Python使用SymPy解决Manim曲线绘制速度不均的问题

from manim import *
import numpy as np

class BadRoseCurve(Scene):
    def construct(self):
        axes = Axes(x_range=[-3, 3], y_range=[-3, 3])
        self.add(axes)

        # 用累积密度法生成非均匀 theta：尖端稀疏(快)，中部密集(慢)
        n_points = 500
        t = np.linspace(0, 2 * PI, n_points)
        # 密度函数：在 cos(5t)=0 处(花瓣中部)密度高，在 |cos(5t)|=1 处(尖端)密度低
        density = 1 + 3 * np.sin(5 * t) ** 2
        theta = np.cumsum(density)
        theta = theta / theta[-1] * PI  # 归一化到 [0, π]（k为奇数时只需π即可画完）
        r = np.cos(5 * theta)
        x = r * 2 * np.cos(theta)
        y = r * 2 * np.sin(theta)

        points = [axes.c2p(x[i], y[i]) for i in range(n_points)]
        curve = VMobject(color=PINK, stroke_width=3)
        curve.set_points_as_corners(points)

        # 用 Create 画曲线——速度明显不均匀！
        self.play(Create(curve), run_time=5, rate_func=linear)
        self.wait()

import sympy as sp

theta = sp.Symbol('theta', real=True)
n = 5  # 五瓣玫瑰线

# 极坐标方程（注意这里放大了2倍，与痛点代码中的 r*2 保持一致）
r = 2 * sp.cos(n * theta)

# 极坐标 → 直角坐标（符号推导，零误差）
x = r * sp.cos(theta)  # x = r(θ)·cos(θ)
y = r * sp.sin(theta)  # y = r(θ)·sin(θ)

# 弧长微元：ds/dθ = sqrt((dx/dθ)² + (dy/dθ)²)
dx_dtheta = sp.diff(x, theta)
dy_dtheta = sp.diff(y, theta)

# 弧长微元表达式（注意：这里不算积分，只保留被积函数）
ds_dtheta = sp.sqrt(dx_dtheta**2 + dy_dtheta**2)
print("弧长微元 ds/dθ =", sp.simplify(ds_dtheta))

# 运行结果：
'''
弧长微元 ds/dθ = 2*sqrt((2*sin(4*theta) + 
                     3*sin(6*theta))**2 + 
                     (2*cos(4*theta) -
                     3*cos(6*theta))**2)
'''

import numpy as np
from scipy.integrate import quad
from scipy.optimize import bisect  # 数值求根，稳定且快

# 把弧长微元转成可数值计算的函数
ds_func = sp.lambdify(theta, ds_dtheta, 'numpy')

# 数值弧长函数：L(t) = ∫₀^t ds
def arc_length(t):
    """计算从 0 到 t 的弧长"""
    result, _ = quad(ds_func, 0, t, limit=100)
    return result

# 计算总弧长
total_length = arc_length(2 * np.pi)
print(f"总弧长: {total_length:.4f}")

# 等分弧长，用数值求根反解对应的 theta
N = 500
s_values = np.linspace(0, total_length, N)
theta_vals = []

# 辅助函数：f(t) = L(t) - s，我们要找 f(t)=0 的根
def f(t, s):
    return arc_length(t) - s

for i, s in enumerate(s_values):
    if i == 0:
        theta_vals.append(0.0)  # s=0 对应 theta=0
        continue

    # 初始搜索区间：用上一个 theta 作为左边界
    # 弧长是单调递增的，所以解一定在 [左边界, 右边界] 之间
    left = theta_vals[-1]  # 上一个已解出的 theta
    right = left + 0.5     # 向右扩展，足够覆盖下一个等分点

    # 扩展右边界直到 f(right, s) > 0（确保根在区间内）
    while f(right, s) < 0:
        right += 0.5

    # 二分法求根（稳定、快速）
    sol = bisect(f, left, right, args=(s,), xtol=1e-8)
    theta_vals.append(float(sol))

theta_vals = np.array(theta_vals)

from manim import *
import numpy as np
import sympy as sp
from scipy.integrate import quad
from scipy.optimize import bisect

class UniformRoseCurve(Scene):
    def construct(self):
        # ===== SymPy + 数值积分计算等弧长参数点 =====
        theta = sp.Symbol("theta", real=True)
        n = 5

        # 极坐标方程 r = 2*cos(5θ)
        r = 2 * sp.cos(n * theta)

        # 极坐标转直角坐标
        x_expr = r * sp.cos(theta)
        y_expr = r * sp.sin(theta)

        # 弧长微元
        dx = sp.diff(x_expr, theta)
        dy = sp.diff(y_expr, theta)
        ds_dtheta = sp.sqrt(dx**2 + dy**2)

        # 数值弧长函数
        ds_func = sp.lambdify(theta, ds_dtheta, "numpy")

        def arc_length(t):
            val, _ = quad(ds_func, 0, t, limit=100)
            return val

        # 总弧长（k为奇数时只需π即可画完）
        total_len = arc_length(np.pi)
        print(f"总弧长: {total_len:.4f}")

        # 用二分法反解等弧长 theta（速度快）
        N = 500
        s_vals = np.linspace(0, total_len, N)
        theta_vals = [0.0]

        def f(t, s):
            return arc_length(t) - s

        for i in range(1, N):
            s = s_vals[i]
            left = theta_vals[-1]
            right = left + 0.5
            # 扩展右边界直到 f(right) > 0
            while f(right, s) < 0:
                right += 0.5
            sol = bisect(f, left, right, args=(s,), xtol=1e-8)
            theta_vals.append(float(sol))

        theta_vals = np.array(theta_vals)

        # 计算直角坐标点
        x_func = sp.lambdify(theta, x_expr, "numpy")
        y_func = sp.lambdify(theta, y_expr, "numpy")

        # ===== Manim 动画 =====
        axes = Axes(x_range=[-3, 3], y_range=[-3, 3])
        self.add(axes)

        points = []
        for t in theta_vals:
            px = float(x_func(t))
            py = float(y_func(t))
            points.append(axes.c2p(px, py))

        curve = VMobject(color=PINK, stroke_width=3)
        curve.set_points_as_corners(points)

        self.play(Create(curve), run_time=5, rate_func=linear)
        self.wait()