如何找到参数化曲线与形状的相交时间？

Question

赏金将在 6 天后到期。 此问题的答案有资格获得+50声望赏金。 mapf正在从有信誉的来源寻找答案。

我有一条由时间参数化的曲线，它与一个形状相交（在这种情况下只是一个矩形）。 遵循这个优雅的建议，我使用shapely来确定对象相交的位置，但是从那里开始，我很难找到一个好的解决方案来解决这种情况。 目前，我通过找到最接近交叉点（在空间上）的曲线点，然后使用它的时间戳来近似时间。

但我相信应该有更好的解决方案，例如通过求解多项式方程，可能使用numpy多项式的root方法。 我只是不确定如何做到这一点，因为我想你需要以某种方式引入公差，因为曲线很可能永远不会假设与 shapely 确定的相交坐标完全相同。

这是我的代码：

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Rectangle, Ellipse
from matplotlib.collections import LineCollection
from shapely.geometry import LineString, Polygon


# the parameterized curve
coeffs = np.array([
    [-2.65053088e-05, 2.76890591e-05],
    [-5.70681576e-02, -2.69415587e-01],
    [7.92564148e+02, 6.88557419e+02],
])
t_fit = np.linspace(-2400, 3600, 1000)
x_fit = np.polyval(coeffs[:, 0], t_fit)
y_fit = np.polyval(coeffs[:, 1], t_fit)
curve = LineString(np.column_stack((x_fit, y_fit)))

# the shape it intersects
area = {'x': [700, 1000], 'y': [1300, 1400]}
area_shape = Polygon([
    (area['x'][0], area['y'][0]),
    (area['x'][1], area['y'][0]),
    (area['x'][1], area['y'][1]),
    (area['x'][0], area['y'][1]),
])

# attempt at finding the time of intersection
intersection = curve.intersection(area_shape).coords[-1]
distances = np.hypot(x_fit-intersection[0], y_fit-intersection[1])
idx = np.where(distances == min(distances))
fit_intersection = x_fit[idx][0], y_fit[idx][0]
t_intersection = t_fit[idx]
print(t_intersection)

# code for visualization
fig, ax = plt.subplots(figsize=(5, 5))
ax.margins(0.4, 0.2)
ax.invert_yaxis()

area_artist = Rectangle(
    (area['x'][0], area['y'][0]),
    width=area['x'][1] - area['x'][0],
    height=area['y'][1] - area['y'][0],
    edgecolor='gray', facecolor='none'
)
ax.add_artist(area_artist)

points = np.array([x_fit, y_fit]).T.reshape(-1, 1, 2)
segments = np.concatenate([points[:-1], points[1:]], axis=1)
z = np.linspace(0, 1, points.shape[0])
norm = plt.Normalize(z.min(), z.max())
lc = LineCollection(
    segments, cmap='autumn', norm=norm, alpha=1,
    linewidths=2, picker=8, capstyle='round',
    joinstyle='round'
)
lc.set_array(z)
ax.add_collection(lc)

ax.autoscale_view()
ax.relim()

trans = (ax.transData + ax.transAxes.inverted()).transform
intersection_point = Ellipse(
    xy=trans(fit_intersection), width=0.02, height=0.02, fc='none',
    ec='black', transform=ax.transAxes, zorder=3,
)
ax.add_artist(intersection_point)

plt.show()

就视觉效果而言，plot 中的问题如下所示：

Answer 1

最好的方法是使用插值函数来计算 (x(t), y(t))。 并使用 function 计算 d(t)：到交叉点的距离。 然后我们在 d(t) 上使用 scipy.optimize.minimize 来找到 d(t) 最小的 t 值。 插值将确保良好的准确性。

因此，我对您的代码进行了一些修改。

1. 插值函数和距离计算的定义
2. 测试是否确实有交集，否则没有意义。
3. 通过最小化计算相交时间

编码：

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Rectangle, Ellipse
from matplotlib.collections import LineCollection
from shapely.geometry import LineString, Polygon

from scipy.optimize import minimize

# Interpolate (x,y) at time t:
def interp_xy(t,tp, fpx,fpy):
        # tp: time grid points, fpx, fpy: the corresponding x,y values
        x=np.interp(t, tp, fpx)
        y=np.interp(t, tp, fpy)
        return x,y

# Compute distance to intersection:
def dist_to_intersect(t,tp, fpx, fpy, intersection):
        x,y = interp_xy(t,tp,fpx,fpy)
        d=np.hypot(x-intersection[0], y-intersection[1])
        return d



# the parameterized curve
coeffs = np.array([
    [-2.65053088e-05, 2.76890591e-05],
    [-5.70681576e-02, -2.69415587e-01],
    [7.92564148e+02, 6.88557419e+02],
])
t_fit = np.linspace(-2400, 3600, 1000)
x_fit = np.polyval(coeffs[:, 0], t_fit)
y_fit = np.polyval(coeffs[:, 1], t_fit)
curve = LineString(np.column_stack((x_fit, y_fit)))


# the shape it intersects
area = {'x': [700, 1000], 'y': [1300, 1400]}
area_shape = Polygon([
    (area['x'][0], area['y'][0]),
    (area['x'][1], area['y'][0]),
    (area['x'][1], area['y'][1]),
    (area['x'][0], area['y'][1]),
])

# attempt at finding the time of intersection
curve_intersection = curve.intersection(area_shape)

# We check if intersection is empty or not:
if not curve_intersection.is_empty:   
    # We can get the coords because intersection is not empty
    intersection=curve_intersection.coords[-1]
    
    distances = np.hypot(x_fit-intersection[0], y_fit-intersection[1])

    print("Looking for minimal distance to intersection: ")
    print('-------------------------------------------------------------------------')
    # Call to minimize:
    # We pass:
    # - the function to be minimized (dist_to_intersect)
    # - a starting value to t (let's say t_fit[0])
    # - arguments, method and tolerance tol. The minimization will succeed when 
    #   dist_to_intersect <  tol=1e-6
    # - option: here -->  verbose
    res = minimize(dist_to_intersect, t_fit[0],  args=(t_fit, x_fit, y_fit, intersection), method='Nelder-Mead',tol = 1e-6, options={ 'disp': True})
    print('-------------------------------------------------------------------------')
    print("Result of the optimization:")
    print(res)
    print('-------------------------------------------------------------------------')
    print("Intersection at time t = ",res.x[0])    
    fit_intersection = interp_xy(res.x[0],t_fit, x_fit,y_fit)
    print("Intersection point : ",fit_intersection)
else:
    print("No intersection.")
    
    
# code for visualization
fig, ax = plt.subplots(figsize=(5, 5))
ax.margins(0.4, 0.2)
ax.invert_yaxis()

area_artist = Rectangle(
    (area['x'][0], area['y'][0]),
    width=area['x'][1] - area['x'][0],
    height=area['y'][1] - area['y'][0],
    edgecolor='gray', facecolor='none'
)
ax.add_artist(area_artist)

points = np.array([x_fit, y_fit]).T.reshape(-1, 1, 2)
segments = np.concatenate([points[:-1], points[1:]], axis=1)
z = np.linspace(0, 1, points.shape[0])
norm = plt.Normalize(z.min(), z.max())
lc = LineCollection(
    segments, cmap='autumn', norm=norm, alpha=1,
    linewidths=2, picker=8, capstyle='round',
    joinstyle='round'
)
lc.set_array(z)
ax.add_collection(lc)

ax.autoscale_view()
ax.relim()

trans = (ax.transData + ax.transAxes.inverted()).transform
# Again, we check that intersection exists because we don't want to draw
# an non-existing point (it would generate an error)
if not curve_intersection.is_empty:
    intersection_point = Ellipse(
        xy=trans(fit_intersection), width=0.02, height=0.02, fc='none',
        ec='black', transform=ax.transAxes, zorder=3,
    )
    ax.add_artist(intersection_point)

plt.show()

OUTPUT：

Looking for minimal distance to intersection: 
-------------------------------------------------------------------------
Optimization terminated successfully.
         Current function value: 0.000000
         Iterations: 31
         Function evaluations: 62
-------------------------------------------------------------------------
Result of the optimization:
 final_simplex: (array([[-1898.91943932],
       [-1898.91944021]]), array([8.44804735e-09, 3.28684898e-07]))
           fun: 8.448047349426054e-09
       message: 'Optimization terminated successfully.'
          nfev: 62
           nit: 31
        status: 0
       success: True
             x: array([-1898.91943932])
-------------------------------------------------------------------------
Intersection at time t =  -1898.919439315796
Intersection point :  (805.3563860471179, 1299.9999999916085)

而您的代码给出的结果不太精确：

t=-1901.5015015 

intersection point: (805.2438793482748,1300.9671136070717)

数字：