Consider the polar equation

*r³ - 3r + a = sin³θ*(where a is a fixed number,

*1.7 ≦ a ≦ 1.8*).

First, we plot the polar equation by plotting the equivalent rectangular equation(s).

The graph (See

**Image 1**) consists of two disjoint closed curves, and the inner curve

looks like a little girl's hair!

Second, since the equation is a cubic equation in the variable r, we can solve it

for r in terms of θ, and plot the three solution functions

*r = r*,

_{1}(θ)*r = r*and

_{2}(θ)*r = r*(See

_{3}(θ)**Image 2**). Note that the union of their graphs is the same as the

graph of the polar equation.

Finally, replace

*r*with its real part

_{3}(θ)*Re(r*and plot again (See

_{3}(θ))**Image 3**).

We see that the union of the graphs of

*r = r*and

_{1}(θ)*r = Re(r*looks like

_{3}(θ))a little girl's head!

**Image 1**-- Graph of

the polar equation

*r³ - 3r + a = sin³θ***Image 2**-- Graphs of

*r = r*,

_{1}(θ)*r = r*,

_{2}(θ)*r = r*(

_{3}(θ)*r∈R*,

*0≦θ≦2π*)

**Image 3**-- Graphs of

*r = r*,

_{1}(θ)*r = r*,

_{2}(θ)*r = Re(r*(

_{3}(θ))*r∈R*,

*0≦θ≦2π*)

###
__gnuplot script for image 1__:

reseta = 1.8

fun1(x,y) = (x**2 + y**2 - 3)*(x**2 + y**2)**2 \

+ a*(x**2 + y**2)*sqrt(x**2 + y**2) - y**3

fun2(x,y) = (x**2 + y**2 - 3)*(x**2 + y**2)**2 \

- a*(x**2 + y**2)*sqrt(x**2 + y**2) - y**3

set xrange [-2.25:2.25]

set yrange [-2.25:2.25]

set size ratio -1

set samples 256

set isosamples 256

set contour base

set cntrparam levels discrete 0.0

unset key

unset surface

set table 'fun1.dat'

splot fun1(x,y)

unset table

set table 'fun2.dat'

splot fun2(x,y)

unset table

plot 'fun1.dat' w l lw 2 lc rgb "red", \

'fun2.dat' w l lw 2 lc rgb "green"

###
__gnuplot script for image 2__:

reseta = 1.8

h(t) = (sin(t)**3 - a + sqrt((sin(t)**3 - a)**2 - 4))**(1.0/3)

s = 2**(1.0/3)

p = (-0.5)*(1 + sqrt(3)*{0,1})

q = (-0.5)*(1 - sqrt(3)*{0,1})

r1(t) = s/h(t) + h(t)/s

r2(t) = p*s/h(t) + q*h(t)/s

r3(t) = q*s/h(t) + p*h(t)/s

set xrange [-2.25:2.25]

set yrange [-2.25:2.25]

set size ratio -1

set samples 4000

set polar

unset key

unset raxis

unset rtics

plot r1(t) w l lw 2 lc rgb "#0099ff", \

r2(t) w l lw 2 lc rgb "green", \

r3(t) w l lw 2 lc rgb "#ff00ff"

###
__gnuplot script for image 3__:

The script is the same as that of (2), except the last line:real(r3(t)) w l lw 2 lc rgb "#ff00ff"

###
__Graph example__: **Head of a Little Girl**

( Mathematical softwares used: Graph, gnuplot )

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