3d implicit function plot with the sign function

Dear all,

I need help with plotting the set of probability functions which have constant entropy:

where c is some positive real number.

I've tried to define a partial function equal to NaN whenever and then use fimplicit3 to plot this -- but failed.

Instead, I've attempted to use the sign function instead: .

So, .

When plotting this function, I obtained plots that just do not look right to me.

After failing for a long time, I finally come to believe that there's a problem with plotting the sign function. Here's my code:

syms x y z
f = @(x,y,z) (-0.1+(sign(x+y+z-1).*sign(x+y+z-1)));
interval = [0 1 0 1 0 1];
fimplicit3(f,interval)
xlabel('P(\omega_1)')
ylabel('P(\omega_2)')
zlabel('P(\omega_3)')
hold off
view([132 48])

This is an implicity plot of all such that . There shouldn't be any such numbers.

However, Matlab plots some such values.

Could someone please help. All help much appreciated.

Best,

Juergen

Answer 1

[Torsten]

MATLAB assumes that the function f you provide is continuous. For x+y+z =1, the function returns -0.1, for all other triples for x,y and z, it returns 0.9. Thus MATLAB assumes that "very near" to x+y+z=1, there is the surface with f(x,y,z) = 0.

[Juergen]

Thanks Torsten.

If MATLAB assumes that the sign function is continuous, then I'm out of luck.

I introduced the sign function precisely for its discontinuous character.

Cheers,

Juergen

[Torsten]

MATLAB does not assume that the sign function is continuous, but that you prescribe a continuous function f if you apply "fimplicit3" to it.

[Juergen]

Thanks Torsten -- again. It doesn't matter to me!, whether MATLAB assumes that the input of fimplicit3 is continuous or that the sign function is continuous.

Either way, I've found a workaround. It's not pretty; but it works -- and that counts! I'm now using fimplicit to solve a two-dimensional projection of the actual problem.

I then store the solutions to the implicit problem as a 2D-array.

I can then calculate the third probability as: 1 minus the sum of the other two probabilities. I then store these third values and then plot the 3D-array. As I said, it ain't pretty -- but it works.

For whatever reason, MATLAB displayed the solution to the final 2D-implicit problem in the figure. It took me a while to google: `set(jp, 'Visible', 'off');' -- this command vanquishes the 2D-plot.

Cheers,

Juergen

clear all
clc
interval = [0 1 0 1];

r = @(x,y) -x.*log(x)-y.*log(y)-(1-x-y).*log(1-x-y)-1.08;
rp=fimplicit(r,interval);
hp = findobj(gca,'Type', 'line');
xd = get(rp, 'XData');
yd = get(rp, 'YData');

for i = 1:length(xd)
    rData(i,1)=xd(i);
    rData(i,2)=yd(i);
    rData(i,3)=1-xd(i)-yd(i);
end

l = @(x,y) -x.*log(x)-y.*log(y)-(1-x-y).*log(1-x-y)-1.05;
lp=fimplicit(l,interval);
hp = findobj(gca,'Type', 'line');
xd = get(lp, 'XData');
yd = get(lp, 'YData');

for i = 1:length(xd)
    lData(i,1)=xd(i);
    lData(i,2)=yd(i);
    lData(i,3)=1-xd(i)-yd(i);
end

f = @(x,y) -x.*log(x)-y.*log(y)-(1-x-y).*log(1-x-y)-1;
fp=fimplicit(f,interval);
hp = findobj(gca,'Type', 'line');
xd = get(fp, 'XData');
yd = get(fp, 'YData');

for i = 1:length(xd)
    fData(i,1)=xd(i);
    fData(i,2)=yd(i);
    fData(i,3)=1-xd(i)-yd(i);
end

g = @(x,y) -x.*log(x)-y.*log(y)-(1-x-y).*log(1-x-y)-0.8;
gp=fimplicit(g,interval);
hp = findobj(gca,'Type', 'line');
xd = get(gp, 'XData');
yd = get(gp, 'YData');

for i = 1:length(xd)
    gData(i,1)=xd(i);
    gData(i,2)=yd(i);
    gData(i,3)=1-xd(i)-yd(i);
end

hh = @(x,y) -x.*log(x)-y.*log(y)-(1-x-y).*log(1-x-y)-0.6;
hhp=fimplicit(hh,interval);
hp = findobj(gca,'Type', 'line');
xd = get(hhp, 'XData');
yd = get(hhp, 'YData');

for i = 1:length(xd)
    hData(i,1)=xd(i);
    hData(i,2)=yd(i);
    hData(i,3)=1-xd(i)-yd(i);
end

j = @(x,y) -x.*log(x)-y.*log(y)-(1-x-y).*log(1-x-y)-0.4;
jp=fimplicit(j,interval); set(jp, 'Visible', 'off');
hp = findobj(gca,'Type', 'line');
xd = get(jp, 'XData');
yd = get(jp, 'YData');

for i = 1:length(xd)
    jData(i,1)=xd(i);
    jData(i,2)=yd(i);
    jData(i,3)=1-xd(i)-yd(i);
end

hold on
plot3(rData(:,1),rData(:,2),rData(:,3))
plot3(lData(:,1),lData(:,2),lData(:,3))
plot3(fData(:,1),fData(:,2),fData(:,3))
plot3(gData(:,1),gData(:,2),gData(:,3))
plot3(hData(:,1),hData(:,2),hData(:,3))
plot3(jData(:,1),jData(:,2),jData(:,3))
view([132 48])
xlabel('P(\omega_1)') 
ylabel('P(\omega_2)')
zlabel('P(\omega_3)') 
hold off