source: lmdz_wrf/trunk/tools/geometry_tools.py @ 2495

# Python tools to manage netCDF files.
# L. Fita, CIMA. Mrch 2019
# More information at: http://www.xn--llusfb-5va.cat/python/PyNCplot
#
# pyNCplot and its component geometry_tools.py comes with ABSOLUTELY NO WARRANTY. 
# This work is licendes under a Creative Commons 
#   Attribution-ShareAlike 4.0 International License (http://creativecommons.org/licenses/by-sa/4.0)
#
## Script for geometry calculations and operations as well as definition of different 
###    standard objects and shapes

import numpy as np
import matplotlib as mpl
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
import os
import generic_tools as gen

errormsg = 'ERROR -- error -- ERROR -- error'
infmsg = 'INFORMATION -- information -- INFORMATION -- information'

####### Contents:
# deg_deci: Function to pass from degrees [deg, minute, sec] to decimal angles [rad]
# dist_points: Function to provide the distance between two points
# max_coords_poly: Function to provide the extremes of the coordinates of a polygon
# mirror_polygon: Function to reflex a polygon for a given axis
# position_sphere: Function to tranform fom a point in lon, lat deg coordinates to 
#   cartesian coordinates over an sphere
# read_join_poly: Function to read an ASCII file with the combination of polygons
# rotate_2D: Function to rotate a vector by a certain angle in the plain
# rotate_polygon_2D: Function to rotate 2D plain the vertices of a polygon
# rotate_line2D: Function to rotate a line given by 2 pairs of x,y coordinates by a 
#   certain angle in the plain
# rotate_lines2D: Function to rotate multiple lines given by mulitple pars of x,y 
#   coordinates by a certain angle in the plain
# spheric_line: Function to transform a series of locations in lon, lat coordinates 
#   to x,y,z over an 3D spaceFunction to provide coordinates of a line  on a 3D space
# write_join_poly: Function to write an ASCII file with the combination of polygons

## Shapes/objects
# circ_sec: Function union of point A and B by a section of a circle
# ellipse_polar: Function to determine an ellipse from its center and polar coordinates
# p_doubleArrow: Function to provide an arrow with double lines
# p_circle: Function to get a polygon of a circle
# p_reg_polygon: Function to provide a regular polygon of Nv vertices
# p_reg_star: Function to provide a regular star of Nv vertices
# p_sinusiode: Function to get coordinates of a sinusoidal curve
# p_square: Function to get a polygon square
# p_spiral: Function to provide a polygon of an Archimedean spiral
# p_triangle: Function to provide the polygon of a triangle from its 3 vertices
# surface_sphere: Function to provide an sphere as matrix of x,y,z coordinates
# z_boat: Function to define an schematic boat from the z-plane
# zsailing_boat: Function to define an schematic sailing boat from the z-plane with sails

## Plotting
# plot_sphere: Function to plot an sphere and determine which standard lines will be 
#   also drawn

def deg_deci(angle):
    """ Function to pass from degrees [deg, minute, sec] to decimal angles [rad]
      angle: list of [deg, minute, sec] to pass
    >>> deg_deci([41., 58., 34.])
    0.732621346072
    """
    fname = 'deg_deci'

    deg = np.abs(angle[0]) + np.abs(angle[1])/60. + np.abs(angle[2])/3600.

    if angle[0] < 0.: deg = -deg*np.pi/180.
    else: deg = deg*np.pi/180.

    return deg

def position_sphere(radii, alpha, beta):
    """ Function to tranform fom a point in lon, lat deg coordinates to cartesian  
          coordinates over an sphere
      radii: radii of the sphere
      alpha: longitude of the point
      beta: latitude of the point
    >>> position_sphere(10., 30., 45.)
    (0.81031678432964027, -5.1903473778327376, 8.5090352453411846
    """
    fname = 'position_sphere'

    xpt = radii*np.cos(beta)*np.cos(alpha)
    ypt = radii*np.cos(beta)*np.sin(alpha)
    zpt = radii*np.sin(beta)

    return xpt, ypt, zpt

def spheric_line(radii,lon,lat):
    """ Function to transform a series of locations in lon, lat coordinates to x,y,z 
          over an 3D space
      radii: radius of the sphere
      lon: array of angles along longitudes
      lat: array of angles along latitudes
    """
    fname = 'spheric_line'

    Lint = lon.shape[0]
    coords = np.zeros((Lint,3), dtype=np.float)

    for iv in range(Lint):
        coords[iv,:] = position_sphere(radii, lon[iv], lat[iv])

    return coords

def rotate_2D(vector, angle):
    """ Function to rotate a vector by a certain angle in the plain
      vector= vector to rotate [y, x]
      angle= angle to rotate [rad]
    >>> rotate_2D(np.array([1.,0.]), np.pi/4.)
    [ 0.70710678 -0.70710678]
    """
    fname = 'rotate_2D'

    rotmat = np.zeros((2,2), dtype=np.float)

    rotmat[0,0] = np.cos(angle)
    rotmat[0,1] = -np.sin(angle)
    rotmat[1,0] = np.sin(angle)
    rotmat[1,1] = np.cos(angle)

    rotvector = np.zeros((2), dtype=np.float)

    vecv = np.zeros((2), dtype=np.float)

    # Unifying vector
    modvec = vector[0]**2+vector[1]**2
    if modvec != 0: 
        vecv[0] = vector[1]/modvec
        vecv[1] = vector[0]/modvec

        rotvec = np.matmul(rotmat, vecv)
        rotvec = np.where(np.abs(rotvec) < 1.e-7, 0., rotvec)

        rotvector[0] = rotvec[1]*modvec
        rotvector[1] = rotvec[0]*modvec

    return rotvector

def rotate_polygon_2D(vectors, angle):
    """ Function to rotate 2D plain the vertices of a polygon
      line= matrix of vectors to rotate
      angle= angle to rotate [rad]
    >>> square = np.zeros((4,2), dtype=np.float)
    >>> square[0,:] = [-0.5,-0.5]
    >>> square[1,:] = [0.5,-0.5]
    >>> square[2,:] = [0.5,0.5]
    >>> square[3,:] = [-0.5,0.5]
    >>> rotate_polygon_2D(square, np.pi/4.)
    [[-0.70710678  0.        ]
     [ 0.         -0.70710678]
     [ 0.70710678  0.        ]
     [ 0.          0.70710678]]
    """
    fname = 'rotate_polygon_2D'

    rotvecs = np.zeros(vectors.shape, dtype=np.float)

    Nvecs = vectors.shape[0]
    for iv in range(Nvecs):
        rotvecs[iv,:] = rotate_2D(vectors[iv,:], angle)

    return rotvecs

def rotate_line2D(line, angle):
    """ Function to rotate a line given by 2 pairs of x,y coordinates by a certain 
          angle in the plain
      line= line to rotate as couple of points [[y0,x0], [y1,x1]]
      angle= angle to rotate [rad]
    >>> rotate_line2D(np.array([[0.,0.], [1.,0.]]), np.pi/4.)
    [[ 0.          0.        ]
     [0.70710678  -0.70710678]]
    """
    fname = 'rotate_2D'

    rotline = np.zeros((2,2), dtype=np.float)
    rotline[0,:] = rotate_2D(line[0,:], angle)
    rotline[1,:] = rotate_2D(line[1,:], angle)

    return rotline

def rotate_lines2D(lines, angle):
    """ Function to rotate multiple lines given by mulitple pars of x,y coordinates  
          by a certain angle in the plain
      line= matrix of N couples of points [N, [y0,x0], [y1,x1]]
      angle= angle to rotate [rad]
    >>> square = np.zeros((4,2,2), dtype=np.float)
    >>> square[0,0,:] = [-0.5,-0.5]
    >>> square[0,1,:] = [0.5,-0.5]
    >>> square[1,0,:] = [0.5,-0.5]
    >>> square[1,1,:] = [0.5,0.5]
    >>> square[2,0,:] = [0.5,0.5]
    >>> square[2,1,:] = [-0.5,0.5]
    >>> square[3,0,:] = [-0.5,0.5]
    >>> square[3,1,:] = [-0.5,-0.5]
    >>> rotate_lines2D(square, np.pi/4.)
    [[[-0.70710678  0.        ]
      [ 0.         -0.70710678]]

     [[ 0.         -0.70710678]
      [ 0.70710678  0.        ]]

     [[ 0.70710678  0.        ]
      [ 0.          0.70710678]]

     [[ 0.          0.70710678]
      [-0.70710678  0.        ]]]
    """
    fname = 'rotate_lines2D'

    rotlines = np.zeros(lines.shape, dtype=np.float)

    Nlines = lines.shape[0]
    for il in range(Nlines):
        line = np.zeros((2,2), dtype=np.float)
        line[0,:] = lines[il,0,:]
        line[1,:] = lines[il,1,:]

        rotlines[il,:,:] = rotate_line2D(line, angle)

    return rotlines

def dist_points(ptA, ptB):
    """ Function to provide the distance between two points
      ptA: coordinates of the point A [yA, xA]
      ptB: coordinates of the point B [yB, xB]
    >>> dist_points([1.,1.], [-1.,-1.])
    2.82842712475
    """
    fname = 'dist_points'

    dist = np.sqrt( (ptA[0]-ptB[0])**2 + (ptA[1]-ptB[1])**2)

    return dist

def max_coords_poly(polygon):
    """ Function to provide the extremes of the coordinates of a polygon
      polygon: coordinates [Nvertexs, 2] of a polygon
    >>> square = np.zeros((4,2), dtype=np.float)
    >>> square[0,:] = [-0.5,-0.5]
    >>> square[1,:] = [0.5,-0.5]
    >>> square[2,:] = [0.5,0.5]
    >>> square[3,:] = [-0.5,0.5]
    >>> max_coords_poly(square)
    [-0.5, 0.5], [-0.5, 0.5], [0.5, 0.5], 0.5
    """
    fname = 'max_coords_poly'

    # x-coordinate min/max
    nx = np.min(polygon[:,1])
    xx = np.max(polygon[:,1])

    # y-coordinate min/max
    ny = np.min(polygon[:,0])
    xy = np.max(polygon[:,0])

    # x/y-coordinate maximum of absolute values
    axx = np.max(np.abs(polygon[:,1]))
    ayx = np.max(np.abs(polygon[:,0]))

    # absolute maximum
    xyx = np.max([axx, ayx])

    return [nx, xx], [ny, xy], [ayx, axx], xyx

def mirror_polygon(polygon,axis):
    """ Function to reflex a polygon for a given axis
      polygon: polygon to mirror
      axis: axis at which mirror is located ('x' or 'y')
    """
    fname = 'mirror_polygon'

    reflex = np.zeros(polygon.shape, dtype=np.float)

    N = polygon.shape[0]
    if axis == 'x':
        for iv in range(N):
            reflex[iv,:] = [-polygon[iv,0], polygon[iv,1]]
    elif axis == 'y':
        for iv in range(N):
            reflex[iv,:] = [polygon[iv,0], -polygon[iv,1]]

    return reflex

####### ###### ##### #### ### ## #
# Shapes/objects

def surface_sphere(radii,Npts):
    """ Function to provide an sphere as matrix of x,y,z coordinates
      radii: radii of the sphere
      Npts: number of points to discretisize longitues (half for latitudes)
    """
    fname = 'surface_sphere'

    sphereup = np.zeros((3,Npts/2,Npts), dtype=np.float)
    spheredown = np.zeros((3,Npts/2,Npts), dtype=np.float)
    for ia in range(Npts):
        alpha = ia*2*np.pi/(Npts-1)
        for ib in range(Npts/2):
            beta = ib*np.pi/(2.*(Npts/2-1))
            sphereup[:,ib,ia] = position_sphere(radii, alpha, beta)
        for ib in range(Npts/2):
            beta = -ib*np.pi/(2.*(Npts/2-1))
            spheredown[:,ib,ia] = position_sphere(radii, alpha, beta)

    return sphereup, spheredown

def ellipse_polar(c, a, b, Nang=100):
    """ Function to determine an ellipse from its center and polar coordinates
        FROM: https://en.wikipedia.org/wiki/Ellipse
      c= coordinates of the center
      a= distance major axis
      b= distance minor axis
      Nang= number of angles to use
    """
    fname = 'ellipse_polar'

    if np.mod(Nang,2) == 0: Nang=Nang+1
  
    dtheta = 2*np.pi/(Nang-1)

    ellipse = np.zeros((Nang,2), dtype=np.float)
    for ia in range(Nang):
        theta = dtheta*ia
        rad = a*b/np.sqrt( (b*np.cos(theta))**2 + (a*np.sin(theta))**2 )
        x = rad*np.cos(theta)
        y = rad*np.sin(theta)
        ellipse[ia,:] = [y+c[0],x+c[1]]

    return ellipse

def hyperbola_polar(a, b, Nang=100):
    """ Fcuntion to determine an hyperbola in polar coordinates
        FROM: https://en.wikipedia.org/wiki/Hyperbola#Polar_coordinates
          x^2/a^2 - y^2/b^2 = 1
      a= x-parameter
      y= y-parameter
      Nang= number of angles to use
      DOES NOT WORK!!!!
    """
    fname = 'hyperbola_polar'

    dtheta = 2.*np.pi/(Nang-1)

    # Positive branch
    hyperbola_p = np.zeros((Nang,2), dtype=np.float)
    for ia in range(Nang):
        theta = dtheta*ia
        x = a*np.cosh(theta)
        y = b*np.sinh(theta)
        hyperbola_p[ia,:] = [y,x]

    # Negative branch
    hyperbola_n = np.zeros((Nang,2), dtype=np.float)
    for ia in range(Nang):
        theta = dtheta*ia
        x = -a*np.cosh(theta)
        y = b*np.sinh(theta)
        hyperbola_n[ia,:] = [y,x]

    return hyperbola_p, hyperbola_n

def circ_sec(ptA, ptB, radii, Nang=100):
    """ Function union of point A and B by a section of a circle
      ptA= coordinates od the point A [yA, xA]
      ptB= coordinates od the point B [yB, xB]
      radii= radi of the circle to use to unite the points
      Nang= amount of angles to use
    """
    fname = 'circ_sec'

    distAB = dist_points(ptA,ptB)

    if distAB > radii:
        print errormsg
        print '  ' + fname + ': radii=', radii, " too small for the distance " +     \
          "between points !!"
        print '    distance between points:', distAB
        quit(-1)

    # Coordinate increments
    dAB = np.abs(ptA-ptB)

    # angle of the circular section joining points
    alpha = 2.*np.arcsin((distAB/2.)/radii)

    # center along coincident bisection of the union
    xcc = -radii
    ycc = 0.

    # Getting the arc of the circle at the x-axis
    dalpha = alpha/(Nang-1)
    circ_sec = np.zeros((Nang,2), dtype=np.float)
    for ia in range(Nang):
        alpha = dalpha*ia
        x = radii*np.cos(alpha)
        y = radii*np.sin(alpha)

        circ_sec[ia,:] = [y+ycc,x+xcc]
    
    # Angle of the points
    theta = np.arctan2(ptB[0]-ptA[0],ptB[1]-ptA[1])

    # rotating angle of the circ
    rotangle = theta + 3.*np.pi/2. - alpha/2.

    #print 'alpha:', alpha*180./np.pi, 'theta:', theta*180./np.pi, 'rotangle:', rotangle*180./np.pi
  

    # rotating the arc along the x-axis
    rotcirc_sec = rotate_polygon_2D(circ_sec, rotangle)

    # Moving arc to the ptA
    circ_sec = rotcirc_sec + ptA

    return circ_sec

def p_square(face, N=5):
    """ Function to get a polygon square
      face: length of the face of the square
      N: number of points of the polygon
    """
    fname = 'p_square'

    square = np.zeros((N,2), dtype=np.float)

    f2 = face/2.
    N4 = N/4
    df = face/(N4)
    # SW-NW
    for ip in range(N4):
        square[ip,:] = [-f2+ip*df,-f2]
    # NW-NE
    for ip in range(N4):
        square[ip+N4,:] = [f2,-f2+ip*df]
    # NE-SE
    for ip in range(N4):
        square[ip+2*N4,:] = [f2-ip*df,f2]
    N42 = N-3*N4-1
    df = face/(N42)
    # SE-SW
    for ip in range(N42):
        square[ip+3*N4,:] = [-f2,f2-ip*df]
    square[N-1,:] = [-f2,-f2]

    return square

def p_circle(radii, N=50):
    """ Function to get a polygon of a circle
      radii: length of the radii of the circle
      N: number of points of the polygon
    """
    fname = 'p_circle'

    circle = np.zeros((N,2), dtype=np.float)

    dangle = 2.*np.pi/(N-1)

    for ia in range(N):
        circle[ia,:] = [radii*np.sin(ia*dangle), radii*np.cos(ia*dangle)]

    circle[N-1,:] = [0., radii]

    return circle

def p_triangle(p1, p2, p3, N=4):
    """ Function to provide the polygon of a triangle from its 3 vertices
      p1: vertex 1 [y,x]
      p2: vertex 2 [y,x]
      p3: vertex 3 [y,x]
      N: number of vertices of the triangle
    """
    fname = 'p_triangle'

    triangle = np.zeros((N,2), dtype=np.float)

    N3 = N / 3
    # 1-2
    dx = (p2[1]-p1[1])/N3
    dy = (p2[0]-p1[0])/N3
    for ip in range(N3):
        triangle[ip,:] = [p1[0]+ip*dy,p1[1]+ip*dx]
    # 2-3
    dx = (p3[1]-p2[1])/N3
    dy = (p3[0]-p2[0])/N3
    for ip in range(N3):
        triangle[ip+N3,:] = [p2[0]+ip*dy,p2[1]+ip*dx]
    # 3-1
    N32 = N - 2*N/3
    dx = (p1[1]-p3[1])/N32
    dy = (p1[0]-p3[0])/N32
    for ip in range(N32):
        triangle[ip+2*N3,:] = [p3[0]+ip*dy,p3[1]+ip*dx]

    triangle[N-1,:] = p1

    return triangle

def p_spiral(loops, eradii, N=1000):
    """ Function to provide a polygon of an Archimedean spiral
        FROM: https://en.wikipedia.org/wiki/Spiral
      loops: number of loops of the spiral
      eradii: length of the radii of the final spiral
      N: number of points of the polygon
    """
    fname = 'p_spiral'

    spiral = np.zeros((N,2), dtype=np.float)

    dangle = 2.*np.pi*loops/(N-1)
    dr = eradii*1./(N-1)

    for ia in range(N):
        radii = dr*ia
        spiral[ia,:] = [radii*np.sin(ia*dangle), radii*np.cos(ia*dangle)]

    return spiral

def p_reg_polygon(Nv, lf, N=50):
    """ Function to provide a regular polygon of Nv vertices
      Nv: number of vertices
      lf: length of the face
      N: number of points
    """
    fname = 'p_reg_polygon'

    reg_polygon = np.zeros((N,2), dtype=np.float)

    # Number of points per vertex
    Np = N/Nv
    # Angle incremental between vertices
    da = 2.*np.pi/Nv
    # Radii of the circle according to lf
    radii = lf*Nv/(2*np.pi)

    iip = 0
    for iv in range(Nv-1):
        # Characteristics between vertices iv and iv+1
        av1 = da*iv
        v1 = [radii*np.sin(av1), radii*np.cos(av1)]
        av2 = da*(iv+1)
        v2 = [radii*np.sin(av2), radii*np.cos(av2)]
        dx = (v2[1]-v1[1])/Np
        dy = (v2[0]-v1[0])/Np
        for ip in range(Np):
            reg_polygon[ip+iv*Np,:] = [v1[0]+dy*ip,v1[1]+dx*ip]

    # Characteristics between vertices Nv and 1

    # Number of points per vertex
    Np2 = N - Np*(Nv-1)

    av1 = da*Nv
    v1 = [radii*np.sin(av1), radii*np.cos(av1)]
    av2 = 0.
    v2 = [radii*np.sin(av2), radii*np.cos(av2)]
    dx = (v2[1]-v1[1])/Np2
    dy = (v2[0]-v1[0])/Np2
    for ip in range(Np2):
        reg_polygon[ip+(Nv-1)*Np,:] = [v1[0]+dy*ip,v1[1]+dx*ip]

    return reg_polygon

def p_reg_star(Nv, lf, freq, vs=0, N=50):
    """ Function to provide a regular star of Nv vertices
      Nv: number of vertices
      lf: length of the face of the regular polygon
      freq: frequency of union of vertices ('0', for just centered to zero arms)
      vs: vertex from which start (0 being first [0,lf])
      N: number of points
    """
    fname = 'p_reg_star'

    reg_star = np.zeros((N,2), dtype=np.float)

    # Number of arms of the star
    if freq != 0 and np.mod(Nv,freq) == 0: 
        Na = Nv/freq + 1
    else:
        Na = Nv

    # Number of points per arm
    Np = N/Na
    # Angle incremental between vertices
    da = 2.*np.pi/Nv
    # Radii of the circle according to lf
    radii = lf*Nv/(2*np.pi)

    iip = 0
    av1 = vs*da
    for iv in range(Na-1):
        # Characteristics between vertices iv and iv+1
        v1 = [radii*np.sin(av1), radii*np.cos(av1)]
        if freq != 0:
            av2 = av1 + da*freq
            v2 = [radii*np.sin(av2), radii*np.cos(av2)]
        else:
            v2 = [0., 0.]
            av2 = av1 + da
        dx = (v2[1]-v1[1])/(Np-1)
        dy = (v2[0]-v1[0])/(Np-1)
        for ip in range(Np):
            reg_star[ip+iv*Np,:] = [v1[0]+dy*ip,v1[1]+dx*ip]
        if av2 > 2.*np.pi: av1 = av2 - 2.*np.pi
        else: av1 = av2 + 0.

    iv = Na-1
    # Characteristics between vertices Na and 1
    Np2 = N-Np*iv
    v1 = [radii*np.sin(av1), radii*np.cos(av1)]
    if freq != 0:
        av2 = vs*da
        v2 = [radii*np.sin(av2), radii*np.cos(av2)]
    else:
        v2 = [0., 0.]
    dx = (v2[1]-v1[1])/(Np2-1)
    dy = (v2[0]-v1[0])/(Np2-1)
    for ip in range(Np2):
        reg_star[ip+iv*Np,:] = [v1[0]+dy*ip,v1[1]+dx*ip]

    return reg_star

def p_sinusiode(length=10., amp=5., lamb=3., ival=0., func='sin', N=100):
    """ Function to get coordinates of a sinusoidal curve
      length: length of the line (default 10.)
      amp: amplitude of the peaks (default 5.)
      lamb: wave longitude (defalult 3.)
      ival: initial angle (default 0. in degree)
      func: function to use: (default sinus)
        'sin': sinus
        'cos': cosinus
      N: number of points (default 100)
    """
    fname = 'p_sinusiode'
    availfunc = ['sin', 'cos']

    dx = length/(N-1)
    ia = ival*np.pi/180.
    da = 2*np.pi*dx/lamb

    sinusoide = np.zeros((N,2), dtype=np.float)
    if func == 'sin':
        for ix in range(N):
            sinusoide[ix,:] = [amp*np.sin(ia+da*ix),dx*ix]
    elif func == 'cos':
        for ix in range(N):
            sinusoide[ix,:] = [amp*np.cos(ia+da*ix),dx*ix]
    else:
        print errormsg
        print '  ' + fname + ": function '" + func + "' not ready !!"
        print '    available ones:', availfunc
        quit(-1)

    sinusoidesecs = ['sinusoide']
    sinusoidedic = {'sinusoide': [sinusoide, '-', '#000000', 1.]}

    return sinusoide, sinusoidesecs, sinusoidedic

def p_doubleArrow(length=5., angle=45., width=1., alength=0.10, N=50):
    """ Function to provide an arrow with double lines
      length: length of the arrow (5. default)
      angle: angle of the head of the arrow (45., default)
      width: separation between the two lines (2., default)
      alength: length of the head (as percentage in excess of width, 0.1 default)
      N: number of points (50, default)
    """
    import numpy.ma as ma
    function = 'p_doubleArrow'

    doubleArrow = np.zeros((50,2), dtype=np.float)
    N4 = int((N-3)/4)

    doublearrowdic = {}

    # Arms
    dx = length/(N4-1)
    for ix in range(N4-1):
        doubleArrow[ix,:] = [dx*ix,-width/2.]
    doublearrowdic['leftarm'] = [doubleArrow[0:N4-1,:], '-', '#000000', 2.]
    doubleArrow[N4-1,:] = [gen.fillValueF,gen.fillValueF]
    for ix in range(N4-1):
        doubleArrow[N4+ix,:] = [dx*ix,width/2.]
    doublearrowdic['rightarm'] = [doubleArrow[N4:2*N4-1,:], '-', '#000000', 2.]
    doubleArrow[2*N4-1,:] = [gen.fillValueF,gen.fillValueF]

    # Head
    N42 = int((N-2 - 2*N4)/2)
    dx = width*(1.+alength)/(N42-1)
    for ix in range(N42):
        doubleArrow[2*N4+ix,:] = [length-dx*ix,-dx*ix]
    doublearrowdic['lefthead'] = [doubleArrow[2*N4:2*N4+N42,:], '-', '#000000', 2.]
    doubleArrow[2*N4+N42,:] = [gen.fillValueF,gen.fillValueF]

    N43 = N-2 - 2*N4 - N42 + 1
    dx = width*(1.+alength)/(N43-1)
    for ix in range(N43):
        doubleArrow[2*N4+N42+1+ix,:] = [length-dx*ix,dx*ix]
    doublearrowdic['rightthead'] = [doubleArrow[2*N4+N42:51,:], '-', '#000000', 2.]

    doubleArrow = ma.masked_equal(doubleArrow, gen.fillValueF)
    doublearrowsecs = ['leftarm', 'rightarm', 'lefthead', 'righthead']

    return doubleArrow, doublearrowsecs, doublearrowdic 

# Combined objects
##

# FROM: http://www.photographers1.com/Sailing/NauticalTerms&Nomenclature.html
def zboat(length=10., beam=1., lbeam=0.4, sternbp=0.5):
    """ Function to define an schematic boat from the z-plane
      length: length of the boat (without stern, default 10)
      beam: beam of the boat (default 1)
      lbeam: length at beam (as percentage of length, default 0.4)
      sternbp: beam at stern (as percentage of beam, default 0.5)
    """
    fname = 'zboat'

    bow = np.array([length, 0.])
    maxportside = np.array([length*lbeam, -beam])
    maxstarboardside = np.array([length*lbeam, beam])
    portside = np.array([0., -beam*sternbp])
    starboardside = np.array([0., beam*sternbp])

    # forward section
    fportside = circ_sec(bow,maxportside, length*2)
    fstarboardside = circ_sec(maxstarboardside, bow, length*2)
    # aft section
    aportside = circ_sec(maxportside, portside, length*2)
    astarboardside = circ_sec(starboardside, maxstarboardside, length*2)
    # stern
    stern = circ_sec(portside, starboardside, length*2)

    dpts = stern.shape[0]
    boat = np.zeros((dpts*5,2), dtype=np.float)

    boat[0:dpts,:] = fportside
    boat[dpts:2*dpts,:] = aportside
    boat[2*dpts:3*dpts,:] = stern
    boat[3*dpts:4*dpts,:] = astarboardside
    boat[4*dpts:5*dpts,:] = fstarboardside

    fname = 'boat_L' + str(int(length*100.)) + '_B' + str(int(beam*100.)) + '_lb' +  \
      str(int(lbeam*100.)) + '_sb' + str(int(sternbp*100.)) + '.dat'
    if not os.path.isfile(fname):
        print infmsg
        print '  ' + fname + ": writting boat coordinates file '" + fname + "' !!"
        of = open(fname, 'w')
        of.write('# boat file with Length: ' + str(length) +' max_beam: '+str(beam)+ \
          'length_at_max_beam:' + str(lbeam) + '% beam at stern: ' + str(sternbp)+   \
          ' %\n')
        for ip in range(dpts*5):
            of.write(str(boat[ip,0]) + ' ' + str(boat[ip,1]) + '\n')
        
        of.close()
        print fname + ": Successfull written '" + fname + "' !!"
 
    return boat

def zsailing_boat(length=10., beam=1., lbeam=0.4, sternbp=0.5, lmast=0.6, wmast=0.1, \
  hsd=5., msd=5., lheads=0.38, lmains=0.55):
    """ Function to define an schematic sailing boat from the z-plane with sails
      length: length of the boat (without stern, default 10)
      beam: beam of the boat (default 1)
      lbeam: length at beam (as percentage of length, default 0.4)
      sternbp: beam at stern (as percentage of beam, default 0.5)
      lmast: position of the mast (as percentage of length, default 0.6)
      wmast: width of the mast (default 0.1)
      hsd: head sail direction respect to center line (default 5., -999.99 for upwind)
      msd: main sail direction respect to center line (default 5., -999.99 for upwind)
      lheads: length of head sail (as percentage of legnth, defaul 0.38)
      lmains: length of main sail (as percentage of legnth, defaul 0.55)
    """
    import numpy.ma as ma
    fname = 'zsailing_boat'

    bow = np.array([length, 0.])
    maxportside = np.array([length*lbeam, -beam])
    maxstarboardside = np.array([length*lbeam, beam])
    portside = np.array([0., -beam*sternbp])
    starboardside = np.array([0., beam*sternbp])

    # forward section
    fportside = circ_sec(bow,maxportside, length*2)
    fstarboardside = circ_sec(maxstarboardside, bow, length*2)
    dpts = fportside.shape[0]

    # aft section
    aportside = circ_sec(maxportside, portside, length*2)
    astarboardside = circ_sec(starboardside, maxstarboardside, length*2)
    # stern
    stern = circ_sec(portside, starboardside, length*2)
    # mast
    mast = p_circle(wmast,N=dpts)
    mast = mast + [length*lmast, 0.]
    # head sails
    lsail = lheads*length
    if hsd != -999.99:
        sailsa = np.pi/2. - np.pi*hsd/180.
        endsail = np.array([lsail*np.sin(sailsa), lsail*np.cos(sailsa)])
        endsail[0] = length - endsail[0]
        if bow[1] < endsail[1]:
            hsail = circ_sec(endsail, bow, lsail*2.15)
        else:
            hsail = circ_sec(bow, endsail, lsail*2.15)
    else:
        hsail0 = p_sinusiode(length=lsail, amp=0.2, lamb=0.75, N=dpts)
        hsail = np.zeros((dpts,2), dtype=np.float)
        hsail[:,0] = hsail0[:,1]
        hsail[:,1] = hsail0[:,0]
        hsail = bow - hsail

    # main sails
    lsail = lmains*length
    if msd != -999.99:
        sailsa = np.pi/2. - np.pi*msd/180.
        begsail = np.array([length*lmast, 0.])
        endsail = np.array([lsail*np.sin(sailsa), lsail*np.cos(sailsa)])
        endsail[0] = length*lmast - endsail[0]
        if endsail[1] < begsail[1]:
            msail = circ_sec(begsail, endsail, lsail*2.15)
        else:
            msail = circ_sec(endsail, begsail, lsail*2.15)
    else:
        msail0 = p_sinusiode(length=lsail, amp=0.25, lamb=1., N=dpts)
        msail = np.zeros((dpts,2), dtype=np.float)
        msail[:,0] = msail0[:,1]
        msail[:,1] = msail0[:,0]
        msail = [length*lmast,0] - msail

    sailingboat = np.zeros((dpts*8+4,2), dtype=np.float)

    sailingboat[0:dpts,:] = fportside
    sailingboat[dpts:2*dpts,:] = aportside
    sailingboat[2*dpts:3*dpts,:] = stern
    sailingboat[3*dpts:4*dpts,:] = astarboardside
    sailingboat[4*dpts:5*dpts,:] = fstarboardside
    sailingboat[5*dpts,:] = [gen.fillValueF, gen.fillValueF]
    sailingboat[5*dpts+1:6*dpts+1,:] = mast
    sailingboat[6*dpts+1,:] = [gen.fillValueF, gen.fillValueF]
    sailingboat[6*dpts+2:7*dpts+2,:] = hsail
    sailingboat[7*dpts+2,:] = [gen.fillValueF, gen.fillValueF]
    sailingboat[7*dpts+3:8*dpts+3,:] = msail
    sailingboat[8*dpts+3,:] = [gen.fillValueF, gen.fillValueF]

    sailingboat = ma.masked_equal(sailingboat, gen.fillValueF)

    # Center line extending [fcl] percentage from length on aft and stern
    fcl = 0.15
    centerline = np.zeros((dpts,2), dtype=np.float)
    dl = length*(1.+fcl*2.)/(dpts-1)
    centerline[:,0] = np.arange(-length*fcl, length*(1. + fcl)+dl, dl)

    # correct order of sections
    sailingboatsecs = ['fportside', 'aportside', 'stern', 'astarboardside',          \
      'fstarboardside', 'mast', 'hsail', 'msail', 'centerline']
    # dictionary with sections [polygon_vertices, line_type, line_color, line_width]
    dicsailingboat = {'fportside': [fportside, '-', '#8A5900', 2.],                  \
      'aportside': [aportside, '-', '#8A5900', 2.],                                  \
      'stern': [stern, '-', '#8A5900', 2.],                                          \
      'astarboardside': [astarboardside, '-', '#8A5900', 2.],                        \
      'fstarboardside': [fstarboardside, '-', '#8A5900', 2.],                        \
      'mast': [mast, '-', '#8A5900', 2.], 'hsail': [hsail, '-', '#AAAAAA', 1.],      \
      'msail': [msail, '-', '#AAAAAA', 1.],                                          \
      'centerline': [centerline, '-.', '#AA6464', 1.5]}
    
    fname = 'sailboat_L' + str(int(length*100.)) + '_B' + str(int(beam*100.)) +      \
      '_lb' + str(int(lbeam*100.)) + '_sb' + str(int(sternbp*100.)) +                \
      '_lm' + str(int(lmast*100.)) + '_wm' + str(int(wmast)) +                       \
      '_hsd' + str(int(hsd)) + '_hs' + str(int(lheads*100.)) +                       \
      '_ms' + str(int(lheads*100.)) + '_msd' + str(int(msd)) +'.dat'
    if not os.path.isfile(fname):
        print infmsg
        print '  ' + fname + ": writting boat coordinates file '" + fname + "' !!"
        of = open(fname, 'w')
        of.write('# boat file with Length: ' + str(length) +' max_beam: '+str(beam)+ \
          'length_at_max_beam:' + str(lbeam) + '% beam at stern: ' + str(sternbp)+   \
          ' % mast position: '+ str(lmast) + ' % mast width: ' + str(wmast) + ' ' +  \
          ' head sail direction:' + str(hsd) + ' head sail length: ' + str(lheads) + \
          ' %' + ' main sail length' + str(lmains) + ' main sail direction:' +       \
          str(msd) +'\n')
        for ip in range(dpts*5):
            of.write(str(sailingboat[ip,0]) + ' ' + str(sailingboat[ip,1]) + '\n')
        
        of.close()
        print fname + ": Successfull written '" + fname + "' !!"
 
    return sailingboat, sailingboatsecs, dicsailingboat

def write_join_poly(polys, flname='join_polygons.dat'):
    """ Function to write an ASCII file with the combination of polygons
      polys: dictionary with the names of the different polygons
      flname: name of the ASCII file
    """
    fname = 'write_join_poly'

    of = open(flname, 'w')

    for polyn in polys.keys():
        vertices = polys[polyn]
        Npts = vertices.shape[0]
        for ip in range(Npts):
            of.write(polyn+' '+str(vertices[ip,1]) + ' ' + str(vertices[ip,0]) + '\n')

    of.close()

    return

def read_join_poly(flname='join_polygons.dat'):
    """ Function to read an ASCII file with the combination of polygons
      flname: name of the ASCII file
    """
    fname = 'read_join_poly'

    of = open(flname, 'r')

    polys = {}
    polyn = ''
    poly = []
    for line in of:
        if len(line) > 1: 
            linevals = line.replace('\n','').split(' ')
            if polyn != linevals[0]:
                if len(poly) > 1:
                    polys[polyn] = np.array(poly)
                polyn = linevals[0]
                poly = []
                poly.append([np.float(linevals[2]), np.float(linevals[1])])
            else:
                poly.append([np.float(linevals[2]), np.float(linevals[1])])

    of.close()
    polys[polyn] = np.array(poly)

    return polys

####### ####### ##### #### ### ## #
# Plotting

def plot_sphere(iazm=-60., iele=30., dist=10., Npts=100, radii=10,                   \
  drwsfc=[True,True], colsfc=['#AAAAAA','#646464'],                                  \
  drwxline = True, linex=[':','b',2.], drwyline = True, liney=[':','r',2.],          \
  drwzline = True, linez=['-.','g',2.], drwxcline=[True,True],                       \
  linexc=[['-','#646400',1.],['--','#646400',1.]],                                   \
  drwequator=[True,True], lineeq=[['-','#AA00AA',1.],['--','#AA00AA',1.]],           \
  drwgreeenwhich=[True,True], linegw=[['-','k',1.],['--','k',1.]]):
    """ Function to plot an sphere and determine which standard lines will be also 
        drawn
      iazm: azimut of the camera form the sphere
      iele: elevation of the camera form the sphere
      dist: distance of the camera form the sphere
      Npts: Resolution for the sphere
      radii: radius of the sphere
      drwsfc: whether 'up' and 'down' portions of the sphere should be drawn
      colsfc: colors of the surface of the sphere portions ['up', 'down']
      drwxline: whether x-axis line should be drawn
      linex: properties of the x-axis line ['type', 'color', 'wdith']
      drwyline: whether y-axis line should be drawn
      liney: properties of the y-axis line ['type', 'color', 'wdith']
      drwzline: whether z-axis line should be drawn
      linez: properties of the z-axis line ['type', 'color', 'wdith']
      drwequator: whether 'front' and 'back' portions of the Equator should be drawn
      lineeq: properties of the lines 'front' and 'back' of the Equator
      drwgreeenwhich: whether 'front', 'back' portions of Greenqhich should be drawn
      linegw: properties of the lines 'front' and 'back' Greenwhich
      drwxcline: whether 'front', 'back' 90 line (lon=90., lon=270.) should be drawn
      linexc: properties of the lines 'front' and 'back' for the 90 line
    """
    fname = 'plot_sphere'

    iazmrad = iazm*np.pi/180.
    ielerad = iele*np.pi/180.

    # 3D surface Sphere
    sfcsphereu, sfcsphered = surface_sphere(radii,Npts)
    
    # greenwhich
    if iazmrad > np.pi/2. and iazmrad < 3.*np.pi/2.:
        ia=np.pi-ielerad
    else:
        ia=0.-ielerad
    ea=ia+np.pi
    da = (ea-ia)/(Npts-1)
    beta = np.arange(ia,ea+da,da)[0:Npts]
    alpha = np.zeros((Npts), dtype=np.float)
    greenwhichc = spheric_line(radii,alpha,beta)
    ia=ea+0.
    ea=ia+np.pi
    da = (ea-ia)/(Npts-1)
    beta = np.arange(ia,ea+da,da)[0:Npts]
    greenwhichd = spheric_line(radii,alpha,beta)

    # Equator
    ia=np.pi-iazmrad/2.
    ea=ia+np.pi
    da = (ea-ia)/(Npts-1)
    alpha = np.arange(ia,ea+da,da)[0:Npts]
    beta = np.zeros((Npts), dtype=np.float)
    equatorc = spheric_line(radii,alpha,beta)
    ia=ea+0.
    ea=ia+np.pi
    da = (ea-ia)/(Npts-1)
    alpha = np.arange(ia,ea+da,da)[0:Npts]
    equatord = spheric_line(radii,alpha,beta)

    # 90 line
    if iazmrad > np.pi and iazmrad < 2.*np.pi:
        ia=3.*np.pi/2. + ielerad
    else:
        ia=np.pi/2. - ielerad
    if ielerad < 0.:
        ia = ia + np.pi
    ea=ia+np.pi
    da = (ea-ia)/(Npts-1)
    beta = np.arange(ia,ea+da,da)[0:Npts]
    alpha = np.ones((Npts), dtype=np.float)*np.pi/2.
    xclinec = spheric_line(radii,alpha,beta)
    ia=ea+0.
    ea=ia+np.pi
    da = (ea-ia)/(Npts-1)
    beta = np.arange(ia,ea+da,da)[0:Npts]
    xclined = spheric_line(radii,alpha,beta)

    # x line
    xline = np.zeros((2,3), dtype=np.float)
    xline[0,:] = position_sphere(radii, 0., 0.)
    xline[1,:] = position_sphere(radii, np.pi, 0.)

    # y line
    yline = np.zeros((2,3), dtype=np.float)
    yline[0,:] = position_sphere(radii, np.pi/2., 0.)
    yline[1,:] = position_sphere(radii, 3*np.pi/2., 0.)

    # z line
    zline = np.zeros((2,3), dtype=np.float)
    zline[0,:] = position_sphere(radii, 0., np.pi/2.)
    zline[1,:] = position_sphere(radii, 0., -np.pi/2.)

    fig = plt.figure()
    ax = fig.gca(projection='3d')

    # Sphere surface
    if drwsfc[0]:
        ax.plot_surface(sfcsphereu[0,:,:], sfcsphereu[1,:,:], sfcsphereu[2,:,:],     \
          color=colsfc[0])
    if drwsfc[1]:
        ax.plot_surface(sfcsphered[0,:,:], sfcsphered[1,:,:], sfcsphered[2,:,:],     \
          color=colsfc[1])

    # greenwhich
    linev = linegw[0]
    if drwgreeenwhich[0]:
        ax.plot(greenwhichc[:,0], greenwhichc[:,1], greenwhichc[:,2], linev[0],      \
          color=linev[1], linewidth=linev[2],  label='Greenwich')
    linev = linegw[1]
    if drwgreeenwhich[1]:
        ax.plot(greenwhichd[:,0], greenwhichd[:,1], greenwhichd[:,2], linev[0],      \
          color=linev[1], linewidth=linev[2])

    # Equator
    linev = lineeq[0]
    if drwequator[0]:
        ax.plot(equatorc[:,0], equatorc[:,1], equatorc[:,2], linev[0],               \
          color=linev[1], linewidth=linev[2], label='Equator')
    linev = lineeq[1]
    if drwequator[1]:
        ax.plot(equatord[:,0], equatord[:,1], equatord[:,2], linev[0],               \
          color=linev[1], linewidth=linev[2])

    # 90line
    linev = linexc[0]
    if drwxcline[0]:
        ax.plot(xclinec[:,0], xclinec[:,1], xclinec[:,2], linev[0], color=linev[1],  \
          linewidth=linev[2], label='90-line')
    linev = linexc[1]
    if drwxcline[1]:
        ax.plot(xclined[:,0], xclined[:,1], xclined[:,2], linev[0], color=linev[1],  \
          linewidth=linev[2])

    # x line
    linev = linex
    if drwxline:
        ax.plot([xline[0,0],xline[1,0]], [xline[0,1],xline[1,1]],                    \
          [xline[0,2],xline[1,2]], linev[0], color=linev[1], linewidth=linev[2],  label='xline')

    # y line
    linev = liney
    if drwyline:
        ax.plot([yline[0,0],yline[1,0]], [yline[0,1],yline[1,1]],                    \
          [yline[0,2],yline[1,2]], linev[0], color=linev[1], linewidth=linev[2],  label='yline')

    # z line
    linev = linez
    if drwzline:
        ax.plot([zline[0,0],zline[1,0]], [zline[0,1],zline[1,1]],                    \
          [zline[0,2],zline[1,2]], linev[0], color=linev[1], linewidth=linev[2],  label='zline')

    plt.legend()

    return fig, ax