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fig09.py
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fig09.py
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from mpl_toolkits.mplot3d import axes3d, proj3d
import matplotlib as mpl
import matplotlib.pyplot as plt
from pylab import np
###patch start### http://stackoverflow.com/questions/16488182/removing-axes-margins-in-3d-plot
from mpl_toolkits.mplot3d.axis3d import Axis
if not hasattr(Axis, "_get_coord_info_old"):
def _get_coord_info_new(self, renderer):
mins, maxs, centers, deltas, tc, highs = self._get_coord_info_old(renderer)
mins += deltas / 4
maxs -= deltas / 4
return mins, maxs, centers, deltas, tc, highs
Axis._get_coord_info_old = Axis._get_coord_info
Axis._get_coord_info = _get_coord_info_new
###patch end###
# Figure size
mpl.rcParams['figure.figsize'] = 10, 10.5
# Text and numbers are substituded by Inkscape's PDF+Latex export option.
# By varying the sizes we can adjust the positioning
# http://stackoverflow.com/questions/30201310/use-of-hyphen-or-minus-sign-in-matplotlib-versus-compatibility-with-latex
mpl.rcParams['axes.unicode_minus'] = False
mpl.rcParams.update({'font.size': 15})
mpl.rc('xtick', labelsize=20)
mpl.rc('ytick', labelsize=20)
#
# This is important for font substitution
mpl.rcParams['svg.fonttype'] = 'none'
def orthogonal_proj(zfront, zback):
a = (zfront+zback)/(zfront-zback)
b = -2*(zfront*zback)/(zfront-zback)
# -0.0001 added for numerical stability as suggested in:
# http://stackoverflow.com/questions/23840756
return np.array([[1,0,0,0],
[0,1,0,0],
[0,0,a,b],
[0,0,-0.0001,zback]])
# Propblem parameters
beta = np.radians(30)
alpha = np.radians(60) # 90 - beta
betas = np.radians(10)
delta = np.radians(30)
lK = 100
Rg = 25
Rk = 20
Rko = 35
le = 40
N = 4
ot0 = 40 # offset
ot = np.array([ 0, 90, 180, 270 ])
# Derived trigonometric data
omegat = np.arange(0, 361, 1)
def rB(R, omegat):
cot = np.cos(np.radians(omegat))
sot = np.sin(np.radians(omegat))
rBx = cot*R
rBy = sot*R
rBz = 0
return np.array([ rBx, rBy, rBz ])
def rA(lK, R, alpha, betas, beta, delta, omegat):
cb = np.cos(beta)
sb = np.sin(beta)
ca = np.cos(alpha)
sa = np.sin(alpha)
cbs = np.cos(betas)
sbs = np.sin(betas)
cotd = np.cos(np.radians(omegat)+delta)
sotd = np.sin(np.radians(omegat)+delta)
rAx = (ca*cbs*cotd - sbs*sotd)*R + cb*lK
rAy = (ca*sbs*cotd + cbs*sotd)*R
rAz = ( -sa*cotd )*R + sb*lK
return np.array([ rAx, rAy, rAz ])
def rK(lK, beta):
cb = np.cos(beta)
sb = np.sin(beta)
rKx = cb*lK
rKy = 0
rKz = sb*lK
return np.array([ rKx, rKy, rKz ])
def ew():
return np.array([[ 1, 0, 0],
[ 0, 1, 0],
[ 0, 0, 1]])
def ek(alpha, betas):
ca = np.cos(alpha)
sa = np.sin(alpha)
cbs = np.cos(betas)
sbs = np.sin(betas)
return np.array([[ ca*cbs, -sbs, sa*cbs],
[ ca*sbs, cbs, sa*sbs],
[-sa, 0, ca]])
def es(betas):
cbs = np.cos(betas)
sbs = np.sin(betas)
return np.array([[ cbs, -sbs, 0],
[ sbs, cbs, 0],
[ 0, 0, 1]])
def ea(alpha, betas, delta, omegat):
cotd = np.cos(np.radians(omegat)+delta)
sotd = np.sin(np.radians(omegat)+delta)
T = ek(alpha, betas)
e = np.array([[ cotd,-sotd, 0],
[ sotd, cotd, 0],
[ 0, 0, 1]])
return T.dot(e)
def eb(omegat):
cot = np.cos(np.radians(omegat))
sot = np.sin(np.radians(omegat))
return np.array([[ cot,-sot, 0],
[ sot, cot, 0],
[ 0, 0, 1]])
def arc(r0, R, e, n, phi0, phi):
e1 = e/np.sqrt(np.sum(e*e)) # normalize
en = n/np.sqrt(np.sum(n*n)) # normalize
ip = np.argmax(phi>phi0) # find end index
e2 = np.cross(en, e1)
cp = np.cos(np.radians(phi[:ip]))
sp = np.sin(np.radians(phi[:ip]))
# r = cp*e1+sp*e2
r = np.zeros((3, ip))
r[0,:] = r0[0]+R*(cp*e1[0]+sp*e2[0])
r[1,:] = r0[1]+R*(cp*e1[1]+sp*e2[1])
r[2,:] = r0[2]+R*(cp*e1[2]+sp*e2[2])
return r
fig = plt.figure()
ax = fig.gca(projection='3d')
#ax.set_axis_off()
#ax.w_zaxis.line.set_lw(0.)
#ax.set_zticks([])
for elt in ax.w_zaxis.get_ticklines() + ax.w_zaxis.get_ticklabels():
elt.set_visible(False)
ax.w_zaxis.gridlines.set_visible(False)
ax.w_zaxis.pane.set_visible(False)
ax.w_zaxis.line.set_visible(False)
ax.w_xaxis.set_pane_color((1.0, 1.0, 1.0, 0.0))
ax.w_yaxis.set_pane_color((1.0, 1.0, 1.0, 0.0))
ax.w_zaxis.set_pane_color((1.0, 1.0, 1.0, 0.0))
# Ground rotor
r = rB(Rg, omegat)
ax.plot(r[0], r[1], r[2], linewidth=1.2, color='b')
ax.scatter(0, 0, 0, zorder=20, s=100, linewidth=1.2, edgecolor='b', facecolor='b')
r = rB(Rg, ot0+ot[0])
ax.scatter(r[0], r[1], r[2], zorder=20, s=100, linewidth=1.2, edgecolor='b', facecolor='w')
r = rB(Rg, ot0+ot[1])
ax.scatter(r[0], r[1], r[2], zorder=20, s=100, linewidth=1.2, edgecolor='b', facecolor='w')
r = rB(Rg, ot0+ot[2])
ax.scatter(r[0], r[1], r[2], zorder=20, s=100, linewidth=1.2, edgecolor='b', facecolor='w')
r = rB(Rg, ot0+ot[3])
ax.scatter(r[0], r[1], r[2], zorder=20, s=100, linewidth=1.2, edgecolor='b', facecolor='w')
# vector base wind reference frame
e = le*ew()
ewx = e[:,0]
ax.plot([0, ewx[0]], [0, ewx[1]], [0, ewx[2]], linewidth=3, color='k')
ewy = e[:,1]
ax.plot([0, ewy[0]], [0, ewy[1]], [0, ewy[2]], linewidth=3, color='k')
ewz = e[:,2]
ax.plot([0, ewz[0]], [0, ewz[1]], [0, ewz[2]], linewidth=3, color='k')
# vector base rotating reference frame
e = le*eb(ot0)
ebx = e[:,0]
ax.plot([0, ebx[0]], [0, ebx[1]], [0, ebx[2]], linewidth=3, color='b')
eby = e[:,1]
ax.plot([0, eby[0]], [0, eby[1]], [0, eby[2]], linewidth=3, color='b')
ebz = e[:,2]
#ax.plot([0, ebz[0]], [0, ebz[1]], [0, ebz[2]], linewidth=2, color='b')
# elevation angle
r0 = np.array([0, 0, 0])
R = 32
e = ewx
n =-ewy
r = arc(r0, R, e, n, np.degrees(beta), omegat)
ax.plot(r[0,:], r[1,:], r[2,:], zorder=20, linewidth=2, color='r')
# rotor spinning
R = 20
e = ewx
n = ewz
r = arc(r0, R, e, n, ot0, omegat)
ax.plot(r[0,:], r[1,:], r[2,:], zorder=20, linewidth=2, color='r')
# Flying rotor
r = rA(lK, Rk, alpha, betas, beta, delta, omegat)
ax.plot(r[0], r[1], r[2], zorder=-10, linewidth=1.2, color='#00ff00')
r = rA(lK, Rk, alpha, betas, beta, delta, ot0+ot[0])
ax.scatter(r[0], r[1], r[2], zorder=10, s=100, linewidth=1.2, edgecolor='#00ff00', facecolor='w')
r = rA(lK, Rk, alpha, betas, beta, delta, ot0+ot[1])
ax.scatter(r[0], r[1], r[2], zorder=10, s=100, linewidth=1.2, edgecolor='#00ff00', facecolor='w')
r = rA(lK, Rk, alpha, betas, beta, delta, ot0+ot[2])
ax.scatter(r[0], r[1], r[2], zorder=10, s=100, linewidth=1.2, edgecolor='#00ff00', facecolor='w')
r = rA(lK, Rk, alpha, betas, beta, delta, ot0+ot[3])
ax.scatter(r[0], r[1], r[2], zorder=10, s=100, linewidth=1.2, edgecolor='#00ff00', facecolor='w')
r = rA(lK, Rko, alpha, betas, beta, delta, omegat)
ax.plot(r[0], r[1], r[2], linewidth=1.2, color='#00ff00')
r = rK(lK, beta)
ax.scatter(r[0], r[1], r[2], zorder=5, s=100, linewidth=1.2, edgecolor='#00ff00', facecolor='#00ff00')
# vector base wind reference frame at kite location
ax.plot([r[0], r[0]+ewx[0]], [r[1], r[1]+ewx[1]], [r[2], r[2]+ewx[2]], zorder=-20, linewidth=3, color='0.7')
ax.plot([r[0], r[0]+ewy[0]], [r[1], r[1]+ewy[1]], [r[2], r[2]+ewy[2]], zorder=-20, linewidth=3, color='0.7')
#ax.plot([r[0], r[0]+ewz[0]], [r[1], r[1]+ewz[1]], [r[2], r[2]+ewz[2]], zorder=-20, linewidth=2, color='0.7')
# vector base intermediate reference frame
e = le*es(betas)
esx = e[:,0]
ax.plot([r[0], r[0]+esx[0]], [r[1], r[1]+esx[1]], [r[2], r[2]+esx[2]], linewidth=3, color='0.7')
esy = e[:,1]
ax.plot([r[0], r[0]+esy[0]], [r[1], r[1]+esy[1]], [r[2], r[2]+esy[2]], linewidth=3, color='0.7')
esz = e[:,2]
ax.plot([r[0], r[0]+esz[0]], [r[1], r[1]+esz[1]], [r[2], r[2]+esz[2]], linewidth=3, color='0.7')
# vector base kite reference frame
e = le*ek(alpha, betas)
ekx = e[:,0]
ax.plot([r[0], r[0]+ekx[0]], [r[1], r[1]+ekx[1]], [r[2], r[2]+ekx[2]], linewidth=3, color='k')
eky = e[:,1]
ax.plot([r[0], r[0]+eky[0]], [r[1], r[1]+eky[1]], [r[2], r[2]+eky[2]], linewidth=3, color='k')
ekz = e[:,2]
ax.plot([r[0], r[0]+ekz[0]], [r[1], r[1]+ekz[1]], [r[2], r[2]+ekz[2]], linewidth=3, color='k')
# vector base kite rotating reference frame
e = le*ea(alpha, betas, delta, ot0)
eax = e[:,0]
ax.plot([r[0], r[0]+eax[0]], [r[1], r[1]+eax[1]], [r[2], r[2]+eax[2]], zorder=-20, linewidth=3, color='#00ff00')
eay = e[:,1]
ax.plot([r[0], r[0]+eay[0]], [r[1], r[1]+eay[1]], [r[2], r[2]+eay[2]], zorder=-20, linewidth=3, color='#00ff00')
eaz = e[:,2]
#ax.plot([r[0], r[0]+eaz[0]], [r[1], r[1]+eaz[1]], [r[2], r[2]+eaz[2]], zorder=-20, linewidth=2, color='#00ff00')
# additional base vectors
#ax.plot([r[0], r[0]+ewz[0]], [r[1], r[1]+ewz[1]], [r[2], r[2]+ewz[2]], linewidth=2, color='0.7')
# lK
ax.plot([0, r[0]], [0, r[1]], [0, r[2]], linewidth=1.22, color='k')
# kite reference frame rotation
r0 = r
R = 35
e = ewx
n = ewz
r = arc(r0, R, e, n, np.degrees(betas), omegat)
ax.plot(r[0,:], r[1,:], r[2,:], linewidth=2, color='r')
e = ewy
r = arc(r0, R, e, n, np.degrees(betas), omegat)
ax.plot(r[0,:], r[1,:], r[2,:], linewidth=2, color='r')
R = 35
e = esx
n = esy
r = arc(r0, R, e, n, np.degrees(alpha), omegat)
ax.plot(r[0,:], r[1,:], r[2,:], linewidth=2, color='r')
e = esz
r = arc(r0, R, e, n, np.degrees(alpha), omegat)
ax.plot(r[0,:], r[1,:], r[2,:], linewidth=2, color='r')
# rotor spinning
R = 15
e = ekx
n = ekz
r = arc(r0, R, e, n, ot0, omegat)
ax.plot(r[0,:], r[1,:], r[2,:], linewidth=2, color='r')
r = arc(r0, R, e, n, ot0+np.degrees(delta), omegat)
ax.plot(r[0,:], r[1,:], r[2,:], linewidth=2, color='r')
# tethers
ra = rA(lK, Rk, alpha, betas, beta, delta, ot0+ot[0])
rb = rB(Rg, ot0+ot[0])
ax.plot([rb[0], ra[0]], [rb[1], ra[1]], [rb[2], ra[2]], zorder=-20, linewidth=1.2, color='k')
#
ra = rA(lK, Rk, alpha, betas, beta, delta, ot0+ot[1])
rb = rB(Rg, ot0+ot[1])
ax.plot([rb[0], ra[0]], [rb[1], ra[1]], [rb[2], ra[2]], zorder=-20, linewidth=1.2, color='0.8')
#
ra = rA(lK, Rk, alpha, betas, beta, delta, ot0+ot[2])
rb = rB(Rg, ot0+ot[2])
ax.plot([rb[0], ra[0]], [rb[1], ra[1]], [rb[2], ra[2]], zorder=-20, linewidth=1.2, color='0.8')
#
ra = rA(lK, Rk, alpha, betas, beta, delta, ot0+ot[3])
rb = rB(Rg, ot0+ot[3])
ax.plot([rb[0], ra[0]], [rb[1], ra[1]], [rb[2], ra[2]], zorder=-20, linewidth=1.2, color='0.8')
# kite angular velocities
rk0 = rK(lK, beta)
r0 = rk0+0.8*ekz
R = 5
e = 0.5*ekx-eky
n = ekz
r = arc(r0, R, e, n, 320, omegat)
ax.plot(r[0,:], r[1,:], r[2,:], linewidth=2, color='r')
ax.scatter(r0[0], r0[1], r0[2], s=10, linewidth=1.2, edgecolor='r', facecolor='w')
r0 = rk0+1.2*ekx
R = 5
e =-eky-ekz
n = ekx
r = arc(r0, R, e, n, 320, omegat)
ax.plot(r[0,:], r[1,:], r[2,:], linewidth=2, color='r')
ax.scatter(r0[0], r0[1], r0[2], s=10, linewidth=1.2, edgecolor='r', facecolor='w')
r0 = rk0+1.2*eky
R = 5
e =-0.5*ekx+ekz
n = eky
r = arc(r0, R, e, n, 320, omegat)
ax.plot(r[0,:], r[1,:], r[2,:], linewidth=2, color='r')
ax.scatter(r0[0], r0[1], r0[2], s=10, linewidth=1.2, edgecolor='r', facecolor='w')
ax.set_xlim3d(-25, 75)
ax.set_ylim3d(-50, 50)
ax.set_zlim3d( 0, 100)
#ax.axis('equal')
#ax.set_aspect('equal')
ax.view_init(20, 225)
ax.set_axis_off()
# orthogonal projection
proj3d.persp_transformation = orthogonal_proj
plt.tight_layout()
plt.subplots_adjust(left=-0.31, right=1.12, top=1.4, bottom=-0.3)
#plt.subplots_adjust(left=-0.31, right=1.12, top=1.2, bottom=-0.5) # to get the top arc
plt.savefig("fig09_diagram.svg")
plt.show()