"""Mobjects that represent coordinate systems."""
from __future__ import annotations
__all__ = [
"CoordinateSystem",
"Axes",
"ThreeDAxes",
"NumberPlane",
"PolarPlane",
"ComplexPlane",
]
import fractions as fr
import numbers
from typing import TYPE_CHECKING, Any, Callable, Iterable, Sequence, TypeVar, overload
import numpy as np
from typing_extensions import Self
from manim import config
from manim.constants import *
from manim.mobject.geometry.arc import Circle, Dot
from manim.mobject.geometry.line import Arrow, DashedLine, Line
from manim.mobject.geometry.polygram import Polygon, Rectangle, RegularPolygon
from manim.mobject.graphing.functions import ImplicitFunction, ParametricFunction
from manim.mobject.graphing.number_line import NumberLine
from manim.mobject.graphing.scale import LinearBase
from manim.mobject.opengl.opengl_compatibility import ConvertToOpenGL
from manim.mobject.opengl.opengl_surface import OpenGLSurface
from manim.mobject.text.tex_mobject import MathTex
from manim.mobject.three_d.three_dimensions import Surface
from manim.mobject.types.vectorized_mobject import (
VDict,
VectorizedPoint,
VGroup,
VMobject,
)
from manim.utils.color import (
BLACK,
BLUE,
BLUE_D,
GREEN,
WHITE,
YELLOW,
ManimColor,
ParsableManimColor,
color_gradient,
invert_color,
)
from manim.utils.config_ops import merge_dicts_recursively, update_dict_recursively
from manim.utils.simple_functions import binary_search
from manim.utils.space_ops import angle_of_vector
if TYPE_CHECKING:
from manim.mobject.mobject import Mobject
from manim.typing import ManimFloat, Point2D, Point3D, Vector3
LineType = TypeVar("LineType", bound=Line)
[docs]class CoordinateSystem:
r"""Abstract base class for Axes and NumberPlane.
Examples
--------
.. manim:: CoordSysExample
:save_last_frame:
class CoordSysExample(Scene):
def construct(self):
# the location of the ticks depends on the x_range and y_range.
grid = Axes(
x_range=[0, 1, 0.05], # step size determines num_decimal_places.
y_range=[0, 1, 0.05],
x_length=9,
y_length=5.5,
axis_config={
"numbers_to_include": np.arange(0, 1 + 0.1, 0.1),
"font_size": 24,
},
tips=False,
)
# Labels for the x-axis and y-axis.
y_label = grid.get_y_axis_label("y", edge=LEFT, direction=LEFT, buff=0.4)
x_label = grid.get_x_axis_label("x")
grid_labels = VGroup(x_label, y_label)
graphs = VGroup()
for n in np.arange(1, 20 + 0.5, 0.5):
graphs += grid.plot(lambda x: x ** n, color=WHITE)
graphs += grid.plot(
lambda x: x ** (1 / n), color=WHITE, use_smoothing=False
)
# Extra lines and labels for point (1,1)
graphs += grid.get_horizontal_line(grid.c2p(1, 1, 0), color=BLUE)
graphs += grid.get_vertical_line(grid.c2p(1, 1, 0), color=BLUE)
graphs += Dot(point=grid.c2p(1, 1, 0), color=YELLOW)
graphs += Tex("(1,1)").scale(0.75).next_to(grid.c2p(1, 1, 0))
title = Title(
# spaces between braces to prevent SyntaxError
r"Graphs of $y=x^{ {1}\over{n} }$ and $y=x^n (n=1,2,3,...,20)$",
include_underline=False,
font_size=40,
)
self.add(title, graphs, grid, grid_labels)
"""
def __init__(
self,
x_range: Sequence[float] | None = None,
y_range: Sequence[float] | None = None,
x_length: float | None = None,
y_length: float | None = None,
dimension: int = 2,
) -> None:
self.dimension = dimension
default_step = 1
if x_range is None:
x_range = [
round(-config["frame_x_radius"]),
round(config["frame_x_radius"]),
default_step,
]
elif len(x_range) == 2:
x_range = [*x_range, default_step]
if y_range is None:
y_range = [
round(-config["frame_y_radius"]),
round(config["frame_y_radius"]),
default_step,
]
elif len(y_range) == 2:
y_range = [*y_range, default_step]
self.x_range = x_range
self.y_range = y_range
self.x_length = x_length
self.y_length = y_length
self.num_sampled_graph_points_per_tick = 10
def coords_to_point(self, *coords: Sequence[ManimFloat]):
raise NotImplementedError()
def point_to_coords(self, point: Point3D):
raise NotImplementedError()
[docs] def polar_to_point(self, radius: float, azimuth: float) -> Point2D:
r"""Gets a point from polar coordinates.
Parameters
----------
radius
The coordinate radius (:math:`r`).
azimuth
The coordinate azimuth (:math:`\theta`).
Returns
-------
numpy.ndarray
The point.
Examples
--------
.. manim:: PolarToPointExample
:ref_classes: PolarPlane Vector
:save_last_frame:
class PolarToPointExample(Scene):
def construct(self):
polarplane_pi = PolarPlane(azimuth_units="PI radians", size=6)
polartopoint_vector = Vector(polarplane_pi.polar_to_point(3, PI/4))
self.add(polarplane_pi)
self.add(polartopoint_vector)
"""
return self.coords_to_point(radius * np.cos(azimuth), radius * np.sin(azimuth))
[docs] def point_to_polar(self, point: np.ndarray) -> Point2D:
r"""Gets polar coordinates from a point.
Parameters
----------
point
The point.
Returns
-------
Tuple[:class:`float`, :class:`float`]
The coordinate radius (:math:`r`) and the coordinate azimuth (:math:`\theta`).
"""
x, y = self.point_to_coords(point)
return np.sqrt(x**2 + y**2), np.arctan2(y, x)
[docs] def c2p(
self, *coords: float | Sequence[float] | Sequence[Sequence[float]] | np.ndarray
) -> np.ndarray:
"""Abbreviation for :meth:`coords_to_point`"""
return self.coords_to_point(*coords)
[docs] def p2c(self, point: Point3D):
"""Abbreviation for :meth:`point_to_coords`"""
return self.point_to_coords(point)
[docs] def pr2pt(self, radius: float, azimuth: float) -> np.ndarray:
"""Abbreviation for :meth:`polar_to_point`"""
return self.polar_to_point(radius, azimuth)
[docs] def pt2pr(self, point: np.ndarray) -> tuple[float, float]:
"""Abbreviation for :meth:`point_to_polar`"""
return self.point_to_polar(point)
def get_axes(self):
raise NotImplementedError()
def get_axis(self, index: int) -> Mobject:
return self.get_axes()[index]
[docs] def get_origin(self) -> np.ndarray:
"""Gets the origin of :class:`~.Axes`.
Returns
-------
np.ndarray
The center point.
"""
return self.coords_to_point(0, 0)
def get_x_axis(self) -> Mobject:
return self.get_axis(0)
def get_y_axis(self) -> Mobject:
return self.get_axis(1)
def get_z_axis(self) -> Mobject:
return self.get_axis(2)
def get_x_unit_size(self) -> float:
return self.get_x_axis().get_unit_size()
def get_y_unit_size(self) -> float:
return self.get_y_axis().get_unit_size()
[docs] def get_x_axis_label(
self,
label: float | str | Mobject,
edge: Sequence[float] = UR,
direction: Sequence[float] = UR,
buff: float = SMALL_BUFF,
**kwargs,
) -> Mobject:
"""Generate an x-axis label.
Parameters
----------
label
The label. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
edge
The edge of the x-axis to which the label will be added, by default ``UR``.
direction
Allows for further positioning of the label from an edge, by default ``UR``.
buff
The distance of the label from the line.
Returns
-------
:class:`~.Mobject`
The positioned label.
Examples
--------
.. manim:: GetXAxisLabelExample
:save_last_frame:
class GetXAxisLabelExample(Scene):
def construct(self):
ax = Axes(x_range=(0, 8), y_range=(0, 5), x_length=8, y_length=5)
x_label = ax.get_x_axis_label(
Tex("$x$-values").scale(0.65), edge=DOWN, direction=DOWN, buff=0.5
)
self.add(ax, x_label)
"""
return self._get_axis_label(
label, self.get_x_axis(), edge, direction, buff=buff, **kwargs
)
[docs] def get_y_axis_label(
self,
label: float | str | Mobject,
edge: Sequence[float] = UR,
direction: Sequence[float] = UP * 0.5 + RIGHT,
buff: float = SMALL_BUFF,
**kwargs,
) -> Mobject:
"""Generate a y-axis label.
Parameters
----------
label
The label. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
edge
The edge of the x-axis to which the label will be added, by default ``UR``.
direction
Allows for further positioning of the label from an edge, by default ``UR``
buff
The distance of the label from the line.
Returns
-------
:class:`~.Mobject`
The positioned label.
Examples
--------
.. manim:: GetYAxisLabelExample
:save_last_frame:
class GetYAxisLabelExample(Scene):
def construct(self):
ax = Axes(x_range=(0, 8), y_range=(0, 5), x_length=8, y_length=5)
y_label = ax.get_y_axis_label(
Tex("$y$-values").scale(0.65).rotate(90 * DEGREES),
edge=LEFT,
direction=LEFT,
buff=0.3,
)
self.add(ax, y_label)
"""
return self._get_axis_label(
label, self.get_y_axis(), edge, direction, buff=buff, **kwargs
)
[docs] def _get_axis_label(
self,
label: float | str | Mobject,
axis: Mobject,
edge: Sequence[float],
direction: Sequence[float],
buff: float = SMALL_BUFF,
) -> Mobject:
"""Gets the label for an axis.
Parameters
----------
label
The label. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
axis
The axis to which the label will be added.
edge
The edge of the axes to which the label will be added. ``RIGHT`` adds to the right side of the axis
direction
Allows for further positioning of the label.
buff
The distance of the label from the line.
Returns
-------
:class:`~.Mobject`
The positioned label along the given axis.
"""
label = self.x_axis._create_label_tex(label)
label.next_to(axis.get_edge_center(edge), direction=direction, buff=buff)
label.shift_onto_screen(buff=MED_SMALL_BUFF)
return label
def get_axis_labels(self):
raise NotImplementedError()
[docs] def add_coordinates(
self,
*axes_numbers: Iterable[float] | None | dict[float, str | float | Mobject],
**kwargs: Any,
) -> Self:
"""Adds labels to the axes. Use ``Axes.coordinate_labels`` to
access the coordinates after creation.
Parameters
----------
axes_numbers
The numbers to be added to the axes. Use ``None`` to represent an axis with default labels.
Examples
--------
.. code-block:: python
ax = ThreeDAxes()
x_labels = range(-4, 5)
z_labels = range(-4, 4, 2)
ax.add_coordinates(x_labels, None, z_labels) # default y labels, custom x & z labels
ax.add_coordinates(x_labels) # only x labels
You can also specifically control the position and value of the labels using a dict.
.. code-block:: python
ax = Axes(x_range=[0, 7])
x_pos = [x for x in range(1, 8)]
# strings are automatically converted into a Tex mobject.
x_vals = ["Monday", "Tuesday", "Wednesday", "Thursday", "Friday", "Saturday", "Sunday"]
x_dict = dict(zip(x_pos, x_vals))
ax.add_coordinates(x_dict)
"""
self.coordinate_labels = VGroup()
# if nothing is passed to axes_numbers, produce axes with default labelling
if not axes_numbers:
axes_numbers = [None for _ in range(self.dimension)]
for axis, values in zip(self.axes, axes_numbers):
if isinstance(values, dict):
axis.add_labels(values, **kwargs)
labels = axis.labels
elif values is None and axis.scaling.custom_labels:
tick_range = axis.get_tick_range()
axis.add_labels(
dict(zip(tick_range, axis.scaling.get_custom_labels(tick_range)))
)
labels = axis.labels
else:
axis.add_numbers(values, **kwargs)
labels = axis.numbers
self.coordinate_labels.add(labels)
return self
# overload necessary until https://github.com/python/mypy/issues/3737 is supported
@overload
def get_line_from_axis_to_point(
self,
index: int,
point: Sequence[float],
line_config: dict | None = ...,
color: ParsableManimColor | None = ...,
stroke_width: float = ...,
) -> DashedLine:
...
@overload
def get_line_from_axis_to_point(
self,
index: int,
point: Sequence[float],
line_func: type[LineType],
line_config: dict | None = ...,
color: ParsableManimColor | None = ...,
stroke_width: float = ...,
) -> LineType:
...
[docs] def get_line_from_axis_to_point( # type: ignore[no-untyped-def]
self,
index,
point,
line_func=DashedLine,
line_config=None,
color=None,
stroke_width=2,
):
"""Returns a straight line from a given axis to a point in the scene.
Parameters
----------
index
Specifies the axis from which to draw the line. `0 = x_axis`, `1 = y_axis`
point
The point to which the line will be drawn.
line_func
The function of the :class:`~.Line` mobject used to construct the line.
line_config
Optional arguments to passed to :attr:`line_func`.
color
The color of the line.
stroke_width
The stroke width of the line.
Returns
-------
:class:`~.Line`
The line from an axis to a point.
.. seealso::
:meth:`~.CoordinateSystem.get_vertical_line`
:meth:`~.CoordinateSystem.get_horizontal_line`
"""
line_config = line_config if line_config is not None else {}
if color is None:
color = VMobject().color
line_config["color"] = ManimColor.parse(color)
line_config["stroke_width"] = stroke_width
axis = self.get_axis(index)
line = line_func(axis.get_projection(point), point, **line_config)
return line
[docs] def get_vertical_line(self, point: Sequence[float], **kwargs: Any) -> Line:
"""A vertical line from the x-axis to a given point in the scene.
Parameters
----------
point
The point to which the vertical line will be drawn.
kwargs
Additional parameters to be passed to :class:`get_line_from_axis_to_point`.
Returns
-------
:class:`Line`
A vertical line from the x-axis to the point.
Examples
--------
.. manim:: GetVerticalLineExample
:save_last_frame:
class GetVerticalLineExample(Scene):
def construct(self):
ax = Axes().add_coordinates()
point = ax.coords_to_point(-3.5, 2)
dot = Dot(point)
line = ax.get_vertical_line(point, line_config={"dashed_ratio": 0.85})
self.add(ax, line, dot)
"""
return self.get_line_from_axis_to_point(0, point, **kwargs)
[docs] def get_horizontal_line(self, point: Sequence[float], **kwargs) -> Line:
"""A horizontal line from the y-axis to a given point in the scene.
Parameters
----------
point
The point to which the horizontal line will be drawn.
kwargs
Additional parameters to be passed to :class:`get_line_from_axis_to_point`.
Returns
-------
:class:`Line`
A horizontal line from the y-axis to the point.
Examples
--------
.. manim:: GetHorizontalLineExample
:save_last_frame:
class GetHorizontalLineExample(Scene):
def construct(self):
ax = Axes().add_coordinates()
point = ax.c2p(-4, 1.5)
dot = Dot(point)
line = ax.get_horizontal_line(point, line_func=Line)
self.add(ax, line, dot)
"""
return self.get_line_from_axis_to_point(1, point, **kwargs)
[docs] def get_lines_to_point(self, point: Sequence[float], **kwargs) -> VGroup:
"""Generate both horizontal and vertical lines from the axis to a point.
Parameters
----------
point
A point on the scene.
kwargs
Additional parameters to be passed to :meth:`get_line_from_axis_to_point`
Returns
-------
:class:`~.VGroup`
A :class:`~.VGroup` of the horizontal and vertical lines.
.. seealso::
:meth:`~.CoordinateSystem.get_vertical_line`
:meth:`~.CoordinateSystem.get_horizontal_line`
Examples
--------
.. manim:: GetLinesToPointExample
:save_last_frame:
class GetLinesToPointExample(Scene):
def construct(self):
ax = Axes()
circ = Circle(radius=0.5).move_to([-4, -1.5, 0])
lines_1 = ax.get_lines_to_point(circ.get_right(), color=GREEN_B)
lines_2 = ax.get_lines_to_point(circ.get_corner(DL), color=BLUE_B)
self.add(ax, lines_1, lines_2, circ)
"""
return VGroup(
self.get_horizontal_line(point, **kwargs),
self.get_vertical_line(point, **kwargs),
)
# graphing
[docs] def plot(
self,
function: Callable[[float], float],
x_range: Sequence[float] | None = None,
use_vectorized: bool = False,
**kwargs: Any,
) -> ParametricFunction:
"""Generates a curve based on a function.
Parameters
----------
function
The function used to construct the :class:`~.ParametricFunction`.
x_range
The range of the curve along the axes. ``x_range = [x_min, x_max, x_step]``.
use_vectorized
Whether to pass in the generated t value array to the function. Only use this if your function supports it.
Output should be a numpy array of shape ``[y_0, y_1, ...]``
kwargs
Additional parameters to be passed to :class:`~.ParametricFunction`.
Returns
-------
:class:`~.ParametricFunction`
The plotted curve.
.. warning::
This method may not produce accurate graphs since Manim currently relies on interpolation between
evenly-spaced samples of the curve, instead of intelligent plotting.
See the example below for some solutions to this problem.
Examples
--------
.. manim:: PlotExample
:save_last_frame:
class PlotExample(Scene):
def construct(self):
# construct the axes
ax_1 = Axes(
x_range=[0.001, 6],
y_range=[-8, 2],
x_length=5,
y_length=3,
tips=False,
)
ax_2 = ax_1.copy()
ax_3 = ax_1.copy()
# position the axes
ax_1.to_corner(UL)
ax_2.to_corner(UR)
ax_3.to_edge(DOWN)
axes = VGroup(ax_1, ax_2, ax_3)
# create the logarithmic curves
def log_func(x):
return np.log(x)
# a curve without adjustments; poor interpolation.
curve_1 = ax_1.plot(log_func, color=PURE_RED)
# disabling interpolation makes the graph look choppy as not enough
# inputs are available
curve_2 = ax_2.plot(log_func, use_smoothing=False, color=ORANGE)
# taking more inputs of the curve by specifying a step for the
# x_range yields expected results, but increases rendering time.
curve_3 = ax_3.plot(
log_func, x_range=(0.001, 6, 0.001), color=PURE_GREEN
)
curves = VGroup(curve_1, curve_2, curve_3)
self.add(axes, curves)
"""
t_range = np.array(self.x_range, dtype=float)
if x_range is not None:
t_range[: len(x_range)] = x_range
if x_range is None or len(x_range) < 3:
# if t_range has a defined step size, increase the number of sample points per tick
t_range[2] /= self.num_sampled_graph_points_per_tick
# For axes, the third coordinate of x_range indicates
# tick frequency. But for functions, it indicates a
# sample frequency
graph = ParametricFunction(
lambda t: self.coords_to_point(t, function(t)),
t_range=t_range,
scaling=self.x_axis.scaling,
use_vectorized=use_vectorized,
**kwargs,
)
graph.underlying_function = function
return graph
[docs] def plot_implicit_curve(
self,
func: Callable[[float, float], float],
min_depth: int = 5,
max_quads: int = 1500,
**kwargs: Any,
) -> ImplicitFunction:
"""Creates the curves of an implicit function.
Parameters
----------
func
The function to graph, in the form of f(x, y) = 0.
min_depth
The minimum depth of the function to calculate.
max_quads
The maximum number of quads to use.
kwargs
Additional parameters to pass into :class:`ImplicitFunction`.
Examples
--------
.. manim:: ImplicitExample
:save_last_frame:
class ImplicitExample(Scene):
def construct(self):
ax = Axes()
a = ax.plot_implicit_curve(
lambda x, y: y * (x - y) ** 2 - 4 * x - 8, color=BLUE
)
self.add(ax, a)
"""
x_scale = self.get_x_axis().scaling
y_scale = self.get_y_axis().scaling
graph = ImplicitFunction(
func=(lambda x, y: func(x_scale.function(x), y_scale.function(y))),
x_range=self.x_range[:2],
y_range=self.y_range[:2],
min_depth=min_depth,
max_quads=max_quads,
**kwargs,
)
(
graph.stretch(self.get_x_unit_size(), 0, about_point=ORIGIN)
.stretch(self.get_y_unit_size(), 1, about_point=ORIGIN)
.shift(self.get_origin())
)
return graph
[docs] def plot_parametric_curve(
self,
function: Callable[[float], np.ndarray],
use_vectorized: bool = False,
**kwargs: Any,
) -> ParametricFunction:
"""A parametric curve.
Parameters
----------
function
A parametric function mapping a number to a point in the
coordinate system.
use_vectorized
Whether to pass in the generated t value array to the function. Only use this if your function supports it.
kwargs
Any further keyword arguments are passed to :class:`.ParametricFunction`.
Example
-------
.. manim:: ParametricCurveExample
:save_last_frame:
class ParametricCurveExample(Scene):
def construct(self):
ax = Axes()
cardioid = ax.plot_parametric_curve(
lambda t: np.array(
[
np.exp(1) * np.cos(t) * (1 - np.cos(t)),
np.exp(1) * np.sin(t) * (1 - np.cos(t)),
0,
]
),
t_range=[0, 2 * PI],
color="#0FF1CE",
)
self.add(ax, cardioid)
"""
dim = self.dimension
graph = ParametricFunction(
lambda t: self.coords_to_point(*function(t)[:dim]),
use_vectorized=use_vectorized,
**kwargs,
)
graph.underlying_function = function
return graph
[docs] def plot_polar_graph(
self,
r_func: Callable[[float], float],
theta_range: Sequence[float] | None = None,
**kwargs: Any,
) -> ParametricFunction:
"""A polar graph.
Parameters
----------
r_func
The function r of theta.
theta_range
The range of theta as ``theta_range = [theta_min, theta_max, theta_step]``.
kwargs
Additional parameters passed to :class:`~.ParametricFunction`.
Examples
--------
.. manim:: PolarGraphExample
:ref_classes: PolarPlane
:save_last_frame:
class PolarGraphExample(Scene):
def construct(self):
plane = PolarPlane()
r = lambda theta: 2 * np.sin(theta * 5)
graph = plane.plot_polar_graph(r, [0, 2 * PI], color=ORANGE)
self.add(plane, graph)
"""
theta_range = theta_range if theta_range is not None else [0, 2 * PI]
graph = ParametricFunction(
function=lambda th: self.pr2pt(r_func(th), th),
t_range=theta_range,
**kwargs,
)
graph.underlying_function = r_func
return graph
[docs] def plot_surface(
self,
function: Callable[[float], float],
u_range: Sequence[float] | None = None,
v_range: Sequence[float] | None = None,
colorscale: Sequence[ParsableManimColor]
| Sequence[tuple[ParsableManimColor, float]]
| None = None,
colorscale_axis: int = 2,
**kwargs: Any,
) -> Surface | OpenGLSurface:
"""Generates a surface based on a function.
Parameters
----------
function
The function used to construct the :class:`~.Surface`.
u_range
The range of the ``u`` variable: ``(u_min, u_max)``.
v_range
The range of the ``v`` variable: ``(v_min, v_max)``.
colorscale
Colors of the surface. Passing a list of colors will color the surface by z-value.
Passing a list of tuples in the form ``(color, pivot)`` allows user-defined pivots
where the color transitions.
colorscale_axis
Defines the axis on which the colorscale is applied (0 = x, 1 = y, 2 = z), default
is z-axis (2).
kwargs
Additional parameters to be passed to :class:`~.Surface`.
Returns
-------
:class:`~.Surface`
The plotted surface.
Examples
--------
.. manim:: PlotSurfaceExample
:save_last_frame:
class PlotSurfaceExample(ThreeDScene):
def construct(self):
resolution_fa = 16
self.set_camera_orientation(phi=75 * DEGREES, theta=-60 * DEGREES)
axes = ThreeDAxes(x_range=(-3, 3, 1), y_range=(-3, 3, 1), z_range=(-5, 5, 1))
def param_trig(u, v):
x = u
y = v
z = 2 * np.sin(x) + 2 * np.cos(y)
return z
trig_plane = axes.plot_surface(
param_trig,
resolution=(resolution_fa, resolution_fa),
u_range = (-3, 3),
v_range = (-3, 3),
colorscale = [BLUE, GREEN, YELLOW, ORANGE, RED],
)
self.add(axes, trig_plane)
"""
if config.renderer == RendererType.CAIRO:
surface = Surface(
lambda u, v: self.c2p(u, v, function(u, v)),
u_range=u_range,
v_range=v_range,
**kwargs,
)
if colorscale:
surface.set_fill_by_value(
axes=self.copy(),
colorscale=colorscale,
axis=colorscale_axis,
)
elif config.renderer == RendererType.OPENGL:
surface = OpenGLSurface(
lambda u, v: self.c2p(u, v, function(u, v)),
u_range=u_range,
v_range=v_range,
axes=self.copy(),
colorscale=colorscale,
colorscale_axis=colorscale_axis,
**kwargs,
)
return surface
[docs] def i2gc(self, x: float, graph: ParametricFunction) -> tuple[float, float]:
"""Alias for :meth:`input_to_graph_coords`."""
return self.input_to_graph_coords(x, graph)
[docs] def i2gp(self, x: float, graph: ParametricFunction) -> np.ndarray:
"""Alias for :meth:`input_to_graph_point`."""
return self.input_to_graph_point(x, graph)
[docs] def get_graph_label(
self,
graph: ParametricFunction,
label: float | str | Mobject = "f(x)",
x_val: float | None = None,
direction: Sequence[float] = RIGHT,
buff: float = MED_SMALL_BUFF,
color: ParsableManimColor | None = None,
dot: bool = False,
dot_config: dict[str, Any] | None = None,
) -> Mobject:
"""Creates a properly positioned label for the passed graph, with an optional dot.
Parameters
----------
graph
The curve.
label
The label for the function's curve. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
x_val
The x_value along the curve that positions the label.
direction
The cartesian position, relative to the curve that the label will be at --> ``LEFT``, ``RIGHT``.
buff
The distance between the curve and the label.
color
The color of the label. Defaults to the color of the curve.
dot
Whether to add a dot at the point on the graph.
dot_config
Additional parameters to be passed to :class:`~.Dot`.
Returns
-------
:class:`Mobject`
The positioned label and :class:`~.Dot`, if applicable.
Examples
--------
.. manim:: GetGraphLabelExample
:save_last_frame:
class GetGraphLabelExample(Scene):
def construct(self):
ax = Axes()
sin = ax.plot(lambda x: np.sin(x), color=PURPLE_B)
label = ax.get_graph_label(
graph=sin,
label= MathTex(r"\\frac{\\pi}{2}"),
x_val=PI / 2,
dot=True,
direction=UR,
)
self.add(ax, sin, label)
"""
if dot_config is None:
dot_config = {}
if color is None:
color = graph.get_color()
label = self.x_axis._create_label_tex(label).set_color(color)
if x_val is None:
# Search from right to left
for x in np.linspace(self.x_range[1], self.x_range[0], 100):
point = self.input_to_graph_point(x, graph)
if point[1] < config["frame_y_radius"]:
break
else:
point = self.input_to_graph_point(x_val, graph)
label.next_to(point, direction, buff=buff)
label.shift_onto_screen()
if dot:
dot = Dot(point=point, **dot_config)
label.add(dot)
label.dot = dot
return label
# calculus
[docs] def get_riemann_rectangles(
self,
graph: ParametricFunction,
x_range: Sequence[float] | None = None,
dx: float | None = 0.1,
input_sample_type: str = "left",
stroke_width: float = 1,
stroke_color: ParsableManimColor = BLACK,
fill_opacity: float = 1,
color: Iterable[ParsableManimColor] | ParsableManimColor = (BLUE, GREEN),
show_signed_area: bool = True,
bounded_graph: ParametricFunction = None,
blend: bool = False,
width_scale_factor: float = 1.001,
) -> VGroup:
"""Generates a :class:`~.VGroup` of the Riemann Rectangles for a given curve.
Parameters
----------
graph
The graph whose area will be approximated by Riemann rectangles.
x_range
The minimum and maximum x-values of the rectangles. ``x_range = [x_min, x_max]``.
dx
The change in x-value that separates each rectangle.
input_sample_type
Can be any of ``"left"``, ``"right"`` or ``"center"``. Refers to where
the sample point for the height of each Riemann Rectangle
will be inside the segments of the partition.
stroke_width
The stroke_width of the border of the rectangles.
stroke_color
The color of the border of the rectangle.
fill_opacity
The opacity of the rectangles.
color
The colors of the rectangles. Creates a balanced gradient if multiple colors are passed.
show_signed_area
Indicates negative area when the curve dips below the x-axis by inverting its color.
blend
Sets the :attr:`stroke_color` to :attr:`fill_color`, blending the rectangles without clear separation.
bounded_graph
If a secondary graph is specified, encloses the area between the two curves.
width_scale_factor
The factor by which the width of the rectangles is scaled.
Returns
-------
:class:`~.VGroup`
A :class:`~.VGroup` containing the Riemann Rectangles.
Examples
--------
.. manim:: GetRiemannRectanglesExample
:save_last_frame:
class GetRiemannRectanglesExample(Scene):
def construct(self):
ax = Axes(y_range=[-2, 10])
quadratic = ax.plot(lambda x: 0.5 * x ** 2 - 0.5)
# the rectangles are constructed from their top right corner.
# passing an iterable to `color` produces a gradient
rects_right = ax.get_riemann_rectangles(
quadratic,
x_range=[-4, -3],
dx=0.25,
color=(TEAL, BLUE_B, DARK_BLUE),
input_sample_type="right",
)
# the colour of rectangles below the x-axis is inverted
# due to show_signed_area
rects_left = ax.get_riemann_rectangles(
quadratic, x_range=[-1.5, 1.5], dx=0.15, color=YELLOW
)
bounding_line = ax.plot(
lambda x: 1.5 * x, color=BLUE_B, x_range=[3.3, 6]
)
bounded_rects = ax.get_riemann_rectangles(
bounding_line,
bounded_graph=quadratic,
dx=0.15,
x_range=[4, 5],
show_signed_area=False,
color=(MAROON_A, RED_B, PURPLE_D),
)
self.add(
ax, bounding_line, quadratic, rects_right, rects_left, bounded_rects
)
"""
# setting up x_range, overwrite user's third input
if x_range is None:
if bounded_graph is None:
x_range = [graph.t_min, graph.t_max]
else:
x_min = max(graph.t_min, bounded_graph.t_min)
x_max = min(graph.t_max, bounded_graph.t_max)
x_range = [x_min, x_max]
x_range = [*x_range[:2], dx]
rectangles = VGroup()
x_range = np.arange(*x_range)
if isinstance(color, (list, tuple)):
color = [ManimColor(c) for c in color]
else:
color = [ManimColor(color)]
colors = color_gradient(color, len(x_range))
for x, color in zip(x_range, colors):
if input_sample_type == "left":
sample_input = x
elif input_sample_type == "right":
sample_input = x + dx
elif input_sample_type == "center":
sample_input = x + 0.5 * dx
else:
raise ValueError("Invalid input sample type")
graph_point = self.input_to_graph_point(sample_input, graph)
if bounded_graph is None:
y_point = self._origin_shift(self.y_range)
else:
y_point = bounded_graph.underlying_function(x)
points = VGroup(
*list(
map(
VectorizedPoint,
[
self.coords_to_point(x, y_point),
self.coords_to_point(x + width_scale_factor * dx, y_point),
graph_point,
],
),
)
)
rect = Rectangle().replace(points, stretch=True)
rectangles.add(rect)
# checks if the rectangle is under the x-axis
if self.p2c(graph_point)[1] < y_point and show_signed_area:
color = invert_color(color)
# blends rectangles smoothly
if blend:
stroke_color = color
rect.set_style(
fill_color=color,
fill_opacity=fill_opacity,
stroke_color=stroke_color,
stroke_width=stroke_width,
)
return rectangles
[docs] def get_area(
self,
graph: ParametricFunction,
x_range: tuple[float, float] | None = None,
color: ParsableManimColor | Iterable[ParsableManimColor] = (BLUE, GREEN),
opacity: float = 0.3,
bounded_graph: ParametricFunction = None,
**kwargs: Any,
) -> Polygon:
"""Returns a :class:`~.Polygon` representing the area under the graph passed.
Parameters
----------
graph
The graph/curve for which the area needs to be gotten.
x_range
The range of the minimum and maximum x-values of the area. ``x_range = [x_min, x_max]``.
color
The color of the area. Creates a gradient if a list of colors is provided.
opacity
The opacity of the area.
bounded_graph
If a secondary :attr:`graph` is specified, encloses the area between the two curves.
kwargs
Additional parameters passed to :class:`~.Polygon`.
Returns
-------
:class:`~.Polygon`
The :class:`~.Polygon` representing the area.
Raises
------
:exc:`ValueError`
When x_ranges do not match (either area x_range, graph's x_range or bounded_graph's x_range).
Examples
--------
.. manim:: GetAreaExample
:save_last_frame:
class GetAreaExample(Scene):
def construct(self):
ax = Axes().add_coordinates()
curve = ax.plot(lambda x: 2 * np.sin(x), color=DARK_BLUE)
area = ax.get_area(
curve,
x_range=(PI / 2, 3 * PI / 2),
color=(GREEN_B, GREEN_D),
opacity=1,
)
self.add(ax, curve, area)
"""
if x_range is None:
a = graph.t_min
b = graph.t_max
else:
a, b = x_range
if bounded_graph is not None:
if bounded_graph.t_min > b:
raise ValueError(
f"Ranges not matching: {bounded_graph.t_min} < {b}",
)
if bounded_graph.t_max < a:
raise ValueError(
f"Ranges not matching: {bounded_graph.t_max} > {a}",
)
a = max(a, bounded_graph.t_min)
b = min(b, bounded_graph.t_max)
if bounded_graph is None:
points = (
[self.c2p(a), graph.function(a)]
+ [p for p in graph.points if a <= self.p2c(p)[0] <= b]
+ [graph.function(b), self.c2p(b)]
)
else:
graph_points, bounded_graph_points = (
[g.function(a)]
+ [p for p in g.points if a <= self.p2c(p)[0] <= b]
+ [g.function(b)]
for g in (graph, bounded_graph)
)
points = graph_points + bounded_graph_points[::-1]
return Polygon(*points, **kwargs).set_opacity(opacity).set_color(color)
[docs] def angle_of_tangent(
self,
x: float,
graph: ParametricFunction,
dx: float = 1e-8,
) -> float:
"""Returns the angle to the x-axis of the tangent
to the plotted curve at a particular x-value.
Parameters
----------
x
The x-value at which the tangent must touch the curve.
graph
The :class:`~.ParametricFunction` for which to calculate the tangent.
dx
The change in `x` used to determine the angle of the tangent to the curve.
Returns
-------
:class:`float`
The angle of the tangent to the curve.
Examples
--------
.. code-block:: python
ax = Axes()
curve = ax.plot(lambda x: x**2)
ax.angle_of_tangent(x=3, graph=curve)
# 1.4056476493802699
"""
p0 = np.array([*self.input_to_graph_coords(x, graph)])
p1 = np.array([*self.input_to_graph_coords(x + dx, graph)])
return angle_of_vector(p1 - p0)
[docs] def slope_of_tangent(
self, x: float, graph: ParametricFunction, **kwargs: Any
) -> float:
"""Returns the slope of the tangent to the plotted curve
at a particular x-value.
Parameters
----------
x
The x-value at which the tangent must touch the curve.
graph
The :class:`~.ParametricFunction` for which to calculate the tangent.
Returns
-------
:class:`float`
The slope of the tangent with the x axis.
Examples
--------
.. code-block:: python
ax = Axes()
curve = ax.plot(lambda x: x**2)
ax.slope_of_tangent(x=-2, graph=curve)
# -3.5000000259052038
"""
return np.tan(self.angle_of_tangent(x, graph, **kwargs))
[docs] def plot_derivative_graph(
self, graph: ParametricFunction, color: ParsableManimColor = GREEN, **kwargs
) -> ParametricFunction:
"""Returns the curve of the derivative of the passed graph.
Parameters
----------
graph
The graph for which the derivative will be found.
color
The color of the derivative curve.
kwargs
Any valid keyword argument of :class:`~.ParametricFunction`.
Returns
-------
:class:`~.ParametricFunction`
The curve of the derivative.
Examples
--------
.. manim:: DerivativeGraphExample
:save_last_frame:
class DerivativeGraphExample(Scene):
def construct(self):
ax = NumberPlane(y_range=[-1, 7], background_line_style={"stroke_opacity": 0.4})
curve_1 = ax.plot(lambda x: x ** 2, color=PURPLE_B)
curve_2 = ax.plot_derivative_graph(curve_1)
curves = VGroup(curve_1, curve_2)
label_1 = ax.get_graph_label(curve_1, "x^2", x_val=-2, direction=DL)
label_2 = ax.get_graph_label(curve_2, "2x", x_val=3, direction=RIGHT)
labels = VGroup(label_1, label_2)
self.add(ax, curves, labels)
"""
def deriv(x):
return self.slope_of_tangent(x, graph)
return self.plot(deriv, color=color, **kwargs)
[docs] def plot_antiderivative_graph(
self,
graph: ParametricFunction,
y_intercept: float = 0,
samples: int = 50,
use_vectorized: bool = False,
**kwargs: Any,
) -> ParametricFunction:
"""Plots an antiderivative graph.
Parameters
----------
graph
The graph for which the antiderivative will be found.
y_intercept
The y-value at which the graph intercepts the y-axis.
samples
The number of points to take the area under the graph.
use_vectorized
Whether to use the vectorized version of the antiderivative. This means
to pass in the generated t value array to the function. Only use this if your function supports it.
Output should be a numpy array of shape ``[y_0, y_1, ...]``
kwargs
Any valid keyword argument of :class:`~.ParametricFunction`.
Returns
-------
:class:`~.ParametricFunction`
The curve of the antiderivative.
.. note::
This graph is plotted from the values of area under the reference graph.
The result might not be ideal if the reference graph contains uncalculatable
areas from x=0.
Examples
--------
.. manim:: AntiderivativeExample
:save_last_frame:
class AntiderivativeExample(Scene):
def construct(self):
ax = Axes()
graph1 = ax.plot(
lambda x: (x ** 2 - 2) / 3,
color=RED,
)
graph2 = ax.plot_antiderivative_graph(graph1, color=BLUE)
self.add(ax, graph1, graph2)
"""
def antideriv(x):
x_vals = np.linspace(0, x, samples, axis=1 if use_vectorized else 0)
f_vec = np.vectorize(graph.underlying_function)
y_vals = f_vec(x_vals)
return np.trapz(y_vals, x_vals) + y_intercept
return self.plot(antideriv, use_vectorized=use_vectorized, **kwargs)
[docs] def get_secant_slope_group(
self,
x: float,
graph: ParametricFunction,
dx: float | None = None,
dx_line_color: ParsableManimColor = YELLOW,
dy_line_color: ParsableManimColor | None = None,
dx_label: float | str | None = None,
dy_label: float | str | None = None,
include_secant_line: bool = True,
secant_line_color: ParsableManimColor = GREEN,
secant_line_length: float = 10,
) -> VGroup:
"""Creates two lines representing `dx` and `df`, the labels for `dx` and `df`, and
the secant to the curve at a particular x-value.
Parameters
----------
x
The x-value at which the secant intersects the graph for the first time.
graph
The curve for which the secant will be found.
dx
The change in `x` after which the secant exits.
dx_line_color
The color of the line that indicates the change in `x`.
dy_line_color
The color of the line that indicates the change in `y`. Defaults to the color of :attr:`graph`.
dx_label
The label for the `dx` line. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
dy_label
The label for the `dy` line. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
include_secant_line
Whether to include the secant line in the graph,
or just the df/dx lines and labels.
secant_line_color
The color of the secant line.
secant_line_length
The length of the secant line.
Returns
-------
:class:`~.VGroup`
A group containing the elements: `dx_line`, `df_line`, and
if applicable also :attr:`dx_label`, :attr:`df_label`, `secant_line`.
Examples
--------
.. manim:: GetSecantSlopeGroupExample
:save_last_frame:
class GetSecantSlopeGroupExample(Scene):
def construct(self):
ax = Axes(y_range=[-1, 7])
graph = ax.plot(lambda x: 1 / 4 * x ** 2, color=BLUE)
slopes = ax.get_secant_slope_group(
x=2.0,
graph=graph,
dx=1.0,
dx_label=Tex("dx = 1.0"),
dy_label="dy",
dx_line_color=GREEN_B,
secant_line_length=4,
secant_line_color=RED_D,
)
self.add(ax, graph, slopes)
"""
group = VGroup()
dx = dx or float(self.x_range[1] - self.x_range[0]) / 10
dy_line_color = dy_line_color or graph.get_color()
p1 = self.input_to_graph_point(x, graph)
p2 = self.input_to_graph_point(x + dx, graph)
interim_point = p2[0] * RIGHT + p1[1] * UP
group.dx_line = Line(p1, interim_point, color=dx_line_color)
group.df_line = Line(interim_point, p2, color=dy_line_color)
group.add(group.dx_line, group.df_line)
labels = VGroup()
if dx_label is not None:
group.dx_label = self.x_axis._create_label_tex(dx_label)
labels.add(group.dx_label)
group.add(group.dx_label)
if dy_label is not None:
group.df_label = self.x_axis._create_label_tex(dy_label)
labels.add(group.df_label)
group.add(group.df_label)
if len(labels) > 0:
max_width = 0.8 * group.dx_line.width
max_height = 0.8 * group.df_line.height
if labels.width > max_width:
labels.width = max_width
if labels.height > max_height:
labels.height = max_height
if dx_label is not None:
group.dx_label.next_to(
group.dx_line,
np.sign(dx) * DOWN,
buff=group.dx_label.height / 2,
)
group.dx_label.set_color(group.dx_line.get_color())
if dy_label is not None:
group.df_label.next_to(
group.df_line,
np.sign(dx) * RIGHT,
buff=group.df_label.height / 2,
)
group.df_label.set_color(group.df_line.get_color())
if include_secant_line:
group.secant_line = Line(p1, p2, color=secant_line_color)
group.secant_line.scale(
secant_line_length / group.secant_line.get_length(),
)
group.add(group.secant_line)
return group
[docs] def get_vertical_lines_to_graph(
self,
graph: ParametricFunction,
x_range: Sequence[float] | None = None,
num_lines: int = 20,
**kwargs: Any,
) -> VGroup:
"""Obtains multiple lines from the x-axis to the curve.
Parameters
----------
graph
The graph along which the lines are placed.
x_range
A list containing the lower and and upper bounds of the lines: ``x_range = [x_min, x_max]``.
num_lines
The number of evenly spaced lines.
kwargs
Additional arguments to be passed to :meth:`~.CoordinateSystem.get_vertical_line`.
Returns
-------
:class:`~.VGroup`
The :class:`~.VGroup` of the evenly spaced lines.
Examples
--------
.. manim:: GetVerticalLinesToGraph
:save_last_frame:
class GetVerticalLinesToGraph(Scene):
def construct(self):
ax = Axes(
x_range=[0, 8.0, 1],
y_range=[-1, 1, 0.2],
axis_config={"font_size": 24},
).add_coordinates()
curve = ax.plot(lambda x: np.sin(x) / np.e ** 2 * x)
lines = ax.get_vertical_lines_to_graph(
curve, x_range=[0, 4], num_lines=30, color=BLUE
)
self.add(ax, curve, lines)
"""
x_range = x_range if x_range is not None else self.x_range
return VGroup(
*(
self.get_vertical_line(self.i2gp(x, graph), **kwargs)
for x in np.linspace(x_range[0], x_range[1], num_lines)
)
)
[docs] def get_T_label(
self,
x_val: float,
graph: ParametricFunction,
label: float | str | Mobject | None = None,
label_color: ParsableManimColor | None = None,
triangle_size: float = MED_SMALL_BUFF,
triangle_color: ParsableManimColor | None = WHITE,
line_func: type[Line] = Line,
line_color: ParsableManimColor = YELLOW,
) -> VGroup:
"""Creates a labelled triangle marker with a vertical line from the x-axis
to a curve at a given x-value.
Parameters
----------
x_val
The position along the curve at which the label, line and triangle will be constructed.
graph
The :class:`~.ParametricFunction` for which to construct the label.
label
The label of the vertical line and triangle.
label_color
The color of the label.
triangle_size
The size of the triangle.
triangle_color
The color of the triangle.
line_func
The function used to construct the vertical line.
line_color
The color of the vertical line.
Returns
-------
:class:`~.VGroup`
A :class:`~.VGroup` of the label, triangle and vertical line mobjects.
Examples
--------
.. manim:: TLabelExample
:save_last_frame:
class TLabelExample(Scene):
def construct(self):
# defines the axes and linear function
axes = Axes(x_range=[-1, 10], y_range=[-1, 10], x_length=9, y_length=6)
func = axes.plot(lambda x: x, color=BLUE)
# creates the T_label
t_label = axes.get_T_label(x_val=4, graph=func, label=Tex("x-value"))
self.add(axes, func, t_label)
"""
T_label_group = VGroup()
triangle = RegularPolygon(n=3, start_angle=np.pi / 2, stroke_width=0).set_fill(
color=triangle_color,
opacity=1,
)
triangle.height = triangle_size
triangle.move_to(self.coords_to_point(x_val, 0), UP)
if label is not None:
t_label = self.x_axis._create_label_tex(label, color=label_color)
t_label.next_to(triangle, DOWN)
T_label_group.add(t_label)
v_line = self.get_vertical_line(
self.i2gp(x_val, graph),
color=line_color,
line_func=line_func,
)
T_label_group.add(triangle, v_line)
return T_label_group
[docs]class Axes(VGroup, CoordinateSystem, metaclass=ConvertToOpenGL):
"""Creates a set of axes.
Parameters
----------
x_range
The ``(x_min, x_max, x_step)`` values of the x-axis.
y_range
The ``(y_min, y_max, y_step)`` values of the y-axis.
x_length
The length of the x-axis.
y_length
The length of the y-axis.
axis_config
Arguments to be passed to :class:`~.NumberLine` that influences both axes.
x_axis_config
Arguments to be passed to :class:`~.NumberLine` that influence the x-axis.
y_axis_config
Arguments to be passed to :class:`~.NumberLine` that influence the y-axis.
tips
Whether or not to include the tips on both axes.
kwargs
Additional arguments to be passed to :class:`CoordinateSystem` and :class:`~.VGroup`.
Examples
--------
.. manim:: LogScalingExample
:save_last_frame:
class LogScalingExample(Scene):
def construct(self):
ax = Axes(
x_range=[0, 10, 1],
y_range=[-2, 6, 1],
tips=False,
axis_config={"include_numbers": True},
y_axis_config={"scaling": LogBase(custom_labels=True)},
)
# x_min must be > 0 because log is undefined at 0.
graph = ax.plot(lambda x: x ** 2, x_range=[0.001, 10], use_smoothing=False)
self.add(ax, graph)
Styling arguments can be passed to the underlying :class:`.NumberLine`
mobjects that represent the axes:
.. manim:: AxesWithDifferentTips
:save_last_frame:
class AxesWithDifferentTips(Scene):
def construct(self):
ax = Axes(axis_config={'tip_shape': StealthTip})
self.add(ax)
"""
def __init__(
self,
x_range: Sequence[float] | None = None,
y_range: Sequence[float] | None = None,
x_length: float | None = round(config.frame_width) - 2,
y_length: float | None = round(config.frame_height) - 2,
axis_config: dict | None = None,
x_axis_config: dict | None = None,
y_axis_config: dict | None = None,
tips: bool = True,
**kwargs: Any,
) -> None:
VGroup.__init__(self, **kwargs)
CoordinateSystem.__init__(self, x_range, y_range, x_length, y_length)
self.axis_config = {
"include_tip": tips,
"numbers_to_exclude": [0],
}
self.x_axis_config = {}
self.y_axis_config = {"rotation": 90 * DEGREES, "label_direction": LEFT}
self._update_default_configs(
(self.axis_config, self.x_axis_config, self.y_axis_config),
(axis_config, x_axis_config, y_axis_config),
)
self.x_axis_config = merge_dicts_recursively(
self.axis_config,
self.x_axis_config,
)
self.y_axis_config = merge_dicts_recursively(
self.axis_config,
self.y_axis_config,
)
# excluding the origin tick removes a tick at the 0-point of the axis
# This is desired for LinearBase because the 0 point is always the x-axis
# For non-LinearBase, the "0-point" does not have this quality, so it must be included.
# i.e. with LogBase range [-2, 4]:
# it would remove the "0" tick, which is actually 10^0,
# not the lowest tick on the graph (which is 10^-2).
if self.x_axis_config.get("scaling") is None or isinstance(
self.x_axis_config.get("scaling"), LinearBase
):
self.x_axis_config["exclude_origin_tick"] = True
else:
self.x_axis_config["exclude_origin_tick"] = False
if self.y_axis_config.get("scaling") is None or isinstance(
self.y_axis_config.get("scaling"), LinearBase
):
self.y_axis_config["exclude_origin_tick"] = True
else:
self.y_axis_config["exclude_origin_tick"] = False
self.x_axis = self._create_axis(self.x_range, self.x_axis_config, self.x_length)
self.y_axis = self._create_axis(self.y_range, self.y_axis_config, self.y_length)
# Add as a separate group in case various other
# mobjects are added to self, as for example in
# NumberPlane below
self.axes = VGroup(self.x_axis, self.y_axis)
self.add(*self.axes)
# finds the middle-point on each axis
lines_center_point = [
axis.scaling.function((axis.x_range[1] + axis.x_range[0]) / 2)
for axis in self.axes
]
self.shift(-self.coords_to_point(*lines_center_point))
[docs] @staticmethod
def _update_default_configs(
default_configs: tuple[dict[Any, Any]], passed_configs: tuple[dict[Any, Any]]
) -> None:
"""Takes in two tuples of dicts and return modifies the first such that values from
``passed_configs`` overwrite values in ``default_configs``. If a key does not exist
in default_configs, it is added to the dict.
This method is useful for having defaults in a class and being able to overwrite
them with user-defined input.
Parameters
----------
default_configs
The dict that will be updated.
passed_configs
The dict that will be used to update.
Examples
--------
To create a tuple with one dictionary, add a comma after the element:
.. code-block:: python
self._update_default_configs(
(dict_1,)(
dict_2,
)
)
"""
for default_config, passed_config in zip(default_configs, passed_configs):
if passed_config is not None:
update_dict_recursively(default_config, passed_config)
[docs] def _create_axis(
self,
range_terms: Sequence[float],
axis_config: dict[str, Any],
length: float,
) -> NumberLine:
"""Creates an axis and dynamically adjusts its position depending on where 0 is located on the line.
Parameters
----------
range_terms
The range of the the axis : ``(x_min, x_max, x_step)``.
axis_config
Additional parameters that are passed to :class:`~.NumberLine`.
length
The length of the axis.
Returns
-------
:class:`NumberLine`
Returns a number line based on ``range_terms``.
"""
axis_config["length"] = length
axis = NumberLine(range_terms, **axis_config)
# without the call to _origin_shift, graph does not exist when min > 0 or max < 0
# shifts the axis so that 0 is centered
axis.shift(-axis.number_to_point(self._origin_shift([axis.x_min, axis.x_max])))
return axis
[docs] def coords_to_point(
self, *coords: float | Sequence[float] | Sequence[Sequence[float]] | np.ndarray
) -> np.ndarray:
"""Accepts coordinates from the axes and returns a point with respect to the scene.
Parameters
----------
coords
The coordinates. Each coord is passed as a separate argument: ``ax.coords_to_point(1, 2, 3)``.
Also accepts a list of coordinates
``ax.coords_to_point( [x_0, x_1, ...], [y_0, y_1, ...], ... )``
``ax.coords_to_point( [[x_0, y_0, z_0], [x_1, y_1, z_1]] )``
Returns
-------
np.ndarray
A point with respect to the scene's coordinate system.
The shape of the array will be similar to the shape of the input.
Examples
--------
.. code-block:: pycon
>>> from manim import Axes
>>> import numpy as np
>>> ax = Axes()
>>> np.around(ax.coords_to_point(1, 0, 0), 2)
array([0.86, 0. , 0. ])
>>> np.around(ax.coords_to_point([[0, 1], [1, 1], [1, 0]]), 2)
array([[0. , 0.75, 0. ],
[0.86, 0.75, 0. ],
[0.86, 0. , 0. ]])
>>> np.around(
... ax.coords_to_point([0, 1, 1], [1, 1, 0]), 2
... ) # Transposed version of the above
array([[0. , 0.86, 0.86],
[0.75, 0.75, 0. ],
[0. , 0. , 0. ]])
.. manim:: CoordsToPointExample
:save_last_frame:
class CoordsToPointExample(Scene):
def construct(self):
ax = Axes().add_coordinates()
# a dot with respect to the axes
dot_axes = Dot(ax.coords_to_point(2, 2), color=GREEN)
lines = ax.get_lines_to_point(ax.c2p(2,2))
# a dot with respect to the scene
# the default plane corresponds to the coordinates of the scene.
plane = NumberPlane()
dot_scene = Dot((2,2,0), color=RED)
self.add(plane, dot_scene, ax, dot_axes, lines)
"""
coords = np.asarray(coords)
origin = self.x_axis.number_to_point(
self._origin_shift([self.x_axis.x_min, self.x_axis.x_max]),
)
# Is coords in the format ([[x1 y1 z1] [x2 y2 z2] ...])? (True)
# Or is coords in the format (x, y, z) or ([x1 x2 ...], [y1 y2 ...], [z1 z2 ...])? (False)
# The latter is preferred.
are_coordinates_transposed = False
# If coords is in the format ([[x1 y1 z1] [x2 y2 z2] ...]):
if coords.ndim == 3:
# Extract from original tuple: now coords looks like [[x y z]] or [[x1 y1 z1] [x2 y2 z2] ...].
coords = coords[0]
# If there's a single coord (coords = [[x y z]]), extract it so that
# coords = [x y z] and coords_to_point returns a single point.
if coords.shape[0] == 1:
coords = coords[0]
# Else, if coords looks more like [[x1 y1 z1] [x2 y2 z2] ...], transform them (by
# transposing) into the format [[x1 x2 ...] [y1 y2 ...] [z1 z2 ...]] for later processing.
else:
coords = coords.T
are_coordinates_transposed = True
# Otherwise, coords already looked like (x, y, z) or ([x1 x2 ...], [y1 y2 ...], [z1 z2 ...]),
# so no further processing is needed.
# Now coords should either look like [x y z] or [[x1 x2 ...] [y1 y2 ...] [z1 z2 ...]],
# so it can be iterated directly. Each element is either a float representing a single
# coordinate, or a float ndarray of coordinates corresponding to a single axis.
# Although "points" and "nums" are in plural, there might be a single point or number.
points = self.x_axis.number_to_point(coords[0])
other_axes = self.axes.submobjects[1:]
for axis, nums in zip(other_axes, coords[1:]):
points += axis.number_to_point(nums) - origin
# Return points as is, except if coords originally looked like
# ([x1 x2 ...], [y1 y2 ...], [z1 z2 ...]), which is determined by the conditions below. In
# that case, the current implementation requires that the results have to be transposed.
if are_coordinates_transposed or points.ndim == 1:
return points
return points.T
[docs] def point_to_coords(self, point: Sequence[float]) -> np.ndarray:
"""Accepts a point from the scene and returns its coordinates with respect to the axes.
Parameters
----------
point
The point, i.e. ``RIGHT`` or ``[0, 1, 0]``.
Also accepts a list of points as ``[RIGHT, [0, 1, 0]]``.
Returns
-------
np.ndarray[float]
The coordinates on the axes, i.e. ``[4.0, 7.0]``.
Or a list of coordinates if `point` is a list of points.
Examples
--------
.. code-block:: pycon
>>> from manim import Axes, RIGHT
>>> import numpy as np
>>> ax = Axes(x_range=[0, 10, 2])
>>> np.around(ax.point_to_coords(RIGHT), 2)
array([5.83, 0. ])
>>> np.around(ax.point_to_coords([[0, 0, 1], [1, 0, 0]]), 2)
array([[5. , 0. ],
[5.83, 0. ]])
.. manim:: PointToCoordsExample
:save_last_frame:
class PointToCoordsExample(Scene):
def construct(self):
ax = Axes(x_range=[0, 10, 2]).add_coordinates()
circ = Circle(radius=0.5).shift(UR * 2)
# get the coordinates of the circle with respect to the axes
coords = np.around(ax.point_to_coords(circ.get_right()), decimals=2)
label = (
Matrix([[coords[0]], [coords[1]]]).scale(0.75).next_to(circ, RIGHT)
)
self.add(ax, circ, label, Dot(circ.get_right()))
"""
point = np.asarray(point)
result = np.asarray([axis.point_to_number(point) for axis in self.get_axes()])
if point.ndim == 2:
return result.T
return result
[docs] def get_axes(self) -> VGroup:
"""Gets the axes.
Returns
-------
:class:`~.VGroup`
A pair of axes.
"""
return self.axes
[docs] def get_axis_labels(
self,
x_label: float | str | Mobject = "x",
y_label: float | str | Mobject = "y",
) -> VGroup:
"""Defines labels for the x-axis and y-axis of the graph.
For increased control over the position of the labels,
use :meth:`~.CoordinateSystem.get_x_axis_label` and
:meth:`~.CoordinateSystem.get_y_axis_label`.
Parameters
----------
x_label
The label for the x_axis. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
y_label
The label for the y_axis. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
Returns
-------
:class:`~.VGroup`
A :class:`~.VGroup` of the labels for the x_axis and y_axis.
.. seealso::
:meth:`~.CoordinateSystem.get_x_axis_label`
:meth:`~.CoordinateSystem.get_y_axis_label`
Examples
--------
.. manim:: GetAxisLabelsExample
:save_last_frame:
class GetAxisLabelsExample(Scene):
def construct(self):
ax = Axes()
labels = ax.get_axis_labels(
Tex("x-axis").scale(0.7), Text("y-axis").scale(0.45)
)
self.add(ax, labels)
"""
self.axis_labels = VGroup(
self.get_x_axis_label(x_label),
self.get_y_axis_label(y_label),
)
return self.axis_labels
[docs] def plot_line_graph(
self,
x_values: Iterable[float],
y_values: Iterable[float],
z_values: Iterable[float] | None = None,
line_color: ParsableManimColor = YELLOW,
add_vertex_dots: bool = True,
vertex_dot_radius: float = DEFAULT_DOT_RADIUS,
vertex_dot_style: dict[str, Any] | None = None,
**kwargs: Any,
) -> VDict:
"""Draws a line graph.
The graph connects the vertices formed from zipping
``x_values``, ``y_values`` and ``z_values``. Also adds :class:`Dots <.Dot>` at the
vertices if ``add_vertex_dots`` is set to ``True``.
Parameters
----------
x_values
Iterable of values along the x-axis.
y_values
Iterable of values along the y-axis.
z_values
Iterable of values (zeros if z_values is None) along the z-axis.
line_color
Color for the line graph.
add_vertex_dots
Whether or not to add :class:`~.Dot` at each vertex.
vertex_dot_radius
Radius for the :class:`~.Dot` at each vertex.
vertex_dot_style
Style arguments to be passed into :class:`~.Dot` at each vertex.
kwargs
Additional arguments to be passed into :class:`~.VMobject`.
Returns
-------
:class:`~.VDict`
A VDict containing both the line and dots (if specified). The line can be accessed with: ``line_graph["line_graph"]``.
The dots can be accessed with: ``line_graph["vertex_dots"]``.
Examples
--------
.. manim:: LineGraphExample
:save_last_frame:
class LineGraphExample(Scene):
def construct(self):
plane = NumberPlane(
x_range = (0, 7),
y_range = (0, 5),
x_length = 7,
axis_config={"include_numbers": True},
)
plane.center()
line_graph = plane.plot_line_graph(
x_values = [0, 1.5, 2, 2.8, 4, 6.25],
y_values = [1, 3, 2.25, 4, 2.5, 1.75],
line_color=GOLD_E,
vertex_dot_style=dict(stroke_width=3, fill_color=PURPLE),
stroke_width = 4,
)
self.add(plane, line_graph)
"""
x_values, y_values = map(np.array, (x_values, y_values))
if z_values is None:
z_values = np.zeros(x_values.shape)
line_graph = VDict()
graph = VGroup(color=line_color, **kwargs)
vertices = [
self.coords_to_point(x, y, z)
for x, y, z in zip(x_values, y_values, z_values)
]
graph.set_points_as_corners(vertices)
line_graph["line_graph"] = graph
if add_vertex_dots:
vertex_dot_style = vertex_dot_style or {}
vertex_dots = VGroup(
*(
Dot(point=vertex, radius=vertex_dot_radius, **vertex_dot_style)
for vertex in vertices
)
)
line_graph["vertex_dots"] = vertex_dots
return line_graph
[docs] @staticmethod
def _origin_shift(axis_range: Sequence[float]) -> float:
"""Determines how to shift graph mobjects to compensate when 0 is not on the axis.
Parameters
----------
axis_range
The range of the axis : ``(x_min, x_max, x_step)``.
"""
if axis_range[0] > 0:
# min greater than 0
return axis_range[0]
if axis_range[1] < 0:
# max less than 0
return axis_range[1]
else:
return 0
[docs]class ThreeDAxes(Axes):
"""A 3-dimensional set of axes.
Parameters
----------
x_range
The ``[x_min, x_max, x_step]`` values of the x-axis.
y_range
The ``[y_min, y_max, y_step]`` values of the y-axis.
z_range
The ``[z_min, z_max, z_step]`` values of the z-axis.
x_length
The length of the x-axis.
y_length
The length of the y-axis.
z_length
The length of the z-axis.
z_axis_config
Arguments to be passed to :class:`~.NumberLine` that influence the z-axis.
z_normal
The direction of the normal.
num_axis_pieces
The number of pieces used to construct the axes.
light_source
The direction of the light source.
depth
Currently non-functional.
gloss
Currently non-functional.
kwargs
Additional arguments to be passed to :class:`Axes`.
"""
def __init__(
self,
x_range: Sequence[float] | None = (-6, 6, 1),
y_range: Sequence[float] | None = (-5, 5, 1),
z_range: Sequence[float] | None = (-4, 4, 1),
x_length: float | None = config.frame_height + 2.5,
y_length: float | None = config.frame_height + 2.5,
z_length: float | None = config.frame_height - 1.5,
z_axis_config: dict[str, Any] | None = None,
z_normal: Vector3 = DOWN,
num_axis_pieces: int = 20,
light_source: Sequence[float] = 9 * DOWN + 7 * LEFT + 10 * OUT,
# opengl stuff (?)
depth=None,
gloss=0.5,
**kwargs: dict[str, Any],
) -> None:
super().__init__(
x_range=x_range,
x_length=x_length,
y_range=y_range,
y_length=y_length,
**kwargs,
)
self.z_range = z_range
self.z_length = z_length
self.z_axis_config = {}
self._update_default_configs((self.z_axis_config,), (z_axis_config,))
self.z_axis_config = merge_dicts_recursively(
self.axis_config,
self.z_axis_config,
)
self.z_normal = z_normal
self.num_axis_pieces = num_axis_pieces
self.light_source = light_source
self.dimension = 3
if self.z_axis_config.get("scaling") is None or isinstance(
self.z_axis_config.get("scaling"), LinearBase
):
self.z_axis_config["exclude_origin_tick"] = True
else:
self.z_axis_config["exclude_origin_tick"] = False
z_axis = self._create_axis(self.z_range, self.z_axis_config, self.z_length)
# [ax.x_min, ax.x_max] used to account for LogBase() scaling
# where ax.x_range[0] != ax.x_min
z_origin = self._origin_shift([z_axis.x_min, z_axis.x_max])
z_axis.rotate_about_number(z_origin, -PI / 2, UP)
z_axis.rotate_about_number(z_origin, angle_of_vector(self.z_normal))
z_axis.shift(-z_axis.number_to_point(z_origin))
z_axis.shift(
self.x_axis.number_to_point(
self._origin_shift([self.x_axis.x_min, self.x_axis.x_max]),
),
)
self.axes.add(z_axis)
self.add(z_axis)
self.z_axis = z_axis
if config.renderer == RendererType.CAIRO:
self._add_3d_pieces()
self._set_axis_shading()
def _add_3d_pieces(self) -> None:
for axis in self.axes:
axis.pieces = VGroup(*axis.get_pieces(self.num_axis_pieces))
axis.add(axis.pieces)
axis.set_stroke(width=0, family=False)
axis.set_shade_in_3d(True)
def _set_axis_shading(self) -> None:
def make_func(axis):
vect = self.light_source
return lambda: (
axis.get_edge_center(-vect),
axis.get_edge_center(vect),
)
for axis in self:
for submob in axis.family_members_with_points():
submob.get_gradient_start_and_end_points = make_func(axis)
submob.get_unit_normal = lambda a: np.ones(3)
submob.set_sheen(0.2)
[docs] def get_y_axis_label(
self,
label: float | str | Mobject,
edge: Sequence[float] = UR,
direction: Sequence[float] = UR,
buff: float = SMALL_BUFF,
rotation: float = PI / 2,
rotation_axis: Vector3 = OUT,
**kwargs,
) -> Mobject:
"""Generate a y-axis label.
Parameters
----------
label
The label. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
edge
The edge of the y-axis to which the label will be added, by default ``UR``.
direction
Allows for further positioning of the label from an edge, by default ``UR``.
buff
The distance of the label from the line, by default ``SMALL_BUFF``.
rotation
The angle at which to rotate the label, by default ``PI/2``.
rotation_axis
The axis about which to rotate the label, by default ``OUT``.
Returns
-------
:class:`~.Mobject`
The positioned label.
Examples
--------
.. manim:: GetYAxisLabelExample
:save_last_frame:
class GetYAxisLabelExample(ThreeDScene):
def construct(self):
ax = ThreeDAxes()
lab = ax.get_y_axis_label(Tex("$y$-label"))
self.set_camera_orientation(phi=2*PI/5, theta=PI/5)
self.add(ax, lab)
"""
positioned_label = self._get_axis_label(
label, self.get_y_axis(), edge, direction, buff=buff, **kwargs
)
positioned_label.rotate(rotation, axis=rotation_axis)
return positioned_label
[docs] def get_z_axis_label(
self,
label: float | str | Mobject,
edge: Vector3 = OUT,
direction: Vector3 = RIGHT,
buff: float = SMALL_BUFF,
rotation: float = PI / 2,
rotation_axis: Vector3 = RIGHT,
**kwargs: Any,
) -> Mobject:
"""Generate a z-axis label.
Parameters
----------
label
The label. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
edge
The edge of the z-axis to which the label will be added, by default ``OUT``.
direction
Allows for further positioning of the label from an edge, by default ``RIGHT``.
buff
The distance of the label from the line, by default ``SMALL_BUFF``.
rotation
The angle at which to rotate the label, by default ``PI/2``.
rotation_axis
The axis about which to rotate the label, by default ``RIGHT``.
Returns
-------
:class:`~.Mobject`
The positioned label.
Examples
--------
.. manim:: GetZAxisLabelExample
:save_last_frame:
class GetZAxisLabelExample(ThreeDScene):
def construct(self):
ax = ThreeDAxes()
lab = ax.get_z_axis_label(Tex("$z$-label"))
self.set_camera_orientation(phi=2*PI/5, theta=PI/5)
self.add(ax, lab)
"""
positioned_label = self._get_axis_label(
label, self.get_z_axis(), edge, direction, buff=buff, **kwargs
)
positioned_label.rotate(rotation, axis=rotation_axis)
return positioned_label
[docs] def get_axis_labels(
self,
x_label: float | str | Mobject = "x",
y_label: float | str | Mobject = "y",
z_label: float | str | Mobject = "z",
) -> VGroup:
"""Defines labels for the x_axis and y_axis of the graph.
For increased control over the position of the labels,
use :meth:`~.CoordinateSystem.get_x_axis_label`,
:meth:`~.ThreeDAxes.get_y_axis_label`, and
:meth:`~.ThreeDAxes.get_z_axis_label`.
Parameters
----------
x_label
The label for the x_axis. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
y_label
The label for the y_axis. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
z_label
The label for the z_axis. Defaults to :class:`~.MathTex` for ``str`` and ``float`` inputs.
Returns
-------
:class:`~.VGroup`
A :class:`~.VGroup` of the labels for the x_axis, y_axis, and z_axis.
.. seealso::
:meth:`~.CoordinateSystem.get_x_axis_label`
:meth:`~.ThreeDAxes.get_y_axis_label`
:meth:`~.ThreeDAxes.get_z_axis_label`
Examples
--------
.. manim:: GetAxisLabelsExample
:save_last_frame:
class GetAxisLabelsExample(ThreeDScene):
def construct(self):
self.set_camera_orientation(phi=2*PI/5, theta=PI/5)
axes = ThreeDAxes()
labels = axes.get_axis_labels(
Tex("x-axis").scale(0.7), Text("y-axis").scale(0.45), Text("z-axis").scale(0.45)
)
self.add(axes, labels)
"""
self.axis_labels = VGroup(
self.get_x_axis_label(x_label),
self.get_y_axis_label(y_label),
self.get_z_axis_label(z_label),
)
return self.axis_labels
[docs]class NumberPlane(Axes):
"""Creates a cartesian plane with background lines.
Parameters
----------
x_range
The ``[x_min, x_max, x_step]`` values of the plane in the horizontal direction.
y_range
The ``[y_min, y_max, y_step]`` values of the plane in the vertical direction.
x_length
The width of the plane.
y_length
The height of the plane.
background_line_style
Arguments that influence the construction of the background lines of the plane.
faded_line_style
Similar to :attr:`background_line_style`, affects the construction of the scene's background lines.
faded_line_ratio
Determines the number of boxes within the background lines: :code:`2` = 4 boxes, :code:`3` = 9 boxes.
make_smooth_after_applying_functions
Currently non-functional.
kwargs
Additional arguments to be passed to :class:`Axes`.
.. note::
If :attr:`x_length` or :attr:`y_length` are not defined, they are automatically calculated such that
one unit on each axis is one Manim unit long.
Examples
--------
.. manim:: NumberPlaneExample
:save_last_frame:
class NumberPlaneExample(Scene):
def construct(self):
number_plane = NumberPlane(
background_line_style={
"stroke_color": TEAL,
"stroke_width": 4,
"stroke_opacity": 0.6
}
)
self.add(number_plane)
.. manim:: NumberPlaneScaled
:save_last_frame:
class NumberPlaneScaled(Scene):
def construct(self):
number_plane = NumberPlane(
x_range=(-4, 11, 1),
y_range=(-3, 3, 1),
x_length=5,
y_length=2,
).move_to(LEFT*3)
number_plane_scaled_y = NumberPlane(
x_range=(-4, 11, 1),
x_length=5,
y_length=4,
).move_to(RIGHT*3)
self.add(number_plane)
self.add(number_plane_scaled_y)
"""
def __init__(
self,
x_range: Sequence[float]
| None = (
-config["frame_x_radius"],
config["frame_x_radius"],
1,
),
y_range: Sequence[float]
| None = (
-config["frame_y_radius"],
config["frame_y_radius"],
1,
),
x_length: float | None = None,
y_length: float | None = None,
background_line_style: dict[str, Any] | None = None,
faded_line_style: dict[str, Any] | None = None,
faded_line_ratio: int = 1,
make_smooth_after_applying_functions: bool = True,
**kwargs: dict[str, Any],
):
# configs
self.axis_config = {
"stroke_width": 2,
"include_ticks": False,
"include_tip": False,
"line_to_number_buff": SMALL_BUFF,
"label_direction": DR,
"font_size": 24,
}
self.y_axis_config = {"label_direction": DR}
self.background_line_style = {
"stroke_color": BLUE_D,
"stroke_width": 2,
"stroke_opacity": 1,
}
self._update_default_configs(
(self.axis_config, self.y_axis_config, self.background_line_style),
(
kwargs.pop("axis_config", None),
kwargs.pop("y_axis_config", None),
background_line_style,
),
)
# Defaults to a faded version of line_config
self.faded_line_style = faded_line_style
self.faded_line_ratio = faded_line_ratio
self.make_smooth_after_applying_functions = make_smooth_after_applying_functions
# init
super().__init__(
x_range=x_range,
y_range=y_range,
x_length=x_length,
y_length=y_length,
axis_config=self.axis_config,
y_axis_config=self.y_axis_config,
**kwargs,
)
self._init_background_lines()
[docs] def _init_background_lines(self) -> None:
"""Will init all the lines of NumberPlanes (faded or not)"""
if self.faded_line_style is None:
style = dict(self.background_line_style)
# For anything numerical, like stroke_width
# and stroke_opacity, chop it in half
for key in style:
if isinstance(style[key], numbers.Number):
style[key] *= 0.5
self.faded_line_style = style
self.background_lines, self.faded_lines = self._get_lines()
self.background_lines.set_style(
**self.background_line_style,
)
self.faded_lines.set_style(
**self.faded_line_style,
)
self.add_to_back(
self.faded_lines,
self.background_lines,
)
[docs] def _get_lines(self) -> tuple[VGroup, VGroup]:
"""Generate all the lines, faded and not faded.
Two sets of lines are generated: one parallel to the X-axis, and parallel to the Y-axis.
Returns
-------
Tuple[:class:`~.VGroup`, :class:`~.VGroup`]
The first (i.e the non faded lines) and second (i.e the faded lines) sets of lines, respectively.
"""
x_axis = self.get_x_axis()
y_axis = self.get_y_axis()
x_lines1, x_lines2 = self._get_lines_parallel_to_axis(
x_axis,
y_axis,
self.y_axis.x_range[2],
self.faded_line_ratio,
)
y_lines1, y_lines2 = self._get_lines_parallel_to_axis(
y_axis,
x_axis,
self.x_axis.x_range[2],
self.faded_line_ratio,
)
# TODO this was added so that we can run tests on NumberPlane
# In the future these attributes will be tacked onto self.background_lines
self.x_lines = x_lines1
self.y_lines = y_lines1
lines1 = VGroup(*x_lines1, *y_lines1)
lines2 = VGroup(*x_lines2, *y_lines2)
return lines1, lines2
[docs] def _get_lines_parallel_to_axis(
self,
axis_parallel_to: NumberLine,
axis_perpendicular_to: NumberLine,
freq: float,
ratio_faded_lines: int,
) -> tuple[VGroup, VGroup]:
"""Generate a set of lines parallel to an axis.
Parameters
----------
axis_parallel_to
The axis with which the lines will be parallel.
axis_perpendicular_to
The axis with which the lines will be perpendicular.
ratio_faded_lines
The ratio between the space between faded lines and the space between non-faded lines.
freq
Frequency of non-faded lines (number of non-faded lines per graph unit).
Returns
-------
Tuple[:class:`~.VGroup`, :class:`~.VGroup`]
The first (i.e the non-faded lines parallel to `axis_parallel_to`) and second
(i.e the faded lines parallel to `axis_parallel_to`) sets of lines, respectively.
"""
line = Line(axis_parallel_to.get_start(), axis_parallel_to.get_end())
if ratio_faded_lines == 0: # don't show faded lines
ratio_faded_lines = 1 # i.e. set ratio to 1
step = (1 / ratio_faded_lines) * freq
lines1 = VGroup()
lines2 = VGroup()
unit_vector_axis_perp_to = axis_perpendicular_to.get_unit_vector()
# need to unpack all three values
x_min, x_max, _ = axis_perpendicular_to.x_range
# account for different axis scalings (logarithmic), where
# negative values do not exist and [-2 , 4] should output lines
# similar to [0, 6]
if axis_perpendicular_to.x_min > 0 and x_min < 0:
x_min, x_max = (0, np.abs(x_min) + np.abs(x_max))
# min/max used in case range does not include 0. i.e. if (2,6):
# the range becomes (0,4), not (0,6).
ranges = (
[0],
np.arange(step, min(x_max - x_min, x_max), step),
np.arange(-step, max(x_min - x_max, x_min), -step),
)
for inputs in ranges:
for k, x in enumerate(inputs):
new_line = line.copy()
new_line.shift(unit_vector_axis_perp_to * x)
if (k + 1) % ratio_faded_lines == 0:
lines1.add(new_line)
else:
lines2.add(new_line)
return lines1, lines2
def get_vector(self, coords: Sequence[ManimFloat], **kwargs: Any) -> Arrow:
kwargs["buff"] = 0
return Arrow(
self.coords_to_point(0, 0), self.coords_to_point(*coords), **kwargs
)
def prepare_for_nonlinear_transform(self, num_inserted_curves: int = 50) -> Self:
for mob in self.family_members_with_points():
num_curves = mob.get_num_curves()
if num_inserted_curves > num_curves:
mob.insert_n_curves(num_inserted_curves - num_curves)
return self
[docs]class PolarPlane(Axes):
r"""Creates a polar plane with background lines.
Parameters
----------
azimuth_step
The number of divisions in the azimuth (also known as the `angular coordinate` or `polar angle`). If ``None`` is specified then it will use the default
specified by ``azimuth_units``:
- ``"PI radians"`` or ``"TAU radians"``: 20
- ``"degrees"``: 36
- ``"gradians"``: 40
- ``None``: 1
A non-integer value will result in a partial division at the end of the circle.
size
The diameter of the plane.
radius_step
The distance between faded radius lines.
radius_max
The maximum value of the radius.
azimuth_units
Specifies a default labelling system for the azimuth. Choices are:
- ``"PI radians"``: Fractional labels in the interval :math:`\left[0, 2\pi\right]` with :math:`\pi` as a constant.
- ``"TAU radians"``: Fractional labels in the interval :math:`\left[0, \tau\right]` (where :math:`\tau = 2\pi`) with :math:`\tau` as a constant.
- ``"degrees"``: Decimal labels in the interval :math:`\left[0, 360\right]` with a degree (:math:`^{\circ}`) symbol.
- ``"gradians"``: Decimal labels in the interval :math:`\left[0, 400\right]` with a superscript "g" (:math:`^{g}`).
- ``None``: Decimal labels in the interval :math:`\left[0, 1\right]`.
azimuth_compact_fraction
If the ``azimuth_units`` choice has fractional labels, choose whether to
combine the constant in a compact form :math:`\tfrac{xu}{y}` as opposed to
:math:`\tfrac{x}{y}u`, where :math:`u` is the constant.
azimuth_offset
The angle offset of the azimuth, expressed in radians.
azimuth_direction
The direction of the azimuth.
- ``"CW"``: Clockwise.
- ``"CCW"``: Anti-clockwise.
azimuth_label_buff
The buffer for the azimuth labels.
azimuth_label_font_size
The font size of the azimuth labels.
radius_config
The axis config for the radius.
Examples
--------
.. manim:: PolarPlaneExample
:ref_classes: PolarPlane
:save_last_frame:
class PolarPlaneExample(Scene):
def construct(self):
polarplane_pi = PolarPlane(
azimuth_units="PI radians",
size=6,
azimuth_label_font_size=33.6,
radius_config={"font_size": 33.6},
).add_coordinates()
self.add(polarplane_pi)
"""
def __init__(
self,
radius_max: float = config["frame_y_radius"],
size: float | None = None,
radius_step: float = 1,
azimuth_step: float | None = None,
azimuth_units: str | None = "PI radians",
azimuth_compact_fraction: bool = True,
azimuth_offset: float = 0,
azimuth_direction: str = "CCW",
azimuth_label_buff: float = SMALL_BUFF,
azimuth_label_font_size: float = 24,
radius_config: dict[str, Any] | None = None,
background_line_style: dict[str, Any] | None = None,
faded_line_style: dict[str, Any] | None = None,
faded_line_ratio: int = 1,
make_smooth_after_applying_functions: bool = True,
**kwargs: Any,
) -> None:
# error catching
if azimuth_units in ["PI radians", "TAU radians", "degrees", "gradians", None]:
self.azimuth_units = azimuth_units
else:
raise ValueError(
"Invalid azimuth units. Expected one of: PI radians, TAU radians, degrees, gradians or None.",
)
if azimuth_direction in ["CW", "CCW"]:
self.azimuth_direction = azimuth_direction
else:
raise ValueError("Invalid azimuth units. Expected one of: CW, CCW.")
# configs
self.radius_config = {
"stroke_width": 2,
"include_ticks": False,
"include_tip": False,
"line_to_number_buff": SMALL_BUFF,
"label_direction": DL,
"font_size": 24,
}
self.background_line_style = {
"stroke_color": BLUE_D,
"stroke_width": 2,
"stroke_opacity": 1,
}
self.azimuth_step = (
(
{
"PI radians": 20,
"TAU radians": 20,
"degrees": 36,
"gradians": 40,
None: 1,
}[azimuth_units]
)
if azimuth_step is None
else azimuth_step
)
self._update_default_configs(
(self.radius_config, self.background_line_style),
(radius_config, background_line_style),
)
# Defaults to a faded version of line_config
self.faded_line_style = faded_line_style
self.faded_line_ratio = faded_line_ratio
self.make_smooth_after_applying_functions = make_smooth_after_applying_functions
self.azimuth_offset = azimuth_offset
self.azimuth_label_buff = azimuth_label_buff
self.azimuth_label_font_size = azimuth_label_font_size
self.azimuth_compact_fraction = azimuth_compact_fraction
# init
super().__init__(
x_range=np.array((-radius_max, radius_max, radius_step)),
y_range=np.array((-radius_max, radius_max, radius_step)),
x_length=size,
y_length=size,
axis_config=self.radius_config,
**kwargs,
)
self._init_background_lines()
[docs] def _init_background_lines(self) -> None:
"""Will init all the lines of NumberPlanes (faded or not)"""
if self.faded_line_style is None:
style = dict(self.background_line_style)
# For anything numerical, like stroke_width
# and stroke_opacity, chop it in half
for key in style:
if isinstance(style[key], numbers.Number):
style[key] *= 0.5
self.faded_line_style = style
self.background_lines, self.faded_lines = self._get_lines()
self.background_lines.set_style(
**self.background_line_style,
)
self.faded_lines.set_style(
**self.faded_line_style,
)
self.add_to_back(
self.faded_lines,
self.background_lines,
)
[docs] def _get_lines(self) -> tuple[VGroup, VGroup]:
"""Generate all the lines and circles, faded and not faded.
Returns
-------
Tuple[:class:`~.VGroup`, :class:`~.VGroup`]
The first (i.e the non faded lines and circles) and second (i.e the faded lines and circles) sets of lines and circles, respectively.
"""
center = self.get_origin()
ratio_faded_lines = self.faded_line_ratio
offset = self.azimuth_offset
if ratio_faded_lines == 0: # don't show faded lines
ratio_faded_lines = 1 # i.e. set ratio to 1
rstep = (1 / ratio_faded_lines) * self.x_axis.x_range[2]
astep = (1 / ratio_faded_lines) * (TAU * (1 / self.azimuth_step))
rlines1 = VGroup()
rlines2 = VGroup()
alines1 = VGroup()
alines2 = VGroup()
rinput = np.arange(0, self.x_axis.x_range[1] + rstep, rstep)
ainput = np.arange(0, TAU, astep)
unit_vector = self.x_axis.get_unit_vector()[0]
for k, x in enumerate(rinput):
new_line = Circle(radius=x * unit_vector)
if k % ratio_faded_lines == 0:
alines1.add(new_line)
else:
alines2.add(new_line)
line = Line(center, self.get_x_axis().get_end())
for k, x in enumerate(ainput):
new_line = line.copy()
new_line.rotate(x + offset, about_point=center)
if k % ratio_faded_lines == 0:
rlines1.add(new_line)
else:
rlines2.add(new_line)
lines1 = VGroup(*rlines1, *alines1)
lines2 = VGroup(*rlines2, *alines2)
return lines1, lines2
[docs] def get_axes(self) -> VGroup:
"""Gets the axes.
Returns
-------
:class:`~.VGroup`
A pair of axes.
"""
return self.axes
def get_vector(self, coords: Sequence[ManimFloat], **kwargs: Any) -> Arrow:
kwargs["buff"] = 0
return Arrow(
self.coords_to_point(0, 0), self.coords_to_point(*coords), **kwargs
)
def prepare_for_nonlinear_transform(self, num_inserted_curves: int = 50) -> Self:
for mob in self.family_members_with_points():
num_curves = mob.get_num_curves()
if num_inserted_curves > num_curves:
mob.insert_n_curves(num_inserted_curves - num_curves)
return self
[docs] def get_coordinate_labels(
self,
r_values: Iterable[float] | None = None,
a_values: Iterable[float] | None = None,
**kwargs: Any,
) -> VDict:
"""Gets labels for the coordinates
Parameters
----------
r_values
Iterable of values along the radius, by default None.
a_values
Iterable of values along the azimuth, by default None.
Returns
-------
VDict
Labels for the radius and azimuth values.
"""
if r_values is None:
r_values = [r for r in self.get_x_axis().get_tick_range() if r >= 0]
if a_values is None:
a_values = np.arange(0, 1, 1 / self.azimuth_step)
r_mobs = self.get_x_axis().add_numbers(r_values)
if self.azimuth_direction == "CCW":
d = 1
elif self.azimuth_direction == "CW":
d = -1
else:
raise ValueError("Invalid azimuth direction. Expected one of: CW, CCW")
a_points = [
{
"label": i,
"point": np.array(
[
self.get_right()[0]
* np.cos(d * (i * TAU) + self.azimuth_offset),
self.get_right()[0]
* np.sin(d * (i * TAU) + self.azimuth_offset),
0,
],
),
}
for i in a_values
]
if self.azimuth_units == "PI radians" or self.azimuth_units == "TAU radians":
a_tex = [
self.get_radian_label(
i["label"],
font_size=self.azimuth_label_font_size,
).next_to(
i["point"],
direction=i["point"],
aligned_edge=i["point"],
buff=self.azimuth_label_buff,
)
for i in a_points
]
elif self.azimuth_units == "degrees":
a_tex = [
MathTex(
f'{360 * i["label"]:g}' + r"^{\circ}",
font_size=self.azimuth_label_font_size,
).next_to(
i["point"],
direction=i["point"],
aligned_edge=i["point"],
buff=self.azimuth_label_buff,
)
for i in a_points
]
elif self.azimuth_units == "gradians":
a_tex = [
MathTex(
f'{400 * i["label"]:g}' + r"^{g}",
font_size=self.azimuth_label_font_size,
).next_to(
i["point"],
direction=i["point"],
aligned_edge=i["point"],
buff=self.azimuth_label_buff,
)
for i in a_points
]
elif self.azimuth_units is None:
a_tex = [
MathTex(
f'{i["label"]:g}',
font_size=self.azimuth_label_font_size,
).next_to(
i["point"],
direction=i["point"],
aligned_edge=i["point"],
buff=self.azimuth_label_buff,
)
for i in a_points
]
a_mobs = VGroup(*a_tex)
self.coordinate_labels = VGroup(r_mobs, a_mobs)
return self.coordinate_labels
[docs] def add_coordinates(
self,
r_values: Iterable[float] | None = None,
a_values: Iterable[float] | None = None,
) -> Self:
"""Adds the coordinates.
Parameters
----------
r_values
Iterable of values along the radius, by default None.
a_values
Iterable of values along the azimuth, by default None.
"""
self.add(self.get_coordinate_labels(r_values, a_values))
return self
def get_radian_label(self, number, font_size: float = 24, **kwargs: Any) -> MathTex:
constant_label = {"PI radians": r"\pi", "TAU radians": r"\tau"}[
self.azimuth_units
]
division = number * {"PI radians": 2, "TAU radians": 1}[self.azimuth_units]
frac = fr.Fraction(division).limit_denominator(max_denominator=100)
if frac.numerator == 0 & frac.denominator == 0:
string = r"0"
elif frac.numerator == 1 and frac.denominator == 1:
string = constant_label
elif frac.numerator == 1:
if self.azimuth_compact_fraction:
string = (
r"\tfrac{" + constant_label + r"}{" + str(frac.denominator) + "}"
)
else:
string = r"\tfrac{1}{" + str(frac.denominator) + "}" + constant_label
elif frac.denominator == 1:
string = str(frac.numerator) + constant_label
else:
if self.azimuth_compact_fraction:
string = (
r"\tfrac{"
+ str(frac.numerator)
+ constant_label
+ r"}{"
+ str(frac.denominator)
+ r"}"
)
else:
string = (
r"\tfrac{"
+ str(frac.numerator)
+ r"}{"
+ str(frac.denominator)
+ r"}"
+ constant_label
)
return MathTex(string, font_size=font_size, **kwargs)
[docs]class ComplexPlane(NumberPlane):
"""A :class:`~.NumberPlane` specialized for use with complex numbers.
Examples
--------
.. manim:: ComplexPlaneExample
:save_last_frame:
:ref_classes: Dot MathTex
class ComplexPlaneExample(Scene):
def construct(self):
plane = ComplexPlane().add_coordinates()
self.add(plane)
d1 = Dot(plane.n2p(2 + 1j), color=YELLOW)
d2 = Dot(plane.n2p(-3 - 2j), color=YELLOW)
label1 = MathTex("2+i").next_to(d1, UR, 0.1)
label2 = MathTex("-3-2i").next_to(d2, UR, 0.1)
self.add(
d1,
label1,
d2,
label2,
)
"""
def __init__(self, **kwargs: Any) -> None:
super().__init__(
**kwargs,
)
[docs] def number_to_point(self, number: float | complex) -> np.ndarray:
"""Accepts a float/complex number and returns the equivalent point on the plane.
Parameters
----------
number
The number. Can be a float or a complex number.
Returns
-------
np.ndarray
The point on the plane.
"""
number = complex(number)
return self.coords_to_point(number.real, number.imag)
[docs] def n2p(self, number: float | complex) -> np.ndarray:
"""Abbreviation for :meth:`number_to_point`."""
return self.number_to_point(number)
[docs] def point_to_number(self, point: Point3D) -> complex:
"""Accepts a point and returns a complex number equivalent to that point on the plane.
Parameters
----------
point
The point in manim's coordinate-system
Returns
-------
complex
A complex number consisting of real and imaginary components.
"""
x, y = self.point_to_coords(point)
return complex(x, y)
[docs] def p2n(self, point: Point3D) -> complex:
"""Abbreviation for :meth:`point_to_number`."""
return self.point_to_number(point)
[docs] def _get_default_coordinate_values(self) -> list[float | complex]:
"""Generate a list containing the numerical values of the plane's labels.
Returns
-------
List[float | complex]
A list of floats representing the x-axis and complex numbers representing the y-axis.
"""
x_numbers = self.get_x_axis().get_tick_range()
y_numbers = self.get_y_axis().get_tick_range()
y_numbers = [complex(0, y) for y in y_numbers if y != 0]
return [*x_numbers, *y_numbers]
[docs] def get_coordinate_labels(
self, *numbers: Iterable[float | complex], **kwargs: Any
) -> VGroup:
"""Generates the :class:`~.DecimalNumber` mobjects for the coordinates of the plane.
Parameters
----------
numbers
An iterable of floats/complex numbers. Floats are positioned along the x-axis, complex numbers along the y-axis.
kwargs
Additional arguments to be passed to :meth:`~.NumberLine.get_number_mobject`, i.e. :class:`~.DecimalNumber`.
Returns
-------
:class:`~.VGroup`
A :class:`~.VGroup` containing the positioned label mobjects.
"""
# TODO: Make this work the same as coord_sys.add_coordinates()
if len(numbers) == 0:
numbers = self._get_default_coordinate_values()
self.coordinate_labels = VGroup()
for number in numbers:
z = complex(number)
if abs(z.imag) > abs(z.real):
axis = self.get_y_axis()
value = z.imag
kwargs["unit"] = "i"
else:
axis = self.get_x_axis()
value = z.real
number_mob = axis.get_number_mobject(value, **kwargs)
self.coordinate_labels.add(number_mob)
return self.coordinate_labels
[docs] def add_coordinates(
self, *numbers: Iterable[float | complex], **kwargs: Any
) -> Self:
"""Adds the labels produced from :meth:`~.NumberPlane.get_coordinate_labels` to the plane.
Parameters
----------
numbers
An iterable of floats/complex numbers. Floats are positioned along the x-axis, complex numbers along the y-axis.
kwargs
Additional arguments to be passed to :meth:`~.NumberLine.get_number_mobject`, i.e. :class:`~.DecimalNumber`.
"""
self.add(self.get_coordinate_labels(*numbers, **kwargs))
return self