# set terminal pngcairo  background "#ffffff" enhanced font "arial,10" fontscale 1.0 size 600, 400 
# set output 'airfoil.3.png'
set dummy t,y
set key inside right top vertical Right noreverse enhanced autotitles box linetype -1 linewidth 1.000
set parametric
set style data lines
set title "Joukowski Airfoil using Complex Variables" 
set timestamp "%a %b %d %H:%M:%S %Y" 
set trange [ 0.00000 : 6.28319 ] noreverse nowriteback
set xlabel "eps = 0.06 real" 
set xrange [ -0.600000 : 0.600000 ] noreverse nowriteback
set yrange [ -0.200000 : 1.80000 ] noreverse nowriteback
bez_d4_i0(x) =     (1 - x)**4
bez_d4_i1(x) = 4 * (1 - x)**3 * x
bez_d4_i2(x) = 6 * (1 - x)**2 * x**2
bez_d4_i3(x) = 4 * (1 - x)**1 * x**3
bez_d4_i4(x) =                  x**4
bez_d8_i0(x) =      (1 - x)**8
bez_d8_i1(x) =  8 * (1 - x)**7 * x
bez_d8_i2(x) = 28 * (1 - x)**6 * x**2
bez_d8_i3(x) = 56 * (1 - x)**5 * x**3
bez_d8_i4(x) = 70 * (1 - x)**4 * x**4
bez_d8_i5(x) = 56 * (1 - x)**3 * x**5
bez_d8_i6(x) = 28 * (1 - x)**2 * x**6
bez_d8_i7(x) =  8 * (1 - x)    * x**7
bez_d8_i8(x) =                   x**8
mean_y(t) = m0 * mm * bez_d4_i0(t) + 	    m1 * mm * bez_d4_i1(t) + 	    m2 * mm * bez_d4_i2(t) + 	    m3 * mm * bez_d4_i3(t) + 	    m4 * mm * bez_d4_i4(t)
mean_x(t) = p0 * bez_d4_i0(t) + 	    p1 * bez_d4_i1(t) + 	    p2 * bez_d4_i2(t) + 	    p3 * bez_d4_i3(t) + 	    p4 * bez_d4_i4(t)
z_x(x) = x0 * bez_d8_i0(x) + x1 * bez_d8_i1(x) + x2 * bez_d8_i2(x) + 	 x3 * bez_d8_i3(x) + x4 * bez_d8_i4(x) + x5 * bez_d8_i5(x) + 	 x6 * bez_d8_i6(x) + x7 * bez_d8_i7(x) + x8 * bez_d8_i8(x)
z_y(x, tk) =    y0 * tk * bez_d8_i0(x) + y1 * tk * bez_d8_i1(x) + y2 * tk * bez_d8_i2(x) +    y3 * tk * bez_d8_i3(x) + y4 * tk * bez_d8_i4(x) + y5 * tk * bez_d8_i5(x) +    y6 * tk * bez_d8_i6(x) + y7 * tk * bez_d8_i7(x) + y8 * tk * bez_d8_i8(x)
airfoil_y1(t, thick) = mean_y(z_x(t)) + z_y(t, thick)
airfoil_y2(t, thick) = mean_y(z_x(t)) - z_y(t, thick)
airfoil_y(t) = mean_y(z_x(t))
airfoil_x(t) = mean_x(z_x(t))
zeta(t) = -eps + (a+eps)*exp(t*{0,1})
eta(t) = zeta(t) + a*a/zeta(t)
mm = 0.0
thick = 0.12
pp = 0.5
x0 = 0.0
y0 = 0.0
x1 = 0.0
y1 = 0.18556
x2 = 0.03571
y2 = 0.34863
x3 = 0.10714
y3 = 0.48919
x4 = 0.21429
y4 = 0.58214
x5 = 0.35714
y5 = 0.55724
x6 = 0.53571
y6 = 0.44992
x7 = 0.75
y7 = 0.30281
x8 = 1.0
y8 = 0.0105
GPFUN_bez_d4_i0 = "bez_d4_i0(x) =     (1 - x)**4"
GPFUN_bez_d4_i1 = "bez_d4_i1(x) = 4 * (1 - x)**3 * x"
GPFUN_bez_d4_i2 = "bez_d4_i2(x) = 6 * (1 - x)**2 * x**2"
GPFUN_bez_d4_i3 = "bez_d4_i3(x) = 4 * (1 - x)**1 * x**3"
GPFUN_bez_d4_i4 = "bez_d4_i4(x) =                  x**4"
GPFUN_bez_d8_i0 = "bez_d8_i0(x) =      (1 - x)**8"
GPFUN_bez_d8_i1 = "bez_d8_i1(x) =  8 * (1 - x)**7 * x"
GPFUN_bez_d8_i2 = "bez_d8_i2(x) = 28 * (1 - x)**6 * x**2"
GPFUN_bez_d8_i3 = "bez_d8_i3(x) = 56 * (1 - x)**5 * x**3"
GPFUN_bez_d8_i4 = "bez_d8_i4(x) = 70 * (1 - x)**4 * x**4"
GPFUN_bez_d8_i5 = "bez_d8_i5(x) = 56 * (1 - x)**3 * x**5"
GPFUN_bez_d8_i6 = "bez_d8_i6(x) = 28 * (1 - x)**2 * x**6"
GPFUN_bez_d8_i7 = "bez_d8_i7(x) =  8 * (1 - x)    * x**7"
GPFUN_bez_d8_i8 = "bez_d8_i8(x) =                   x**8"
m0 = 0.0
m1 = 0.1
m2 = 0.1
m3 = 0.1
m4 = 0.0
GPFUN_mean_y = "mean_y(t) = m0 * mm * bez_d4_i0(t) + \t    m1 * mm * bez_d4_i1(t) + \t    m2 * mm * bez_d4_i2(t) + \t    m3 * mm * bez_d4_i3(t) + \t    m4 * mm * bez_d4_i4(t)"
p0 = 0.0
p1 = 0.2
p2 = 0.4
p3 = 0.7
p4 = 1.0
GPFUN_mean_x = "mean_x(t) = p0 * bez_d4_i0(t) + \t    p1 * bez_d4_i1(t) + \t    p2 * bez_d4_i2(t) + \t    p3 * bez_d4_i3(t) + \t    p4 * bez_d4_i4(t)"
GPFUN_z_x = "z_x(x) = x0 * bez_d8_i0(x) + x1 * bez_d8_i1(x) + x2 * bez_d8_i2(x) + \t x3 * bez_d8_i3(x) + x4 * bez_d8_i4(x) + x5 * bez_d8_i5(x) + \t x6 * bez_d8_i6(x) + x7 * bez_d8_i7(x) + x8 * bez_d8_i8(x)"
GPFUN_z_y = "z_y(x, tk) =    y0 * tk * bez_d8_i0(x) + y1 * tk * bez_d8_i1(x) + y2 * tk * bez_d8_i2(x) +    y3 * tk * bez_d8_i3(x) + y4 * tk * bez_d8_i4(x) + y5 * tk * bez_d8_i5(x) +    y6 * tk * bez_d8_i6(x) + y7 * tk * bez_d8_i7(x) + y8 * tk * bez_d8_i8(x)"
GPFUN_airfoil_y1 = "airfoil_y1(t, thick) = mean_y(z_x(t)) + z_y(t, thick)"
GPFUN_airfoil_y2 = "airfoil_y2(t, thick) = mean_y(z_x(t)) - z_y(t, thick)"
GPFUN_airfoil_y = "airfoil_y(t) = mean_y(z_x(t))"
GPFUN_airfoil_x = "airfoil_x(t) = mean_x(z_x(t))"
eps = 0.06
a = 0.25
GPFUN_zeta = "zeta(t) = -eps + (a+eps)*exp(t*{0,1})"
GPFUN_eta = "eta(t) = zeta(t) + a*a/zeta(t)"
plot real(eta(t)),imag(eta(t))