error region in parametric plot

• To: mathgroup at smc.vnet.net
• Subject: [mg92659] error region in parametric plot
• From: "Cristina Ballantine" <cballant at holycross.edu>
• Date: Fri, 10 Oct 2008 04:31:34 -0400 (EDT)

```In plotting parametric regions sometimes I get cracks, i.e., white regions
that should be colored as part of the region. The code below should
produce a deformed annulus, but it has two cracks in it. Sometimes I am
able to force Mathematica to fill part of the cracks by subdividing the
region with smaller intervals for the angle v. However, this does not work
for the example below.

Any suggestions are much appreciated.

Cristina

a1 := 1/3*Exp[I*Pi/6]
a2 := 2/3*Exp[I*3*Pi/4]
n := 6

a1c := Conjugate[a1]
a2c := Conjugate[a2]
r1 := Abs[a1]
r2 := Abs[a2]
t1 := Arg[a1]
t2 := Arg[a2]

B[z_] := ((a1c/(r1))*(a1 - z)/(1 - a1c*z))^
n*((a2c/(r2))*(a2 - z)/(1 - a2c*z))^n

a := a1*a2 (a1c + a2c) - (a1 + a2)

b := (1 - (r1)^2*(r2)^2 - ((1 - (Abs[a1*a2])^2)^2 - (Abs[
a])^2)^(1/2))/(a1c*(1 - (r2)^2) + a2c*(1 - (r1)^2))
alpha = Arg[N[B[b]]]

u0[rho_, v_] := (rho)^(1/n)*Exp[I*((v + 2*0*Pi)/n )]
u1[rho_, v_] := (rho)^(1/n)*Exp[I*((v + 2*1*Pi)/n )]
u2[rho_, v_] := (rho)^(1/n)*Exp[I*((v + 2*2*Pi)/n )]
u3[rho_, v_] := (rho)^(1/n)*Exp[I*((v + 2*3*Pi)/n )]
u4[rho_, v_] := (rho)^(1/n)*Exp[I*((v + 2*4*Pi)/n )]
u5[rho_, v_] := (rho)^(1/n)*Exp[I*((v + 2*5*Pi)/n )]

sol0 :=
Solve[(1 - u0[rho, v]*r1*r2)*Exp[-I (t1 + t2)]*
z^2 + ((r1*u0[rho, v] - r2)*Exp[-I*t1] + (r2*u0[rho, v] - r1)*
Exp[-I*t2])*z + r1*r2 - u0[rho, v] == 0, z]
sol1 := Solve[(1 - u1[rho, v]*r1*r2)*Exp[-I (t1 + t2)]*
z^2 + ((r1*u1[rho, v] - r2)*Exp[-I*t1] + (r2*u1[rho, v] - r1)*
Exp[-I*t2])*z + r1*r2 - u1[rho, v] == 0, z]
sol2 := Solve[(1 - u2[rho, v]*r1*r2)*Exp[-I (t1 + t2)]*
z^2 + ((r1*u2[rho, v] - r2)*Exp[-I*t1] + (r2*u2[rho, v] - r1)*
Exp[-I*t2])*z + r1*r2 - u2[rho, v] == 0, z]
sol3 := Solve[(1 - u3[rho, v]*r1*r2)*Exp[-I (t1 + t2)]*
z^2 + ((r1*u3[rho, v] - r2)*Exp[-I*t1] + (r2*u3[rho, v] - r1)*
Exp[-I*t2])*z + r1*r2 - u3[rho, v] == 0, z]
sol4 := Solve[(1 - u4[rho, v]*r1*r2)*Exp[-I (t1 + t2)]*
z^2 + ((r1*u4[rho, v] - r2)*Exp[-I*t1] + (r2*u4[rho, v] - r1)*
Exp[-I*t2])*z + r1*r2 - u4[rho, v] == 0, z]
sol5 := Solve[(1 - u5[rho, v]*r1*r2)*Exp[-I (t1 + t2)]*
z^2 + ((r1*u5[rho, v] - r2)*Exp[-I*t1] + (r2*u5[rho, v] - r1)*
Exp[-I*t2])*z + r1*r2 - u5[rho, v] == 0, z]

plotpi0[tmin_, tmax_, c_] :=
ParametricPlot[
Evaluate[{Re[z], Im[z]} /. sol0], {v, 0, 2*Pi}, {rho, tmin, tmax},
ColorFunction -> Function[{x, y, v, rho}, Hue[c, v, rho]],
PlotPoints -> 25, PlotRange -> All]
plotpi1[tmin_, tmax_, c_] :=
ParametricPlot[
Evaluate[{Re[z], Im[z]} /. sol1], {v, 0, 2*Pi}, {rho, tmin, tmax},
ColorFunction -> Function[{x, y, v, rho}, Hue[c, v, rho]],
PlotPoints -> 25, PlotRange -> All]
plotpi2[tmin_, tmax_, c_] :=
ParametricPlot[
Evaluate[{Re[z], Im[z]} /. sol2], {v, 0, 2*Pi}, {rho, tmin, tmax},
ColorFunction -> Function[{x, y, v, rho}, Hue[c, v, rho]],
PlotPoints -> 25, PlotRange -> All]
plotpi3[tmin_, tmax_, c_] :=
ParametricPlot[
Evaluate[{Re[z], Im[z]} /. sol3], {v, 0, 2*Pi}, {rho, tmin, tmax},
ColorFunction -> Function[{x, y, v, rho}, Hue[c, v, rho]],
PlotPoints -> 25, PlotRange -> All]
plotpi4[tmin_, tmax_, c_] :=
ParametricPlot[
Evaluate[{Re[z], Im[z]} /. sol4], {v, 0, 2*Pi}, {rho, tmin, tmax},
ColorFunction -> Function[{x, y, v, rho}, Hue[c, v, rho]],
PlotPoints -> 25, PlotRange -> All]
plotpi5[tmin_, tmax_, c_] :=
ParametricPlot[
Evaluate[{Re[z], Im[z]} /. sol5], {v, 0, 2*Pi}, {rho, tmin, tmax},
ColorFunction -> Function[{x, y, v, rho}, Hue[c, v, rho]],
PlotPoints -> 25, PlotRange -> All]

With[{tmin = .0015, tmax = .01, c = .8},
Show[plotpi0[tmin, tmax, c], plotpi1[tmin, tmax, c],
plotpi2[tmin, tmax, c], plotpi3[tmin, tmax, c],
plotpi4[tmin, tmax, c], plotpi5[tmin, tmax, c], PlotRange -> All]]

```

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