content: #! /usr/bin/python
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  """
  The Bezier library was  implemented as a class project in CIS515,
  Fundamentals of Linear Algebra, taught by Professor Jean Gallier
  in the summer of 2011 at the University of Pennsylvania. For an
  excellent explanation of Cubic Bezier Curves, and the math
  represented in this library, see
  http://www.cis.upenn.edu/~cis515/proj1-12.pdf
  
  ~~~~~~~~~~~~~~~~~~~~~~~~ Bezier ~~~~~~~~~~~~~~~~~~~~~~~~
  A library for computing Bezier Cubic Splines for an arbitrary
  set of control points in R2, R3, up to RN space.
  
  Cubic Segment:
  C(t) = (1 - t)^3*b0 + 3(1 - t)*b1 + 3(1 - t)*t^2*b2 + t^3*b3
  
  Bezier Spline of Cubic Segments:
  B(t) = C_(i)(t-i+1), i-1 <= t <= i
  where C0 continuity exists: C_(i)(1) = C_(i+1)(0)
  where C1 continuity exists: C'_(i)(1) = C'_(i+1)(0)
  and where C2 continuity exists: C"_(i)(1) = C"_(i+1)(0)
  
  ex. usage:
  import numpy
  import bezier
  points_array = numpy.array([[1, 2, 3], [4, 4, 4],
                              [6, 4, 6], [2, 5, 6],
                              [5, 6, 7]])
  d_pts = bezier.de_boor_control_pts(points_array)
  b_coeffs = bezier.bezier_coefficients(points_array, d_pts)
  b_curve = bezier.bezier_curve(b_coeffs, 50)
  #  plotting example
  import matplotlib.pyplot as plt
  from mpl_toolkits.mplot3d import Axes3D
  fig = plt.figure()
  ax = fig.gca(projection='3d')
  #plot bezier curve
  ax.plot(b_curve[:,0], b_curve[:,1], b_curve[:,2])
  #plot specified points
  ax.plot(points_array[:,0], points_array[:,1], points_array[:,2], 'g*')
  ax.set_title("Cubic Bezier Spline")
  ax.set_xlabel("X")
  ax.set_ylabel("Y")
  ax.set_zlabel("Z")
  ax.legend(["Bezier Curve", "Control Points"], loc=2)
  plt.show()
  """
  import numpy as np
  
  
  def de_boor_control_pts(points_array, d0=None,
                          dN=None, natural=True):
      """
      Compute the de Boor control points for a given
      set for control points
  
      params:
          points_array: array of user-supplied control points
              numpy.array of size N by k
              N is the number of input control points
              k is the number of dimensions for each point
  
          d0: the first control point - None if "natural"
              numpy.array of size 1 by k
  
          dN: the last control point - None if "natural"
              numpy.array of size 1 by k
  
          natural: flag to signify natural start/end conditions
              bool
  
      returns:
          d_pts: array of de Boor control points
              numpy.array of size N+3 by k
      """
      # N+3 auxiliary points required to compute d_pts
      # dpts_(-1) = x_(0)
      # dpts_(N+1) = x_(N)
      # so it is only necessary to find N+1 pts, dpts_(0) to to dpts_(N)
      (rows, k) = np.shape(points_array)
      N = rows - 1  # minus 1 because list includes x_(0)
      # Compute A matrix
      if natural:
          if N > 2:
              A = np.zeros((N-1, N-1))
              A[np.ix_([0], [0, 1])] = [4, 1]
              A[np.ix_([N-2], [N-3, N-2])] = [1, 4]
          else:
              A = 4.0
      else:
          if N > 2:
              A = np.zeros((N-1, N-1))
              A[np.ix_([0], [0, 1])] = [3.5, 1]
              A[np.ix_([N-2], [N-3, N-2])] = [1, 3.5]
          else:
              A = 3.5
      for i in range(1, N-2):
          A[np.ix_([i], [i-1, i, i+1])] = [1, 4, 1]
      # Construct de Boor Control Points from A matrix
      d_pts = np.zeros((N+3, k))
      for col in range(0, k):
          x = np.zeros((max(N-1, 1), 1))
          if N > 2:
              # Compute start / end conditions
              if natural:
                  x[N-2, 0] = 6*points_array[-2, col] - points_array[-1, col]
                  x[0, 0] = 6*points_array[1, col] - points_array[0, col]
              else:
                  x[N-2, 0] = 6*points_array[-2, col] - 1.5*dN[0, col]
                  x[0, 0] = 6*points_array[1, col] - 1.5*d0[0, col]
              x[range(1, N-3+1), 0] = 6*points_array[range(2, N-2+1), col]
              # Solve bezier interpolation
              d_pts[2:N+1, col] = np.linalg.solve(A, x).T
          else:
              # Compute start / end conditions
              if natural:
                  x[0, 0] = 6*points_array[1, col] - points_array[0, col]
              else:
                  x[0, 0] = 6*points_array[1, col] - 1.5*d0[col]
              # Solve bezier interpolation
              d_pts[2, col] = x / A
      # Store off start and end positions
      d_pts[0, :] = points_array[0, :]
      d_pts[-1, :] = points_array[-1, :]
      # Compute the second to last de Boor point based on end conditions
      if natural:
          one_third = (1.0/3.0)
          two_thirds = (2.0/3.0)
          d_pts[1, :] = (two_thirds)*points_array[0, :] + (one_third)*d_pts[2, :]
          d_pts[N+1, :] = ((one_third)*d_pts[-3, :] +
                           (two_thirds)*points_array[-1, :])
      else:
          d_pts[1, :] = d0
          d_pts[N+1, :] = dN
      return d_pts
  
  
  def bezier_coefficients(points_array, d_pts):
      """
      Compute the Bezier coefficients for a given
      set for user-supplied control pts and
      de Boor control pts.
  
      These B coeffs are used to compute the cubic
      splines for each cubic spline segment as
      follows (where t is a percentage of time between
      b_coeff segments):
      C(t) = (1 - t)^3*b0 + 3(1 - t)*b1
              + 3(1 - t)*t^2*b2 + t^3*b3
  
      params:
          points_array: array of user-supplied control points
              numpy.array of size N by k
              N is the number of control points
              k is the number of dimensions for each point
  
          d_pts: array of de Boor control points
              numpy.array of size N+3 by k
  
      returns:
          b_coeffs: k-dimensional array of 4 Bezier coefficients
              for every control point
              numpy.array of size N by 4 by k
      """
      (rows, k) = np.shape(points_array)
      N = rows - 1  # N minus 1 because points array includes x_0
      b_coeffs = np.zeros(shape=(k, N, 4))
      for i in range(0, N):
          points_array_i = i+1
          d_pts_i = i + 2
          if i == 0:
              for axis_pos in range(0, k):
                  b_coeffs[axis_pos, i, 0] = points_array[points_array_i - 1,
                                                          axis_pos]
                  b_coeffs[axis_pos, i, 1] = d_pts[d_pts_i - 1, axis_pos]
                  b_coeffs[axis_pos, i, 2] = (0.5 * d_pts[d_pts_i - 1, axis_pos]
                                              + 0.5 * d_pts[d_pts_i, axis_pos])
                  b_coeffs[axis_pos, i, 3] = points_array[points_array_i,
                                                          axis_pos]
          elif i == N-1:
              for axis_pos in range(0, k):
                  b_coeffs[axis_pos, i, 0] = points_array[points_array_i - 1,
                                                          axis_pos]
                  b_coeffs[axis_pos, i, 1] = (0.5 * d_pts[d_pts_i - 1, axis_pos]
                                              + 0.5 * d_pts[d_pts_i, axis_pos])
                  b_coeffs[axis_pos, i, 2] = d_pts[d_pts_i, axis_pos]
                  b_coeffs[axis_pos, i, 3] = points_array[points_array_i,
                                                          axis_pos]
          else:
              for axis_pos in range(0, k):
                  b_coeffs[axis_pos, i, 0] = points_array[points_array_i - 1,
                                                          axis_pos]
                  b_coeffs[axis_pos, i, 1] = (2.0/3.0 * d_pts[d_pts_i - 1,
                                                              axis_pos]
                                              + 1.0/3.0 * d_pts[d_pts_i,
                                                                axis_pos])
                  b_coeffs[axis_pos, i, 2] = (1.0/3.0 * d_pts[d_pts_i - 1,
                                                              axis_pos]
                                              + 2.0/3.0 * d_pts[d_pts_i,
                                                                axis_pos])
                  b_coeffs[axis_pos, i, 3] = points_array[points_array_i,
                                                          axis_pos]
  
      return b_coeffs
  
  
  def _cubic_spline_point(b_coeff, t):
      """
      Internal convenience function for calculating
      a k-dimensional point defined by the supplied
      Bezier coefficients. Finds the point that
      describes the current position along the bezier
      segment for k dimensions.
  
      params:
          b_coeff => b0...b3: Four k-dimensional Bezier
              coefficients each one is a numpy.array
              of size k by 1, so
              b_coeff is a numpy array of size k by 4
              k is the number of dimensions for each
              coefficient
          t: percentage of time elapsed for this segment
              0 <= int <= 1.0
  
      returns:
          current position in k dimensions
              numpy.array of size 1 by k
      """
      return (pow((1-t), 3)*b_coeff[:, 0] +
              3*pow((1-t), 2)*t*b_coeff[:, 1] +
              3*(1-t)*pow(t, 2)*b_coeff[:, 2] +
              pow(t, 3)*b_coeff[:, 3]
              )
  
  
  def bezier_point(b_coeffs, b_index, t):
      """
      Finds the k values that describe the current
      position along the bezier curve for k dimensions.
  
      params:
          b_coeffs: k-dimensional array
              for every control point with 4 Bezier coefficients
              numpy.array of size k by N by 4
              N is the number of control points
              k is the number of dimensions for each point
          b_index: index position out between two of
              the N b_coeffs for this point in time
              int
          t: percentage of time that has passed between
              the two control points
              0 <= int <= 1.0
  
      returns:
          b_point: current position in k dimensions
              numpy.array of size 1 by k
      """
      if b_index <= 0:
          b_point = b_coeffs[:, 0, 0]
      elif b_index > b_coeffs.shape[1]:
          b_point = b_coeffs[:, -1, -1]
      else:
          t = 0.0 if t < 0.0 else t
          t = 1.0 if t > 1.0 else t
          b_coeff_set = b_coeffs[:, b_index-1, range(4)]
          b_point = _cubic_spline_point(b_coeff_set, t)
      return b_point
  
  
  def bezier_curve(b_coeffs, num_intervals):
      """
      Iterpolation of the entire Bezier curve at once,
      using a specified number of intervals between
      control points (encapsulated by b_coeffs).
  
      params:
          b_coeffs: k-dimensional array of 4 Bezier coefficients
              for every control point
              numpy.array of size N by 4 by k
              N is the number of control points
              k is the number of dimensions for each point
          num_intervals: the number of intervals between
              control points
              int > 0
  
      returns:
          b_curve: positions along the bezier curve in k-dimensions
              numpy.array of size N*num_interval+1  by k
              (the +1 is to include the start position on the curve)
      """
      assert num_intervals > 0,\
          "Invalid number of intervals chosen (must be greater than 0)"
      interval = 1.0 / num_intervals
      (num_axes, num_bpts, _) = np.shape(b_coeffs)
      b_curve = np.zeros((num_bpts*num_intervals+1, num_axes))
      # Copy out initial point
      b_curve[0, :] = b_coeffs[:, 0, 0]
      for current_bpt in range(num_bpts):
              b_coeff_set = b_coeffs[:, current_bpt, range(4)]
              for iteration, t in enumerate(np.linspace(interval, 1,
                                                        num_intervals)):
                  b_curve[(current_bpt *
                           num_intervals +
                           iteration+1), :] = _cubic_spline_point(b_coeff_set, t)
      return b_curve
  