def try_numerical_answer(question: str) -> Optional[Tuple[str, str]]:
    if question.endswith('?') or question.endswith('.') or question.endswith('!'):
        question = question[:-1]
    words = NON_WORD_CHARS.split(question)
    starting_idx = None
    for idx, word in enumerate(words):
        if NUMERICAL_EXPRESSION_STARTER.match(word):
            starting_idx = idx
            break
    if starting_idx is None:
        return
    words = words[starting_idx:]
    length = len(words)
    ending_idx = None
    for idx, word in enumerate(reversed(words)):
        if NUMERICAL_EXPRESSION_ENDER.match("".join(reversed(word))):
            ending_idx = length - idx
            break
    expression = " ".join(words[:ending_idx])
    try:
        result = numexpr.evaluate(expression, local_dict={'pi': math.pi, 'tau': math.tau, 'e': math.e}, global_dict={})
    except Exception:
        return
    return expression, str(result) 

def cubic_bezier_from_arc(
        center: Vector = (0, 0), radius: float = 1, start_angle: float = 0, end_angle: float = 360,
        segments: int = 1) -> Iterable[Bezier4P]:
    """
    Returns an approximation for a circular 2D arc by multiple cubic Bézier curves.

    Args:
        center: circle center as :class:`Vector` compatible object
        radius: circle radius
        start_angle: start angle in degrees
        end_angle: end angle in degrees
        segments: count of spline segments, at least one segment for each quarter (90 deg), ``1`` for as few as needed.

    .. versionadded:: 0.13

    """
    center = Vector(center)
    radius = float(radius)
    start_angle = math.radians(start_angle) % math.tau
    end_angle = math.radians(end_angle) % math.tau
    for control_points in cubic_bezier_arc_parameters(start_angle, end_angle, segments):
        defpoints = [center + (p * radius) for p in control_points]
        yield Bezier4P(defpoints) 

def cubic_bezier_from_ellipse(ellipse: 'ConstructionEllipse', segments: int = 1) -> Iterable[Bezier4P]:
    """
    Returns an approximation for an elliptic arc by multiple cubic Bézier curves.

    Args:
        ellipse: ellipse parameters as :class:`~ezdxf.math.ConstructionEllipse` object
        segments: count of spline segments, at least one segment for each quarter (pi/2), ``1`` for as few as needed.

    .. versionadded:: 0.13

    """
    from ezdxf.math import param_to_angle
    start_angle = param_to_angle(ellipse.ratio, ellipse.start_param) % math.tau
    end_angle = param_to_angle(ellipse.ratio, ellipse.end_param) % math.tau

    def transform(points: Iterable[Vector]) -> Iterable[Vector]:
        center = Vector(ellipse.center)
        x_axis = ellipse.major_axis
        y_axis = ellipse.minor_axis
        for p in points:
            yield center + x_axis * p.x + y_axis * p.y

    for defpoints in cubic_bezier_arc_parameters(start_angle, end_angle, segments):
        yield Bezier4P(tuple(transform(defpoints))) 

def test_swap_axis_arbitrary_params():
    random_tests_count = 100
    random.seed(0)

    for _ in range(random_tests_count):
        ellipse = ConstructionEllipse(
            # avoid (0, 0, 0) as major axis
            major_axis=(non_zero_random(), non_zero_random(), 0),
            ratio=2,
            start_param=random.uniform(0, math.tau),
            end_param=random.uniform(0, math.tau),
            extrusion=(0, 0, random.choice((1, -1))),
        )

        # Test if coordinates of start- and end point stay at the same location
        # before and after swapping axis.
        start_point = ellipse.start_point
        end_point = ellipse.end_point
        minor_axis = ellipse.minor_axis
        ellipse.swap_axis()
        assert ellipse.major_axis.isclose(minor_axis, abs_tol=1e-9)
        assert ellipse.start_point.isclose(start_point, abs_tol=1e-9)
        assert ellipse.end_point.isclose(end_point, abs_tol=1e-9) 

def test_params_from_vertices_random():
    center = Vector.random(5)
    major_axis = Vector.random(5)
    extrusion = Vector.random()
    ratio = 0.75
    e = ConstructionEllipse(center, major_axis, extrusion, ratio)

    params = [random.uniform(0.0001, math.tau - 0.0001) for _ in range(20)]
    vertices = e.vertices(params)
    new_params = e.params_from_vertices(vertices)
    for expected, param in zip(params, new_params):
        assert math.isclose(expected, param)

    # This creates the same vertex as v1 and v2
    v1, v2 = e.vertices([0, math.tau])
    assert v1.isclose(v2)

    # This should create the same param for v1 and v2, but
    # floating point inaccuracy produces unpredictable results:
    p1, p2 = e.params_from_vertices((v1, v2))

    assert math.isclose(p1, 0, abs_tol=1e-9) or math.isclose(p1, math.tau, abs_tol=1e-9)
    assert math.isclose(p2, 0, abs_tol=1e-9) or math.isclose(p2, math.tau, abs_tol=1e-9) 

def test_angle_about():
    extrusion = Vector(0, 0, 1)
    a = Vector(1, 0, 0)
    b = Vector(1, 1, 0)
    assert math.isclose(a.angle_between(b), math.pi / 4)
    assert math.isclose(extrusion.angle_about(a, b), math.pi / 4)

    extrusion = Vector(0, 0, -1)
    assert math.isclose(a.angle_between(b), math.pi / 4)
    assert math.isclose(extrusion.angle_about(a, b), (-math.pi / 4) % math.tau)

    extrusion = Vector(0, 0, 1)
    a = Vector(1, 1, 0)
    b = Vector(1, 1, 0)
    assert math.isclose(a.angle_between(b), 0, abs_tol=1e-5)
    assert math.isclose(extrusion.angle_about(a, b), 0)

    extrusion = Vector(0, 1, 0)
    a = Vector(1, 1, 0)
    b = Vector(0, 1, -1)
    assert math.isclose(a.angle_between(b), math.pi / 3, abs_tol=1e-5)
    c = a.cross(b)
    assert math.isclose(a.angle_between(b), c.angle_about(a, b))
    assert math.isclose(extrusion.angle_about(a, b), math.pi / 2) 

def circle(num_t, radius=1.0, phase=0.0):
    """Returns x and y positions of `num_t` points regularly placed around a circle
    of radius `radius`, shifted by `phase` radians.

    Parameters
    ----------
    num_t : int
        the number of points around the circle
    radius : float, default 1.
        the radius of the circle
    phase : float, default 0.0
        angle shift w/r to the x axis in radians

    Returns
    -------
    points : np.Ndarray of shape (num_t, 2), the x, y positions of the points

    """
    if not num_t:
        return np.zeros((1, 2))

    theta = np.arange(0, tau, tau / num_t)
    return np.vstack([radius * np.cos(theta + phase), radius * np.sin(theta + phase)]).T 

def hexa_disk(num_t, radius=1):
    """Returns an arrays of x, y positions of points evenly spread on
    a disk with num_t points on the periphery.

    Parameters
    ----------
    num_t : int
        the number of poitns on the disk periphery, the rest of the disk is
        filled automaticaly
    radius : float, default 1.
        the radius of the disk

    """

    n_circles = int(np.ceil(num_t / tau) + 1)
    if not n_circles:
        return np.zeros((1, 2))

    num_ts = np.linspace(num_t, 0, n_circles, dtype=int)
    rads = radius * num_ts / num_t
    phases = np.pi * num_ts / num_t
    return np.concatenate(
        [circle(n, r, phi) for n, r, phi in zip(num_ts, rads, phases)]
    ) 

def __init__(self, seed):
        self.rnd = np.random.RandomState(seed)
        self.size = 256
        self.mask = int(self.size - 1)
        self.indices = np.arange(self.size, dtype = np.int16)
        self.rnd.shuffle(self.indices)
        theta = np.linspace(0, math.tau, self.size, endpoint=False)
        self.gradients = [np.cos(theta), np.sin(theta)] 

def groupDelay(data: List[Datapoint], index: int) -> float:
    idx0 = clamp_value(index - 1, 0, len(data) - 1)
    idx1 = clamp_value(index + 1, 0, len(data) - 1)
    delta_angle = data[idx1].phase - data[idx0].phase
    delta_freq = data[idx1].freq - data[idx0].freq
    if delta_freq == 0:
        return 0
    if abs(delta_angle) > math.tau:
        if delta_angle > 0:
            delta_angle = delta_angle % math.tau
        else:
            delta_angle = -1 * (delta_angle % math.tau)
    val = -delta_angle / math.tau / delta_freq
    return val 

def screen_shake(self, dist=25):
        """Trigger a screen shake effect.

        The camera will be offset from ``.pos`` by ``dist`` in a random
        direction; then steady itself in a damped harmonic motion.

        """
        theta = np.random.uniform(0, math.tau)
        basis = np.array([theta, + math.pi * 0.5])
        self._cam_offset[:] = dist * np.sin(basis)
        self._xform[-1][:2] = self._cam_offset - self._pos
        clock.schedule_unique(self._steady_cam, 0.01) 

def girdle_coords(radius, mat):
    angle = tau / 64

    return tuple(
        (
            mat @ Vector(
                (
                    sin(i * angle) * radius,
                    cos(i * angle) * radius,
                    0.0,
                )
            )
        ).freeze()
        for i in range(64)
    ) 

def random_angle():
    return random.uniform(0, math.tau) 

def ellipse(major_axis=(1, 0), ratio: float = 0.5, start: float = 0, end: float = math.tau, count: int = 8):
    major_axis = Vector(major_axis).replace(z=0)
    ellipse_ = Ellipse.new(dxfattribs={
        'center': (0, 0, 0),
        'major_axis': major_axis,
        'ratio': min(max(ratio, 1e-6), 1),
        'start_param': start,
        'end_param': end
    }, doc=doc)
    control_vertices = list(ellipse_.vertices(ellipse_.params(count)))
    axis_vertices = list(ellipse_.vertices([0, math.pi / 2, math.pi, math.pi * 1.5]))
    return ellipse_, control_vertices, axis_vertices 

def sine_wave(count: int, scale: float = 1.0):
    for t in linspace(0, math.tau, count):
        yield Vector(t * scale, math.sin(t) * scale) 

def sine_wave(count: int, scale: float = 1.0) -> Iterable[Vector]:
    for t in linspace(0, math.tau, count):
        yield Vector(t * scale, math.sin(t) * scale) 

def rational_spline_from_arc(
        center: Vector = (0, 0), radius: float = 1, start_angle: float = 0, end_angle: float = 360,
        segments: int = 1) -> BSpline:
    """
    Returns a rational B-splines for a circular 2D arc.

    Args:
        center: circle center as :class:`Vector` compatible object
        radius: circle radius
        start_angle: start angle in degrees
        end_angle: end angle in degrees
        segments: count of spline segments, at least one segment for each quarter (90 deg), ``1`` for as few as needed.

    .. versionadded:: 0.13

    """
    center = Vector(center)
    radius = float(radius)
    start_angle = math.radians(start_angle) % math.tau
    end_angle = math.radians(end_angle) % math.tau
    control_points, weights, knots = nurbs_arc_parameters(start_angle, end_angle, segments)
    return BSpline(
        control_points=(center + (p * radius) for p in control_points),
        weights=weights,
        knots=knots,
        order=3,
    ) 

def enclosing_angles(angle, start_angle, end_angle, ccw=True, abs_tol=TOLERANCE):
    isclose = partial(math.isclose, abs_tol=abs_tol)

    s = start_angle % math.tau
    e = end_angle % math.tau
    a = angle % math.tau
    if isclose(s, e):
        return isclose(s, a)

    if s < e:
        r = s < a < e
    else:
        r = not (e < a < s)
    return r if ccw else not r 

def __init__(self, center: 'Vertex' = NULLVEC, major_axis: 'Vertex' = X_AXIS, extrusion: 'Vertex' = Z_AXIS,
                 ratio: float = 1, start_param: float = 0, end_param: float = math.tau, ccw: bool = True):
        self.center = Vector(center)
        self.major_axis = Vector(major_axis)
        self.extrusion = Vector(extrusion)
        self.ratio = float(ratio)
        self.start_param = float(start_param)
        self.end_param = float(end_param)
        if not ccw:
            self.start_param, self.end_param = self.end_param, self.start_param
        self.minor_axis = minor_axis(self.major_axis, self.extrusion, self.ratio) 

def param_span(self) -> float:
        """ Returns params span of ellipse from start- to end param. """
        end = self.end_param
        if end < self.start_param:
            end += math.tau
        return end - self.start_param 

def swap_axis(self) -> None:
        """ Swap axis and adjust start- and end parameter. """
        self.major_axis = self.minor_axis
        ratio = 1.0 / self.ratio
        self.ratio = max(ratio, 1e-6)
        self.minor_axis = minor_axis(self.major_axis, self.extrusion, self.ratio)

        start_param = self.start_param
        end_param = self.end_param
        if math.isclose(start_param, 0) and math.isclose(end_param, math.tau):
            return
        self.start_param = (start_param - HALF_PI) % math.tau
        self.end_param = (end_param - HALF_PI) % math.tau 

def mid_param(start: float, end: float) -> float:
    if end < start:
        end += math.tau
    return (start + end) / 2.0 

def ellipse_edges_to_spline_edges(self, num: int = 32) -> None:
        """
        Convert all ellipse edges to spline edges (approximation).

        Args:
            num: count of control points for a **full** ellipse, partial ellipses have proportional fewer control points
                 but at least 3.

        """

        def to_spline_edge(e: EllipseEdge) -> SplineEdge:
            # No OCS transformation needed, source ellipse and target spline reside in the same OCS.
            # ezdxf stores angles always in counter-clockwise orientation.
            # DXF conversion is done at export, see also ArcEdge.load_tags() for explanation

            ellipse = ConstructionEllipse(
                center=e.center, major_axis=e.major_axis, ratio=e.ratio,
                start_param=e.start_param, end_param=e.end_param,
            )
            count = max(int(float(num) * ellipse.param_span / math.tau), 3)
            tool = BSpline.ellipse_approximation(ellipse, count)
            spline = SplineEdge()
            spline.degree = tool.degree
            if not e.ccw:
                tool = tool.reverse()

            spline.control_points = Vec2.list(tool.control_points)
            spline.knot_values = tool.knots()
            spline.weights = tool.weights()
            return spline

        for path_index, path in enumerate(self.paths):
            if path.PATH_TYPE == 'EdgePath':
                edges = path.edges
                for edge_index, edge in enumerate(edges):
                    if edge.EDGE_TYPE == 'EllipseEdge':
                        edges[edge_index] = to_spline_edge(edge) 

def ellipse_edges_to_line_edges(self, num: int = 64) -> None:
        """ Convert all ellipse edges to line edges (approximation).

        Args:
            num: count of control points for a **full** ellipse, partial ellipses have proportional fewer control points
                 but at least 3.

        """

        def to_line_edges(edge):
            ellipse = ConstructionEllipse(
                center=edge.center,
                major_axis=edge.major_axis,
                ratio=edge.ratio,
                start_param=edge.start_param,
                end_param=edge.end_param,
            )
            segment_count = max(int(float(num) * ellipse.param_span / math.tau), 3)
            params = ellipse.params(segment_count + 1)
            if not edge.ccw:
                params = reversed(list(params))
            vertices = list(ellipse.vertices(params))
            for v1, v2 in zip(vertices[:-1], vertices[1:]):
                line = LineEdge()
                line.start = v1.vec2
                line.end = v2.vec2
                yield line

        for path in self.paths:
            if path.PATH_TYPE == 'EdgePath':
                new_edges = []
                for edge in path.edges:
                    if edge.EDGE_TYPE == 'EllipseEdge':
                        new_edges.extend(to_line_edges(edge))
                    else:
                        new_edges.append(edge)
                path.edges = new_edges 

def params(self, num: int) -> Iterable[float]:
        """ Returns `num` params from start- to end param in counter clockwise order.

        All params are normalized in the range from [0, 2pi).

        """
        start = self.dxf.start_param % math.tau
        end = self.dxf.end_param % math.tau
        yield from ellipse.get_params(start, end, num) 

def draw_elliptic_arc_entity_3d(self, entity: DXFGraphic) -> None:
        dxf, dxftype = entity.dxf, entity.dxftype()
        properties = self._resolve_properties(entity)

        if dxftype in {'CIRCLE', 'ARC'}:
            center = dxf.center  # ocs transformation in .from_arc()
            radius = dxf.radius
            if dxftype == 'CIRCLE':
                start_angle = 0
                end_angle = 360
            else:
                start_angle = dxf.start_angle
                end_angle = dxf.end_angle
            e = ConstructionEllipse.from_arc(center, radius, dxf.extrusion, start_angle, end_angle)
        elif dxftype == 'ELLIPSE':
            e = cast(Ellipse, entity).construction_tool()
        else:
            raise TypeError(dxftype)

        # Approximate as 3D polyline
        segments = int((e.end_param - e.start_param) / math.tau * self.circle_approximation_count)
        points = list(e.vertices(linspace(e.start_param, e.end_param, max(4, segments + 1))))
        self.out.start_path()
        for a, b in zip(points, points[1:]):
            self.out.draw_line(a, b, properties)
        self.out.end_path() 

def _draw_angles_to_start_and_span(draw_angles: Tuple[float, float]) -> Tuple[float, float]:
    a, b = draw_angles
    if b < a:  # arc crosses the discontinuity at n*360
        b += math.tau
    start_angle = -math.degrees(a)
    span_angle = -math.degrees(b - a)
    return start_angle, span_angle 

def test_from_arc():
    ellipse = ConstructionEllipse.from_arc(center=(2, 2, 2), radius=3)
    assert ellipse.center == (2, 2, 2)
    assert ellipse.major_axis == (3, 0, 0)
    assert ellipse.ratio == 1
    assert ellipse.start_param == 0
    assert math.isclose(ellipse.end_param, math.tau) 

def test_angle_to_param():
    random_tests_count = 100
    random.seed(0)

    angle = 1.23
    assert math.isclose(angle_to_param(1.0, angle), angle)

    angle = 1.23 + math.pi / 2
    assert math.isclose(angle_to_param(1.0, angle), angle)

    angle = 1.23 + math.pi
    assert math.isclose(angle_to_param(1.0, angle), angle)

    angle = 1.23 + 3 * math.pi / 2
    assert math.isclose(angle_to_param(1.0, angle), angle)

    angle = math.pi / 2 + 1e-15
    assert math.isclose(angle_to_param(1.0, angle), angle)

    for _ in range(random_tests_count):
        ratio = random.uniform(1e-6, 1)
        angle = random.uniform(0, math.tau)
        param = angle_to_param(ratio, angle)
        ellipse = ConstructionEllipse(
            # avoid (0, 0, 0) as major axis
            major_axis=(non_zero_random(), non_zero_random(), 0),
            ratio=ratio,
            start_param=0,
            end_param=param,
            extrusion=(0, 0, random.choice((1, -1))),
        )
        calculated_angle = ellipse.extrusion.angle_about(ellipse.major_axis, ellipse.end_point)
        calculated_angle_without_direction = ellipse.major_axis.angle_between(ellipse.end_point)
        assert math.isclose(calculated_angle, angle, abs_tol=1e-9)
        assert (math.isclose(calculated_angle, calculated_angle_without_direction) or
                math.isclose(math.tau - calculated_angle, calculated_angle_without_direction)) 

def test_cubic_bezier_arc_parameters():
    parts = list(cubic_bezier_arc_parameters(0, math.tau))
    assert len(parts) == 4

    chk = 4.0 * (math.sqrt(2) - 1.0) / 3.0
    sp, cp1, cp2, ep = parts[0]
    assert sp.isclose((1, 0))
    assert cp1.isclose((1, chk))
    assert cp2.isclose((chk, 1))
    assert ep.isclose((0, 1))

    sp, cp1, cp2, ep = parts[1]
    assert sp.isclose((0, 1))
    assert cp1.isclose((-chk, 1))
    assert cp2.isclose((-1, chk))
    assert ep.isclose((-1, 0))

    sp, cp1, cp2, ep = parts[2]
    assert sp.isclose((-1, 0))
    assert cp1.isclose((-1, -chk))
    assert cp2.isclose((-chk, -1))
    assert ep.isclose((0, -1))

    sp, cp1, cp2, ep = parts[3]
    assert sp.isclose((0, -1))
    assert cp1.isclose((chk, -1))
    assert cp2.isclose((1, -chk))
    assert ep.isclose((1, 0))