How to find the intersection time of a parameterized curve with a shape?

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I have a curve parameterized by time that intersects a shape (in this case just a rectangle). Following this elegant suggestion , I used shapely to determine where the objects intersect, however from there on, I struggle to find a good solution for when that happens. Currently, I am approximating the time awkwardly by finding the point of the curve that is closest (in space) to the intersection, and then using its time stamp.

But I believe there should be a better solution eg by solving the polynomial equation, maybe using the root method of a numpy polynomial. I'm just not sure how to do this, because I guess you would need to somehow introduce tolerances as it is likely that the curve will never assume exactly the same intersection coordinates as determined by shapely.

Here is my code:

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()

And just for the visuals, here is what the problem looks like in a plot:

Answer 1

The best is to use interpolation functions to compute (x(t), y(t)). And use a function to compute d(t): the distance to intersection. Then we use scipy.optimize.minimize on d(t) to find the t value at which d(t) is minimum. Interpolation will ensure good accuracy.

So, I added a few modifications to you code.

1. definitions of interpolation functions and distance calculation
2. Test if there is indeed intersection, otherwise it doesn't make sense.
3. Compute the intersection time by minimization

The code:

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)

Whereas your code gives a much less precise result:

t=-1901.5015015 

intersection point: (805.2438793482748,1300.9671136070717)

Figure: