 0), (Rational(1, 2), sqrt(3)/2, 0), pi/3),
        ]
    )
    def test_static_pathway_on_sphere_length(self, pA_vec, pB_vec, factor):
        pA_vec = self._expand_pos_to_vec(pA_vec, self.N)
        pB_vec = self._expand_pos_to_vec(pB_vec, self.N)
        self.pA.set_pos(self.pO, self.r*pA_vec)
        self.pB.set_pos(self.pO, self.r*pB_vec)
        pathway = WrappingPathway(self.pA, self.pB, self.sphere)
        expected = factor*self.r
        assert simplify(pathway.length - expected) == 0

    @pytest.mark.parametrize(
        'pA_vec, pB_vec, factor',
        [
            ((1, 0, 0), (0, 1, 0), Rational(1, 2)*pi),
            ((1, 0, 0), (-1, 0, 0), pi),
            ((-1, 0, 0), (1, 0, 0), pi),
            ((0, 1, 0), (sqrt(2)/2, -sqrt(2)/2, 0), 5*pi/4),
            ((1, 0, 0), (Rational(1, 2), sqrt(3)/2, 0), pi/3),
            (
                (0, 1, 0),
                (sqrt(2)*Rational(1, 2), -sqrt(2)*Rational(1, 2), 1),
                sqrt(1 + (Rational(5, 4)*pi)**2),
            ),
            (
                (1, 0, 0),
                (Rational(1, 2), sqrt(3)*Rational(1, 2), 1),
                sqrt(1 + (Rational(1, 3)*pi)**2),
            ),
        ]
    )
    def test_static_pathway_on_cylinder_length(self, pA_vec, pB_vec, factor):
        pA_vec = self._expand_pos_to_vec(pA_vec, self.N)
        pB_vec = self._expand_pos_to_vec(pB_vec, self.N)
        self.pA.set_pos(self.pO, self.r*pA_vec)
        self.pB.set_pos(self.pO, self.r*pB_vec)
        pathway = WrappingPathway(self.pA, self.pB, self.cylinder)
        expected = factor*sqrt(self.r**2)
        assert simplify(pathway.length - expected) == 0

    @pytest.mark.parametrize(
        'pA_vec, pB_vec',
        [
            ((1, 0, 0), (0, 1, 0)),
            ((0, 1, 0), (sqrt(2)*Rational(1, 2), -sqrt(2)*Rational(1, 2), 0)),
            ((1, 0, 0), (Rational(1, 2), sqrt(3)*Rational(1, 2), 0)),
        ]
    )
    def test_static_pathway_on_sphere_extension_velocity(self, pA_vec, pB_vec):
        pA_vec = self._expand_pos_to_vec(pA_vec, self.N)
        pB_vec = self._expand_pos_to_vec(pB_vec, self.N)
        self.pA.set_pos(self.pO, self.r*pA_vec)
        self.pB.set_pos(self.pO, self.r*pB_vec)
        pathway = WrappingPathway(self.pA, self.pB, self.sphere)
        assert pathway.extension_velocity == 0

    @pytest.mark.parametrize(
        'pA_vec, pB_vec',
        [
            ((1, 0, 0), (0, 1, 0)),
            ((1, 0, 0), (-1, 0, 0)),
            ((-1, 0, 0), (1, 0, 0)),
            ((0, 1, 0), (sqrt(2)/2, -sqrt(2)/2, 0)),
            ((1, 0, 0), (Rational(1, 2), sqrt(3)/2, 0)),
            ((0, 1, 0), (sqrt(2)*Rational(1, 2), -sqrt(2)/2, 1)),
            ((1, 0, 0), (Rational(1, 2), sqrt(3)/2, 1)),
        ]
    )
    def test_static_pathway_on_cylinder_extension_velocity(self, pA_vec, pB_vec):
        pA_vec = self._expand_pos_to_vec(pA_vec, self.N)
        pB_vec = self._expand_pos_to_vec(pB_vec, self.N)
        self.pA.set_pos(self.pO, self.r*pA_vec)
        self.pB.set_pos(self.pO, self.r*pB_vec)
        pathway = WrappingPathway(self.pA, self.pB, self.cylinder)
        assert pathway.extension_velocity == 0

    @pytest.mark.parametrize(
        'pA_vec, pB_vec, pA_vec_expected, pB_vec_expected, pO_vec_expected',
        (
            ((1, 0, 0), (0, 1, 0), (0, 1, 0), (1, 0, 0), (-1, -1, 0)),
            (
                (0, 1, 0),
                (sqrt(2)/2, -sqrt(2)/2, 0),
                (1, 0, 0),
                (sqrt(2)/2, sqrt(2)/2, 0),
                (-1 - sqrt(2)/2, -sqrt(2)/2, 0)
            ),
            (
                (1, 0, 0),
                (Rational(1, 2), sqrt(3)/2, 0),
                (0, 1, 0),
                (sqrt(3)/2, -Rational(1, 2), 0),
                (-sqrt(3)/2, Rational(1, 2) - 1, 0),
            ),
        )
    )
    def test_static_pathway_on_sphere_to_loads(
        self,
        pA_vec,
        pB_vec,
        pA_vec_expected,
        pB_vec_expected,
        pO_vec_expected,
    ):
        pA_vec = self._expand_pos_to_vec(pA_vec, self.N)
        pB_vec = self._expand_pos_to_vec(pB_vec, self.N)
        self.pA.set_pos(self.pO, self.r*pA_vec)
        self.pB.set_pos(self.pO, self.r*pB_vec)
        pathway = WrappingPathway(self.pA, self.pB, self.sphere)

        pA_vec_expected = sum(
            mag*unit for (mag, unit) in zip(pA_vec_expected, self.N)
        )
        pB_vec_expected = sum(
            mag*unit for (mag, unit) in zip(pB_vec_expected, self.N)
        )
        pO_vec_expected = sum(
            mag*unit for (mag, unit) in zip(pO_vec_expected, self.N)
        )
        expected = [
            Force(self.pA, self.F*(self.r**3/sqrt(self.r**6))*pA_vec_expected),
            Force(self.pB, self.F*(self.r**3/sqrt(self.r**6))*pB_vec_expected),
            Force(self.pO, self.F*(self.r**3/sqrt(self.r**6))*pO_vec_expected),
        ]
        assert pathway.to_loads(self.F) == expected

    @pytest.mark.parametrize(
        'pA_vec, pB_vec, pA_vec_expected, pB_vec_expected, pO_vec_expected',
        (
            ((1, 0, 0), (0, 1, 0), (0, 1, 0), (1, 0, 0), (-1, -1, 0)),
            ((1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, 1, 0), (0, -2, 0)),
            ((-1, 0, 0), (1, 0, 0), (0, -1, 0), (0, -1, 0), (0, 2, 0)),
            (
                (0, 1, 0),
                (sqrt(2)/2, -sqrt(2)/2, 0),
                (-1, 0, 0),
                (-sqrt(2)/2, -sqrt(2)/2, 0),
                (1 + sqrt(2)/2, sqrt(2)/2, 0)
            ),
            (
                (1, 0, 0),
                (Rational(1, 2), sqrt(3)/2, 0),
                (0, 1, 0),
                (sqrt(3)/2, -Rational(1, 2), 0),
                (-sqrt(3)/2, Rational(1, 2) - 1, 0),
            ),
            (
                (1, 0, 0),
                (sqrt(2)/2, sqrt(2)/2, 0),
                (0, 1, 0),
                (sqrt(2)/2, -sqrt(2)/2, 0),
                (-sqrt(2)/2, sqrt(2)/2 - 1, 0),
            ),
            ((0, 1, 0), (0, 1, 1), (0, 0, 1), (0, 0, -1), (0, 0, 0)),
            (
                (0, 1, 0),
                (sqrt(2)/2, -sqrt(2)/2, 1),
                (-5*pi/sqrt(16 + 25*pi**2), 0, 4/sqrt(16 + 25*pi**2)),
                (
                    -5*sqrt(2)*pi/(2*sqrt(16 + 25*pi**2)),
                    -5*sqrt(2)*pi/(2*sqrt(16 + 25*pi**2)),
                    -4/sqrt(16 + 25*pi**2),
                ),
                (
                    5*(sqrt(2) + 2)*pi/(2*sqrt(16 + 25*pi**2)),
                    5*sqrt(2)*pi/(2*sqrt(16 + 25*pi**2)),
                    0,
                ),
            ),
        )
    )
    def test_static_pathway_on_cylinder_to_loads(
        self,
        pA_vec,
        pB_vec,
        pA_vec_expected,
        pB_vec_expected,
        pO_vec_expected,
    ):
        pA_vec = self._expand_pos_to_vec(pA_vec, self.N)
        pB_vec = self._expand_pos_to_vec(pB_vec, self.N)
        self.pA.set_pos(self.pO, self.r*pA_vec)
        self.pB.set_pos(self.pO, self.r*pB_vec)
        pathway = WrappingPathway(self.pA, self.pB, self.cylinder)

        pA_force_expected = self.F*self._expand_pos_to_vec(pA_vec_expected,
                                                           self.N)
        pB_force_expected = self.F*self._expand_pos_to_vec(pB_vec_expected,
                                                           self.N)
        pO_force_expected = self.F*self._expand_pos_to_vec(pO_vec_expected,
                                                           self.N)
        expected = [
            Force(self.pA, pA_force_expected),
            Force(self.pB, pB_force_expected),
            Force(self.pO, pO_force_expected),
        ]
        assert _simplify_loads(pathway.to_loads(self.F)) == expected

    def test_2D_pathway_on_cylinder_length(self):
        q = dynamicsymbols('q')
        pA_pos = self.r*self.N.x
        pB_pos = self.r*(cos(q)*self.N.x + sin(q)*self.N.y)
        self.pA.set_pos(self.pO, pA_pos)
        self.pB.set_pos(self.pO, pB_pos)
        expected = self.r*sqrt(q**2)
        assert simplify(self.pathway.length - expected) == 0

    def test_2D_pathway_on_cylinder_extension_velocity(self):
        q = dynamicsymbols('q')
        qd = dynamicsymbols('q', 1)
        pA_pos = self.r*self.N.x
        pB_pos = self.r*(cos(q)*self.N.x + sin(q)*self.N.y)
        self.pA.set_pos(self.pO, pA_pos)
        self.pB.set_pos(self.pO, pB_pos)
        expected = self.r*(sqrt(q**2)/q)*qd
        assert simplify(self.pathway.extension_velocity - expected) == 0

    def test_2D_pathway_on_cylinder_to_loads(self):
        q = dynamicsymbols('q')
        pA_pos = self.r*self.N.x
        pB_pos = self.r*(cos(q)*self.N.x + sin(q)*self.N.y)
        self.pA.set_pos(self.pO, pA_pos)
        self.pB.set_pos(self.pO, pB_pos)

        pA_force = self.F*self.N.y
        pB_force = self.F*(sin(q)*self.N.x - cos(q)*self.N.y)
        pO_force = self.F*(-sin(q)*self.N.x + (cos(q) - 1)*self.N.y)
        expected = [
            Force(self.pA, pA_force),
            Force(self.pB, pB_force),
            Force(self.pO, pO_force),
        ]

        loads = _simplify_loads(self.pathway.to_loads(self.F))
        assert loads == expected