如何更新sympy图的matplotlib元素？

Question

下面的代码展示了一个使用matplotlib绘制的函数图像，函数是y > 5/x，用户可以在图上平移或缩放时填充图形。

import numpy as np
import matplotlib.pyplot as plt

fig, ax = plt.subplots()

x_range = np.linspace(-10, 10, 400)
y_range = 5 / x_range

line, = ax.plot(x_range, y_range, 'r', linewidth=2, linestyle='--')

ax.fill_between(x_range, y_range, y_range.max(), alpha=0.3, color='gray')

ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_title('Inequality: y > 5 / x')

ax.axhline(0, color='black',linewidth=3)
ax.axvline(0, color='black',linewidth=3)

ax.grid(color='gray', linestyle='--', linewidth=0.5)

def update_limits(event):
    xlim = ax.get_xlim()
    x_range = np.linspace(xlim[0], xlim[1], max(200, int(200 * (xlim[1] - xlim[0]))))
    y_range = 5 / x_range
    line.set_data(x_range, y_range)
    for collection in ax.collections:
        collection.remove()
    ax.fill_between(x_range, y_range, max(y_range.max(), ax.get_ylim()[1]), alpha=0.3, color='gray')
    plt.draw()

fig.canvas.mpl_connect('button_release_event', update_limits)
plt.show()

我一直在尝试把这个概念用代码实现，使用的是sympy模块（通过访问Matplotlib的后端），因为它对代数函数的控制更好。不过，下面的代码在用户平移时似乎没有填充图形。这是为什么呢？我该怎么解决这个问题呢？

import sympy as sp
from sympy.plotting.plot import MatplotlibBackend

x, y = sp.symbols('x y')

# Define the implicit plot
p1 = sp.plot_implicit(sp.And(y > 5 / x), (x, -10, 10), (y, -10, 10), show=False)

mplPlot = MatplotlibBackend(p1)

mplPlot.process_series()
mplPlot.fig.tight_layout()
mplPlot.ax[0].set_xlabel("x-axis")
mplPlot.ax[0].set_ylabel("y-axis")


def update_limits(event):
    global mplPlot
    xmin, xmax = mplPlot.ax[0].get_xlim()
    ymin, ymax = mplPlot.ax[0].get_ylim()

    p1 = sp.plot_implicit(sp.And(y > 5 / x), (x, xmin, xmax), (y, ymin, ymax), show=False)

    mplPlot = MatplotlibBackend(p1)
    mplPlot.process_series()
    mplPlot.fig.tight_layout()

    mplPlot.ax[0].set_xlabel("x-axis")
    mplPlot.ax[0].set_ylabel("y-axis")
    mplPlot.plt.draw()



mplPlot.fig.canvas.mpl_connect('button_release_event', update_limits)
mplPlot.plt.show()

更新：经过一些调试（使用sympy代码），我发现xmax和xmin这两个变量在update_limits函数中只改变了一次，之后在程序运行的整个过程中都保持不变。如果可以的话，我也想知道这是为什么。

更新 2：如果不运行mplPlot.plt.draw()，而是运行mplPlot.plt.show()，会创建一个新窗口，显示正确的图形。但这不是我想要的，因为我希望所有的变化都在同一个窗口中显示。当我这样做时，还发现了另一个问题，就是当我在图的第四象限平移得足够远时，图形似乎变得无响应。这并不是每次都会发生，我也找不到原因。如果有人知道这是为什么，请随时补充到你的回答中！

Answer 1

我打算使用SymPy绘图后端的模块，因为它已经写了很多代码来支持交互功能。在写这个回答的时候，版本3.1.1已经发布，但它还没有实现平移/缩放等事件。不过，我们可以很容易地自己实现这些功能。

你可以先尝试一个隐式绘图：

%matplotlib widget
from sympy import *
from spb import *
from matplotlib.colors import ListedColormap
var("x, y")
     
g = graphics(
    implicit_2d(
        y>5/x, (x, -10, 10), (y, -200, 200), n=500,
        rendering_kw={"alpha": 0.3, "cmap": ListedColormap(["#ffffff00", "gray"])},
        border_kw={"linestyles": "--", "cmap": ListedColormap(["r", "r"])}
    ),
    xlabel="x", ylabel="y", title="y > 5/x", show=False
)

def _update_axis_limits(event):
    xlim = g.ax.get_xlim()
    ylim = g.ax.get_ylim()
    limits = [xlim, ylim]
    all_params = {}
    for s in g.series:
        new_ranges = []
        for r, l in zip(s.ranges, limits):
            new_ranges.append((r[0], *l))
        s.ranges = new_ranges
        # inform the data series that they must generate new data on next update
        s._interactive_ranges = True
        s.is_interactive = True
        # extract any existing parameters
        all_params = g.merge({}, all_params)
    # create new data and update the plot
    g.update_interactive(all_params)

g.show(block=False)
g.fig.canvas.mpl_connect('button_release_event', _update_axis_limits)

这个可视化有个问题：在x=0的时候，函数是未定义的，所以不应该有那条垂直的红色虚线。隐式绘图依赖于Matplotlib的contour功能：在等高线绘图中实现未定义的“点/线”是非常困难的。

我们可以通过使用fill_between来改善可视化，就像你最初的方法那样。然而，绘图模块并没有实现这个功能。我们必须自己创建它，像这样：

import numpy as np
from sympy import *
from spb import *
from spb.series import LineOver1DRangeSeries
from spb.backends.matplotlib.renderers.renderer import MatplotlibRenderer

# Things to know in order to extend the plotting module capabilities:
#
# 1. data is generated by some data series.
# 2. each data series is mapped to a particular renderer

# Let's create a data series.
# Matplotlib's `fill_between`: https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.fill_between.html
# requires the coordinates of the first curve and second curve.
# The first curve is just an ordinary line corresponding to the y-values of your expressions.
# The second curve is the maximum value of the y-coordinate of the first curve.
# So, the data is identical to the one we would use to render a line.
class FillBetweenSeries(LineOver1DRangeSeries):
    pass

# Next, we must create a renderer, which must have two methods:
#
# 1. a `draw` method, where the initial handle will be created.
# 2. an `update` method, where the handle will be updated with new data.
# 
# More information can be found here: https://sympy-plot-backends.readthedocs.io/en/latest/tutorials/tut-6.html
def draw(renderer, data):
    ax = renderer.plot.ax
    s = renderer.series
    merge = renderer.plot.merge
    x, y = data
    rkw = merge({}, {"y2": np.nanmax(y)}, s.rendering_kw)
    handles = [
        ax.fill_between(x, y, **rkw)
    ]
    return handles

def update(renderer, data, handles):
    for h in handles:
        h.remove()
    handles[0] = draw(renderer, data)[0]
    
class FillBetweenRenderer(MatplotlibRenderer):
    draw_update_map = {
        draw: update
    }

# Next, we need to inform the backend that when a FillBetweenSeries is
# ecountered, it will be rendered with FillBetweenRenderer
MB.renderers_map[FillBetweenSeries] = FillBetweenRenderer

# Next, let's create a function similar to the ones exposed by the plotting module.
# Note that it returns a list of series. Depending on the visualization
# that you are trying to create, you may need more than one data
# series...
def fill_between(expr, range, label="", rendering_kw={}, **kwargs):
    """
    Parameters
    ----------
    expr : Expr
        The symbolic expression
    ramge : (symbol, min, max)
        Initial range in which to plot the expression
    label : str
        Eventual label to be shown on the legend
    rendering_kw : dict
        A dictionary containing keys/values which are going to
        be passed to matplotlib.fill_between.
    **kwargs : 
        Keyword arguments related to `line()`.
    """
    series = [
        FillBetweenSeries(expr, range, label, rendering_kw=rendering_kw, **kwargs)
    ]
    return series

# Finally, we are ready to use it
var("x")
g = graphics(
    fill_between(
        5/x, (x, -10, 10), rendering_kw={"color": "gray", "alpha": 0.3},
        exclude=[0] # exclude this point, where the function is undefined
    ),
    line(
        5/x, (x, -10, 10), rendering_kw={"color": "r", "linestyle": "--"},
        exclude=[0] # exclude this point, where the function is undefined
    ),
    xlabel="x", ylabel="y", title="y > 5/x", show=False, legend=False
)

def _update_axis_limits(event):
    xlim = g.ax.get_xlim()
    ylim = g.ax.get_ylim()
    limits = [xlim, ylim]
    all_params = {}
    for s in g.series:
        new_ranges = []
        for r, l in zip(s.ranges, limits):
            new_ranges.append((r[0], *l))
        s.ranges = new_ranges
        # inform the data series that they must generate new data on next update
        s._interactive_ranges = True
        s.is_interactive = True
        # extract any existing parameters
        all_params = g.merge({}, all_params)
    # create new data and update the plot
    g.update_interactive(all_params)

g.show(block=False)
g.fig.canvas.mpl_connect('button_release_event', _update_axis_limits)