Gradient colored polar curve

Question

I'm trying to do the following in gnuplot: color with gradient an ellipse. Each point of the ellipse has its z-value given by the function

charge_density(t,beta) = -sin(t)*beta*(sqrt(1-(beta**2)))/(1-((sin(t)*beta)**2))

The radius function of the ellipse is given by a similar function:

radius(t,beta) = sqrt(1-(beta**2))/sqrt(1-((sin(t)*beta)**2))

Where beta is just a parameter satisfying 0<beta<1, and t is the angle.

Well, I tried to use the "+" special file with the lc rgb variable option, but it doesn't work with polar coordinates.

I also tried the set mapping cylindrical, but nothing happened.

Is this possible only with cartesian coordinates? In this case, I'll have to do 2 graphics and modify the above functions...

Or will I have to create a data file with angle, radius, z data?

I'd like to do this with pm3d and the following palette:

set palette model RGB defined (-1 "blue", 0 "black", 1 "red")

Answer 1

here is the code:

beta =0.5
charge_density(t,beta) = -sin(t)*beta*(sqrt(1-(sin(t)*beta)**2))/(1-((sin(t)*beta)**2))
radius(t,beta) = sin(t*beta) # your function equals 1!

# convert polar to carthesian
r_x(t)=radius(t,beta)*cos(t)
r_y(t)=radius(t,beta)*sin(t)


set palette model RGB defined (-1 "blue", 0 "black", 1 "red")

set size ratio -1 # same unit length in x and y

# number of sample points.
# increase if curve has edges
set samples 100 

#decouple range of "+" and xrange
set parametric

plot  [0:3*pi] "+" u (r_x($1)):(r_y($1)):(charge_density($1,beta))\
    with lines linewidth 3 linecolor palette

and there the result:

NOTE: Your radius equals to 1, so I took another function. Also, your charge_density has one extra closing parenthesis.

Some comments:

• if you plot with lc rgb variable, a 24bit RGB color value is expected:

  (red*256^2 + green*256 + blue) with red, green, blue = 0...255

  If you want gnuplot to use the color according to the palette, write lc palette

• gnuplot 4.6 does not support "+" for polar. Also, the mapping sets the behavior for 3D plots. However, as your formula calculates a radius for an angle, you can easily transform this to carthesian coordinates and plot these. There is still one drawback: The range given in the plot sets the xrange, too. This also means that the "length" of your curve changes when you change the xrange. You can solve this by set parametric which causes gnuplot to use a dedicated variable u instead of x when plotting functions. It is nice (and helps you) that this affects the special file "+", too. I do not know if this is a (positive) bug or a feature.