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)

        # 用累積密度法生成非均勻 theta：尖端稀疏(快)，中部密集(慢)
        n_points = 500
        t = np.linspace(0, 2 * PI, n_points)
        # 密度函數(shù)：在 cos(5t)=0 處(花瓣中部)密度高，在 |cos(5t)|=1 處(尖端)密度低
        density = 1 + 3 * np.sin(5 * t) ** 2
        theta = np.cumsum(density)
        theta = theta / theta[-1] * PI  # 歸一化到 [0, π]（k為奇數(shù)時(shí)只需π即可畫完）
        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 畫曲線——速度明顯不均勻！
        self.play(Create(curve), run_time=5, rate_func=linear)
        self.wait()

import sympy as sp

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

# 極坐標(biāo)方程（注意這里放大了2倍，與痛點(diǎn)代碼中的 r*2 保持一致）
r = 2 * sp.cos(n * theta)

# 極坐標(biāo) → 直角坐標(biāo)（符號(hào)推導(dǎo)，零誤差）
x = r * sp.cos(theta)  # x = r(θ)·cos(θ)
y = r * sp.sin(theta)  # y = r(θ)·sin(θ)

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

# 弧長(zhǎng)微元表達(dá)式（注意：這里不算積分，只保留被積函數(shù)）
ds_dtheta = sp.sqrt(dx_dtheta**2 + dy_dtheta**2)
print("弧長(zhǎng)微元 ds/dθ =", sp.simplify(ds_dtheta))

# 運(yùn)行結(jié)果：
'''
弧長(zhǎng)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  # 數(shù)值求根，穩(wěn)定且快

# 把弧長(zhǎng)微元轉(zhuǎn)成可數(shù)值計(jì)算的函數(shù)
ds_func = sp.lambdify(theta, ds_dtheta, 'numpy')

# 數(shù)值弧長(zhǎng)函數(shù)：L(t) = ∫?^t ds
def arc_length(t):
    """計(jì)算從 0 到 t 的弧長(zhǎng)"""
    result, _ = quad(ds_func, 0, t, limit=100)
    return result

# 計(jì)算總弧長(zhǎng)
total_length = arc_length(2 * np.pi)
print(f"總弧長(zhǎng): {total_length:.4f}")

# 等分弧長(zhǎng)，用數(shù)值求根反解對(duì)應(yīng)的 theta
N = 500
s_values = np.linspace(0, total_length, N)
theta_vals = []

# 輔助函數(shù)：f(t) = L(t) - s，我們要找 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 對(duì)應(yīng) theta=0
        continue

    # 初始搜索區(qū)間：用上一個(gè) theta 作為左邊界
    # 弧長(zhǎng)是單調(diào)遞增的，所以解一定在 [左邊界, 右邊界] 之間
    left = theta_vals[-1]  # 上一個(gè)已解出的 theta
    right = left + 0.5     # 向右擴(kuò)展，足夠覆蓋下一個(gè)等分點(diǎn)

    # 擴(kuò)展右邊界直到 f(right, s) > 0（確保根在區(qū)間內(nèi)）
    while f(right, s) < 0:
        right += 0.5

    # 二分法求根（穩(wěn)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 + 數(shù)值積分計(jì)算等弧長(zhǎng)參數(shù)點(diǎn) =====
        theta = sp.Symbol("theta", real=True)
        n = 5

        # 極坐標(biāo)方程 r = 2*cos(5θ)
        r = 2 * sp.cos(n * theta)

        # 極坐標(biāo)轉(zhuǎn)直角坐標(biāo)
        x_expr = r * sp.cos(theta)
        y_expr = r * sp.sin(theta)

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

        # 數(shù)值弧長(zhǎng)函數(shù)
        ds_func = sp.lambdify(theta, ds_dtheta, "numpy")

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

        # 總弧長(zhǎng)（k為奇數(shù)時(shí)只需π即可畫完）
        total_len = arc_length(np.pi)
        print(f"總弧長(zhǎng): {total_len:.4f}")

        # 用二分法反解等弧長(zhǎng) 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
            # 擴(kuò)展右邊界直到 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)

        # 計(jì)算直角坐標(biāo)點(diǎn)
        x_func = sp.lambdify(theta, x_expr, "numpy")
        y_func = sp.lambdify(theta, y_expr, "numpy")

        # ===== Manim 動(dòng)畫 =====
        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()