Question

python中是否有一个库或函数可以从三个点计算Catmull-Rom样条线？

我最终需要的是沿着样条曲线的点的x，y坐标，只要它们沿着样条曲线始终等于给定量t（例如，样条曲线长度为3个单位，我想要x ，y坐标为样条长度0,1,2和3）

没什么好激动的。我是自己写的，但如果你发现了一些不错的东西，那么测试（或节省时间）会很棒

Answer 1

3分？ Catmull-Rom定义为4个点，比如p_1 p0 p1 p2; 立方曲线从p0到p1，外点p_1和p2确定p0和p1处的斜率。要绘制数组P中某些点的曲线，请执行以下操作：

for j in range( 1, len(P)-2 ):  # skip the ends
    for t in range( 10 ):  # t: 0 .1 .2 .. .9
        p = spline_4p( t/10, P[j-1], P[j], P[j+1], P[j+2] )
        # draw p

def spline_4p( t, p_1, p0, p1, p2 ):
    """ Catmull-Rom
        (Ps can be numpy vectors or arrays too: colors, curves ...)
    """
        # wikipedia Catmull-Rom -> Cubic_Hermite_spline
        # 0 -> p0,  1 -> p1,  1/2 -> (- p_1 + 9 p0 + 9 p1 - p2) / 16
    # assert 0 <= t <= 1
    return (
          t*((2-t)*t - 1)   * p_1
        + (t*t*(3*t - 5) + 2) * p0
        + t*((4 - 3*t)*t + 1) * p1
        + (t-1)*t*t         * p2 ) / 2

一个可以使用分段二次曲线通过3个点 - 见Dodgson, Quadratic Interpolation for Image Resampling。你真的想做什么？

Answer 2

有这个：jj_catmull，似乎是在Python中，也许你可以找到你需要的东西。

Answer 3

如前所述，catmull-rom需要4分，端点是个问题。我正在考虑自己应用这些而不是自然的三次样条（超出已知数据范围的潜在超调在我的应用中是不切实际的）。与@ denis的代码类似，这里有些东西可以帮助你（注意一些事情.1）代码只是随机生成点，我相信你可以使用注释掉的例子来说明如何使用你自己的数据。 2）我创建扩展端点，保留第一个/最后两个点之间的斜率 - 使用域的1％的任意距离。 3）我包括均匀，向心和和弦结参数化用于比较）：

# coding: utf-8

# In[1]:

import numpy
import matplotlib.pyplot as plt
get_ipython().magic(u'pylab inline')


# In[2]:

def CatmullRomSpline(P0, P1, P2, P3, a, nPoints=100):
  """
  P0, P1, P2, and P3 should be (x,y) point pairs that define the Catmull-Rom spline.
  nPoints is the number of points to include in this curve segment.
  """
  # Convert the points to numpy so that we can do array multiplication
  P0, P1, P2, P3 = map(numpy.array, [P0, P1, P2, P3])

  # Calculate t0 to t4
  alpha = a
  def tj(ti, Pi, Pj):
    xi, yi = Pi
    xj, yj = Pj
    return ( ( (xj-xi)**2 + (yj-yi)**2 )**0.5 )**alpha + ti

  t0 = 0
  t1 = tj(t0, P0, P1)
  t2 = tj(t1, P1, P2)
  t3 = tj(t2, P2, P3)

  # Only calculate points between P1 and P2
  t = numpy.linspace(t1,t2,nPoints)

  # Reshape so that we can multiply by the points P0 to P3
  # and get a point for each value of t.
  t = t.reshape(len(t),1)

  A1 = (t1-t)/(t1-t0)*P0 + (t-t0)/(t1-t0)*P1
  A2 = (t2-t)/(t2-t1)*P1 + (t-t1)/(t2-t1)*P2
  A3 = (t3-t)/(t3-t2)*P2 + (t-t2)/(t3-t2)*P3

  B1 = (t2-t)/(t2-t0)*A1 + (t-t0)/(t2-t0)*A2
  B2 = (t3-t)/(t3-t1)*A2 + (t-t1)/(t3-t1)*A3

  C  = (t2-t)/(t2-t1)*B1 + (t-t1)/(t2-t1)*B2
  return C

def CatmullRomChain(P,alpha):
  """
  Calculate Catmull Rom for a chain of points and return the combined curve.
  """
  sz = len(P)

  # The curve C will contain an array of (x,y) points.
  C = []
  for i in range(sz-3):
    c = CatmullRomSpline(P[i], P[i+1], P[i+2], P[i+3],alpha)
    C.extend(c)

  return C


# In[139]:

# Define a set of points for curve to go through
Points = numpy.random.rand(10,2)
#Points=array([array([153.01,722.67]),array([152.73,699.92]),array([152.91,683.04]),array([154.6,643.45]),
#        array([158.07,603.97])])
#Points = array([array([0,92.05330318]),
#               array([2.39580622,29.76345192]),
#               array([10.01564963,16.91470591]),
#               array([15.26219886,71.56301997]),
#               array([15.51234733,73.76834447]),
#               array([24.88468545,50.89432899]),
#               array([27.83934153,81.1341789]),
#               array([36.80443404,56.55810783]),
#               array([43.1404725,16.96946811]),
#               array([45.27824599,15.75903418]),
#               array([51.58871027,90.63583215])])

x1=Points[0][0]
x2=Points[1][0]
y1=Points[0][1]
y2=Points[1][1]
x3=Points[-2][0]
x4=Points[-1][0]
y3=Points[-2][1]
y4=Points[-1][1]
dom=max(Points[:,0])-min(Points[:,0])
rng=max(Points[:,1])-min(Points[:,1])
pctdom=1
pctdom=float(pctdom)/100
prex=x1+sign(x1-x2)*dom*pctdom
prey=(y1-y2)/(x1-x2)*(prex-x1)+y1
endx=x4+sign(x4-x3)*dom*pctdom
endy=(y4-y3)/(x4-x3)*(endx-x4)+y4
print len(Points)
Points=list(Points)
Points.insert(0,array([prex,prey]))
Points.append(array([endx,endy]))
print len(Points)


# In[140]:

#Define alpha
a=0.

# Calculate the Catmull-Rom splines through the points
c = CatmullRomChain(Points,a)

# Convert the Catmull-Rom curve points into x and y arrays and plot
x,y = zip(*c)
plt.plot(x,y,c='green',zorder=10)

a=0.5
c = CatmullRomChain(Points,a)
x,y = zip(*c)
plt.plot(x,y,c='blue')

a=1.
c = CatmullRomChain(Points,a)
x,y = zip(*c)
plt.plot(x,y,c='red')

# Plot the control points
px, py = zip(*Points)
plt.plot(px,py,'o',c='black')

plt.grid(b=True)
plt.show()


# In[141]:

Points


# In[104]:

Catmull-Rom splines在python中

3 个答案: