查找点到复杂曲线的最小距离 [英] Find minimum distance from point to complicated curve

问题描述

我在表中定义了一组复杂的曲线，就像这样(整个表在此处 ):

I have a complicated curve defined as a set of points in a table like so (the full table is here):

#  x   y
1.0577  12.0914
1.0501  11.9946
1.0465  11.9338
...

如果我使用以下命令绘制此表:

If I plot this table with the commands:

plt.plot(x_data, y_data, c='b',lw=1.)
plt.scatter(x_data, y_data, marker='o', color='k', s=10, lw=0.2)

我得到以下信息:

我在其中手动添加了红点和细分.我需要的是一种为每个点计算这些线段的方法，即:一种找到从2D空间中给定点到插值曲线的最小距离的方法.

where I've added the red points and segments manually. What I need is a way to calculate those segments for each of those points, that is: a way to find the minimum distance from a given point in this 2D space to the interpolated curve.

我无法使用到数据点本身的距离(生成蓝色曲线的黑点)，因为它们之间的距离不相等，有时它们很近，有时又很远，这对我的结果产生了深远的影响再往下走.

I can't use the distance to the data points themselves (the black dots that generate the blue curve) since they are not located at equal intervals, sometimes they are close and sometimes they are far apart and this deeply affects my results further down the line.

由于这不是表现良好的曲线，因此我不确定自己能做什么.我尝试使用 UnivariateSpline 进行插值但是它返回的拟合度很差:

Since this is not a well behaved curve I'm not really sure what I could do. I've tried interpolating it with a UnivariateSpline but it returns a very poor fit:

# Sort data according to x.
temp_data = zip(x_data, y_data)
temp_data.sort()
# Unpack sorted data.
x_sorted, y_sorted = zip(*temp_data)

# Generate univariate spline.
s = UnivariateSpline(x_sorted, y_sorted, k=5)
xspl = np.linspace(0.8, 1.1, 100)
yspl = s(xspl)

# Plot.
plt.scatter(xspl, yspl, marker='o', color='r', s=10, lw=0.2)

我也尝试增加插值点的数量，但是一团糟:

I also tried increasing the number of interpolating points but got a mess:

# Sort data according to x.
temp_data = zip(x_data, y_data)
temp_data.sort()
# Unpack sorted data.
x_sorted, y_sorted = zip(*temp_data)

t = np.linspace(0, 1, len(x_sorted))
t2 = np.linspace(0, 1, 100)    
# One-dimensional linear interpolation.
x2 = np.interp(t2, t, x_sorted)
y2 = np.interp(t2, t, y_sorted)
plt.scatter(x2, y2, marker='o', color='r', s=10, lw=0.2)

任何想法/指针将不胜感激.

Any ideas/pointers will be greatly appreciated.

推荐答案

如果您愿意为此使用库，请查看shapely:

If you're open to using a library for this, have a look at shapely: https://github.com/Toblerity/Shapely

作为一个简单的示例(points.txt包含您在问题中链接到的数据):

As a quick example (points.txt contains the data you linked to in your question):

import shapely.geometry as geom
import numpy as np

coords = np.loadtxt('points.txt')

line = geom.LineString(coords)
point = geom.Point(0.8, 10.5)

# Note that "line.distance(point)" would be identical
print point.distance(line)

作为一个交互式示例(这还会绘制您想要的线段):

As an interactive example (this also draws the line segments you wanted):

import numpy as np
import shapely.geometry as geom
import matplotlib.pyplot as plt

class NearestPoint(object):
    def __init__(self, line, ax):
        self.line = line
        self.ax = ax
        ax.figure.canvas.mpl_connect('button_press_event', self)

    def __call__(self, event):
        x, y = event.xdata, event.ydata
        point = geom.Point(x, y)
        distance = self.line.distance(point)
        self.draw_segment(point)
        print 'Distance to line:', distance

    def draw_segment(self, point):
        point_on_line = line.interpolate(line.project(point))
        self.ax.plot([point.x, point_on_line.x], [point.y, point_on_line.y], 
                     color='red', marker='o', scalex=False, scaley=False)
        fig.canvas.draw()

if __name__ == '__main__':
    coords = np.loadtxt('points.txt')

    line = geom.LineString(coords)

    fig, ax = plt.subplots()
    ax.plot(*coords.T)
    ax.axis('equal')
    NearestPoint(line, ax)
    plt.show()

请注意，我已经添加了ax.axis('equal'). shapely在数据所在的坐标系中运行.如果没有等轴坐标图，则视图将变形，并且shapely仍会找到最近的点，但在显示中看起来不太正确:

Note that I've added ax.axis('equal'). shapely operates in the coordinate system that the data is in. Without the equal axis plot, the view will be distorted, and while shapely will still find the nearest point, it won't look quite right in the display:

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