# Plotting two functions that have 3 parameters (against each other)

I have two functions as follows:

-((5 + 2 k - 3 l - 4 x - 2 k x + 2 l x)/(3 (-2 - k + l))) -
(2^(1/3) (-(5 + 2 k - 3 l - 4 x - 2 k x + 2 l x)^2 + 3 (-2 - k + l) (-8 - 4 k + 6 x + 4 k x - 2 l x - 2 x^2 - k x^2 + l x^2)))/
(3 (-2 - k + l) (146 + 60 k - 57 k^2 - 25 k^3 - 18 l + 90 k l + 54 k^2 l - 135 l^2 - 81 k l^2 + 54 l^3 - 84 x - 282 k x - 204 k^2 x - 42 k^3 x - 102 l x +
84 k l x + 60 k^2 l x + 180 l^2 x + 36 k l^2 x - 54 l^3 x + 24 x^2 + 84 k x^2 + 66 k^2 x^2 + 15 k^3 x^2 + 12 l x^2 - 36 k l x^2 - 21 k^2 l x^2 - 30 l^2 x^2 -
3 k l^2 x^2 + 9 l^3 x^2 - 16 x^3 - 24 k x^3 - 12 k^2 x^3 - 2 k^3 x^3 + 24 l x^3 + 24 k l x^3 + 6 k^2 l x^3 - 12 l^2 x^3 - 6 k l^2 x^3 + 2 l^3 x^3 +
Sqrt[(146 + 60 k - 57 k^2 - 25 k^3 - 18 l + 90 k l + 54 k^2 l - 135 l^2 - 81 k l^2 + 54 l^3 - 84 x - 282 k x - 204 k^2 x - 42 k^3 x - 102 l x + 84 k l x +
60 k^2 l x + 180 l^2 x + 36 k l^2 x - 54 l^3 x + 24 x^2 + 84 k x^2 + 66 k^2 x^2 + 15 k^3 x^2 + 12 l x^2 - 36 k l x^2 - 21 k^2 l x^2 - 30 l^2 x^2 -
3 k l^2 x^2 + 9 l^3 x^2 - 16 x^3 - 24 k x^3 - 12 k^2 x^3 - 2 k^3 x^3 + 24 l x^3 + 24 k l x^3 + 6 k^2 l x^3 - 12 l^2 x^3 - 6 k l^2 x^3 + 2 l^3 x^3)^2 +
4 (-(5 + 2 k - 3 l - 4 x - 2 k x + 2 l x)^2 + 3 (-2 - k + l) (-8 - 4 k + 6 x + 4 k x - 2 l x - 2 x^2 - k x^2 + l x^2))^3])^(1/3)) +
(1/(3 2^(1/3) (-2 - k + l))) (146 + 60 k - 57 k^2 - 25 k^3 - 18 l + 90 k l + 54 k^2 l - 135 l^2 - 81 k l^2 + 54 l^3 - 84 x - 282 k x - 204 k^2 x - 42 k^3 x -
102 l x + 84 k l x + 60 k^2 l x + 180 l^2 x + 36 k l^2 x - 54 l^3 x + 24 x^2 + 84 k x^2 + 66 k^2 x^2 + 15 k^3 x^2 + 12 l x^2 - 36 k l x^2 - 21 k^2 l x^2 -
30 l^2 x^2 - 3 k l^2 x^2 + 9 l^3 x^2 - 16 x^3 - 24 k x^3 - 12 k^2 x^3 - 2 k^3 x^3 + 24 l x^3 + 24 k l x^3 + 6 k^2 l x^3 - 12 l^2 x^3 - 6 k l^2 x^3 +
2 l^3 x^3 + Sqrt[(146 + 60 k - 57 k^2 - 25 k^3 - 18 l + 90 k l + 54 k^2 l - 135 l^2 - 81 k l^2 + 54 l^3 - 84 x - 282 k x - 204 k^2 x - 42 k^3 x - 102 l x +
84 k l x + 60 k^2 l x + 180 l^2 x + 36 k l^2 x - 54 l^3 x + 24 x^2 + 84 k x^2 + 66 k^2 x^2 + 15 k^3 x^2 + 12 l x^2 - 36 k l x^2 - 21 k^2 l x^2 -
30 l^2 x^2 - 3 k l^2 x^2 + 9 l^3 x^2 - 16 x^3 - 24 k x^3 - 12 k^2 x^3 - 2 k^3 x^3 + 24 l x^3 + 24 k l x^3 + 6 k^2 l x^3 - 12 l^2 x^3 - 6 k l^2 x^3 +
2 l^3 x^3)^2 + 4 (-(5 + 2 k - 3 l - 4 x - 2 k x + 2 l x)^2 + 3 (-2 - k + l) (-8 - 4 k + 6 x + 4 k x - 2 l x - 2 x^2 - k x^2 + l x^2))^3])^(1/3)

-((5 + 2*k - 3*l - 4*z - 2*k*z + 2*l*z)/(3*(-2 - k + l))) -
(2^(1/3)*(-(5 + 2*k - 3*l - 4*z - 2*k*z + 2*l*z)^2 + 3*(-2 - k + l)*(-8 - 4*k + 6*z + 4*k*z - 2*l*z - 2*z^2 - k*z^2 + l*z^2)))/
(3*(-2 - k + l)*(146 + 60*k - 57*k^2 - 25*k^3 - 18*l + 90*k*l + 54*k^2*l - 135*l^2 - 81*k*l^2 + 54*l^3 - 84*z - 282*k*z - 204*k^2*z - 42*k^3*z - 102*l*z +
84*k*l*z + 60*k^2*l*z + 180*l^2*z + 36*k*l^2*z - 54*l^3*z + 24*z^2 + 84*k*z^2 + 66*k^2*z^2 + 15*k^3*z^2 + 12*l*z^2 - 36*k*l*z^2 - 21*k^2*l*z^2 - 30*l^2*z^2 -
3*k*l^2*z^2 + 9*l^3*z^2 - 16*z^3 - 24*k*z^3 - 12*k^2*z^3 - 2*k^3*z^3 + 24*l*z^3 + 24*k*l*z^3 + 6*k^2*l*z^3 - 12*l^2*z^3 - 6*k*l^2*z^3 + 2*l^3*z^3 +
Sqrt[(146 + 60*k - 57*k^2 - 25*k^3 - 18*l + 90*k*l + 54*k^2*l - 135*l^2 - 81*k*l^2 + 54*l^3 - 84*z - 282*k*z - 204*k^2*z - 42*k^3*z - 102*l*z + 84*k*l*z +
60*k^2*l*z + 180*l^2*z + 36*k*l^2*z - 54*l^3*z + 24*z^2 + 84*k*z^2 + 66*k^2*z^2 + 15*k^3*z^2 + 12*l*z^2 - 36*k*l*z^2 - 21*k^2*l*z^2 - 30*l^2*z^2 -
3*k*l^2*z^2 + 9*l^3*z^2 - 16*z^3 - 24*k*z^3 - 12*k^2*z^3 - 2*k^3*z^3 + 24*l*z^3 + 24*k*l*z^3 + 6*k^2*l*z^3 - 12*l^2*z^3 - 6*k*l^2*z^3 + 2*l^3*z^3)^2 +
4*(-(5 + 2*k - 3*l - 4*z - 2*k*z + 2*l*z)^2 + 3*(-2 - k + l)*(-8 - 4*k + 6*z + 4*k*z - 2*l*z - 2*z^2 - k*z^2 + l*z^2))^3])^(1/3)) +
(1/(3*2^(1/3)*(-2 - k + l)))*(146 + 60*k - 57*k^2 - 25*k^3 - 18*l + 90*k*l + 54*k^2*l - 135*l^2 - 81*k*l^2 + 54*l^3 - 84*z - 282*k*z - 204*k^2*z - 42*k^3*z -
102*l*z + 84*k*l*z + 60*k^2*l*z + 180*l^2*z + 36*k*l^2*z - 54*l^3*z + 24*z^2 + 84*k*z^2 + 66*k^2*z^2 + 15*k^3*z^2 + 12*l*z^2 - 36*k*l*z^2 - 21*k^2*l*z^2 -
30*l^2*z^2 - 3*k*l^2*z^2 + 9*l^3*z^2 - 16*z^3 - 24*k*z^3 - 12*k^2*z^3 - 2*k^3*z^3 + 24*l*z^3 + 24*k*l*z^3 + 6*k^2*l*z^3 - 12*l^2*z^3 - 6*k*l^2*z^3 +
2*l^3*z^3 + Sqrt[(146 + 60*k - 57*k^2 - 25*k^3 - 18*l + 90*k*l + 54*k^2*l - 135*l^2 - 81*k*l^2 + 54*l^3 - 84*z - 282*k*z - 204*k^2*z - 42*k^3*z - 102*l*z +
84*k*l*z + 60*k^2*l*z + 180*l^2*z + 36*k*l^2*z - 54*l^3*z + 24*z^2 + 84*k*z^2 + 66*k^2*z^2 + 15*k^3*z^2 + 12*l*z^2 - 36*k*l*z^2 - 21*k^2*l*z^2 -
30*l^2*z^2 - 3*k*l^2*z^2 + 9*l^3*z^2 - 16*z^3 - 24*k*z^3 - 12*k^2*z^3 - 2*k^3*z^3 + 24*l*z^3 + 24*k*l*z^3 + 6*k^2*l*z^3 - 12*l^2*z^3 - 6*k*l^2*z^3 +
2*l^3*z^3)^2 + 4*(-(5 + 2*k - 3*l - 4*z - 2*k*z + 2*l*z)^2 + 3*(-2 - k + l)*(-8 - 4*k + 6*z + 4*k*z - 2*l*z - 2*z^2 - k*z^2 + l*z^2))^3])^(1/3)

As you see the first one depens on x, k and l and the second one depends on z, k and l. The parameters take values between 0 to 1. I have to plot these two functions against each other as those parameters changes from 0 to 1. I think they should look like symmetric to each other.

Thanks,

• Please come up with a minimalistic example of your problem. E.g. how to plot Sin[x] vs Cos[x] Jan 4, 2017 at 20:30
• "As you see the first one depends on z, k [...]" - you made laugh out loud here! I'll gladly give it a shot as soon as you provide us with a smaller example Jan 4, 2017 at 20:42
• So your have two functions $f(x,k,l)$ and $g(y,k,l)$ and, if I understand correctly, they are defined on the unit cube $[0,1]^3$. What exactly do you want to plot? What do you expect for a simple case, for ex. $f=x+k+l$ and $g=y+k-l$ ?
– A.G.
Jan 5, 2017 at 3:36
• @A.G. Yes I have two functions f(x,k,l) and g(z,k,l). You can think like f=x+k+l and g=z+k+l. I want to plot f vs g with respect to x element of [0,1], k element of [0,1] and l element of [0,1]
– ecco
Jan 5, 2017 at 11:07
• I don't see how you can visualise that many variables in one plot. Jan 5, 2017 at 11:13

With four variables you are dealing with a 4-dimensional region. One way to tackle this problem is to slice the object by fixing two of the variables. If you do this for $x$ and $z$ you may then explore the region by combining Manipulate and RegionPlot:

Manipulate[
ParametricPlot[
{f[x, k, l], g[z, k, l]}, {k, 0, 1}, {l, 0, 1},
PlotRange -> {{0, .8}, {0, .8}},
AxesOrigin -> {0, 0}, AxesLabel -> {"f", "g"},
Frame -> None,
ColorFunction -> "Rainbow"
],
{x, 0, 1}, {z, 0, 1}]

with output:

Of course you can choose to use a different configuration on Manipulate/ParametricPlot variables. Here is another example the exhibits some symetry:

Manipulate[
ParametricPlot[
{f[x, k, l], g[z, k, l]}, {x, 0, 1}, {z, 0, 1},
PlotRange -> {{0, .8}, {0, .8}},
AxesOrigin -> {0, 0}, AxesLabel -> {"f", "g"},
Frame -> None,
ColorFunction -> "Rainbow", AspectRatio -> 1
],
{k, 0, 1}, {l, 0, 1}]

Hope this helps!

For reference here is the rest of the code:

f[x_,k_,l_]:=-((5+2 k-3 l-4 x-2 k x+2 l x)/(3 (-2-k+l)))-(2^(1/3) (-(5+2 k-3 l-4 x-2 k x+2 l x)^2+3 (-2-k+l) (-8-4 k+6 x+4 k x-2 l x-2 x^2-k x^2+l x^2)))/(3 (-2-k+l) (146+60 k-57 k^2-25 k^3-18 l+90 k l+54 k^2 l-135 l^2-81 k l^2+54 l^3-84 x-282 k x-204 k^2 x-42 k^3 x-102 l x+84 k l x+60 k^2 l x+180 l^2 x+36 k l^2 x-54 l^3 x+24 x^2+84 k x^2+66 k^2 x^2+15 k^3 x^2+12 l x^2-36 k l x^2-21 k^2 l x^2-30 l^2 x^2-3 k l^2 x^2+9 l^3 x^2-16 x^3-24 k x^3-12 k^2 x^3-2 k^3 x^3+24 l x^3+24 k l x^3+6 k^2 l x^3-12 l^2 x^3-6 k l^2 x^3+2 l^3 x^3+Sqrt[(146+60 k-57 k^2-25 k^3-18 l+90 k l+54 k^2 l-135 l^2-81 k l^2+54 l^3-84 x-282 k x-204 k^2 x-42 k^3 x-102 l x+84 k l x+60 k^2 l x+180 l^2 x+36 k l^2 x-54 l^3 x+24 x^2+84 k x^2+66 k^2 x^2+15 k^3 x^2+12 l x^2-36 k l x^2-21 k^2 l x^2-30 l^2 x^2-3 k l^2 x^2+9 l^3 x^2-16 x^3-24 k x^3-12 k^2 x^3-2 k^3 x^3+24 l x^3+24 k l x^3+6 k^2 l x^3-12 l^2 x^3-6 k l^2 x^3+2 l^3 x^3)^2+4 (-(5+2 k-3 l-4 x-2 k x+2 l x)^2+3 (-2-k+l) (-8-4 k+6 x+4 k x-2 l x-2 x^2-k x^2+l x^2))^3])^(1/3))+(1/(3 2^(1/3) (-2-k+l))) (146+60 k-57 k^2-25 k^3-18 l+90 k l+54 k^2 l-135 l^2-81 k l^2+54 l^3-84 x-282 k x-204 k^2 x-42 k^3 x-102 l x+84 k l x+60 k^2 l x+180 l^2 x+36 k l^2 x-54 l^3 x+24 x^2+84 k x^2+66 k^2 x^2+15 k^3 x^2+12 l x^2-36 k l x^2-21 k^2 l x^2-30 l^2 x^2-3 k l^2 x^2+9 l^3 x^2-16 x^3-24 k x^3-12 k^2 x^3-2 k^3 x^3+24 l x^3+24 k l x^3+6 k^2 l x^3-12 l^2 x^3-6 k l^2 x^3+2 l^3 x^3+Sqrt[(146+60 k-57 k^2-25 k^3-18 l+90 k l+54 k^2 l-135 l^2-81 k l^2+54 l^3-84 x-282 k x-204 k^2 x-42 k^3 x-102 l x+84 k l x+60 k^2 l x+180 l^2 x+36 k l^2 x-54 l^3 x+24 x^2+84 k x^2+66 k^2 x^2+15 k^3 x^2+12 l x^2-36 k l x^2-21 k^2 l x^2-30 l^2 x^2-3 k l^2 x^2+9 l^3 x^2-16 x^3-24 k x^3-12 k^2 x^3-2 k^3 x^3+24 l x^3+24 k l x^3+6 k^2 l x^3-12 l^2 x^3-6 k l^2 x^3+2 l^3 x^3)^2+4 (-(5+2 k-3 l-4 x-2 k x+2 l x)^2+3 (-2-k+l) (-8-4 k+6 x+4 k x-2 l x-2 x^2-k x^2+l x^2))^3])^(1/3);

g[z_,k_,l_]:=-((5+2*k-3*l-4*z-2*k*z+2*l*z)/(3*(-2-k+l)))-(2^(1/3)*(-(5+2*k-3*l-4*z-2*k*z+2*l*z)^2+3*(-2-k+l)*(-8-4*k+6*z+4*k*z-2*l*z-2*z^2-k*z^2+l*z^2)))/(3*(-2-k+l)*(146+60*k-57*k^2-25*k^3-18*l+90*k*l+54*k^2*l-135*l^2-81*k*l^2+54*l^3-84*z-282*k*z-204*k^2*z-42*k^3*z-102*l*z+84*k*l*z+60*k^2*l*z+180*l^2*z+36*k*l^2*z-54*l^3*z+24*z^2+84*k*z^2+66*k^2*z^2+15*k^3*z^2+12*l*z^2-36*k*l*z^2-21*k^2*l*z^2-30*l^2*z^2-3*k*l^2*z^2+9*l^3*z^2-16*z^3-24*k*z^3-12*k^2*z^3-2*k^3*z^3+24*l*z^3+24*k*l*z^3+6*k^2*l*z^3-12*l^2*z^3-6*k*l^2*z^3+2*l^3*z^3+Sqrt[(146+60*k-57*k^2-25*k^3-18*l+90*k*l+54*k^2*l-135*l^2-81*k*l^2+54*l^3-84*z-282*k*z-204*k^2*z-42*k^3*z-102*l*z+84*k*l*z+60*k^2*l*z+180*l^2*z+36*k*l^2*z-54*l^3*z+24*z^2+84*k*z^2+66*k^2*z^2+15*k^3*z^2+12*l*z^2-36*k*l*z^2-21*k^2*l*z^2-30*l^2*z^2-3*k*l^2*z^2+9*l^3*z^2-16*z^3-24*k*z^3-12*k^2*z^3-2*k^3*z^3+24*l*z^3+24*k*l*z^3+6*k^2*l*z^3-12*l^2*z^3-6*k*l^2*z^3+2*l^3*z^3)^2+4*(-(5+2*k-3*l-4*z-2*k*z+2*l*z)^2+3*(-2-k+l)*(-8-4*k+6*z+4*k*z-2*l*z-2*z^2-k*z^2+l*z^2))^3])^(1/3))+(1/(3*2^(1/3)*(-2-k+l)))*(146+60*k-57*k^2-25*k^3-18*l+90*k*l+54*k^2*l-135*l^2-81*k*l^2+54*l^3-84*z-282*k*z-204*k^2*z-42*k^3*z-102*l*z+84*k*l*z+60*k^2*l*z+180*l^2*z+36*k*l^2*z-54*l^3*z+24*z^2+84*k*z^2+66*k^2*z^2+15*k^3*z^2+12*l*z^2-36*k*l*z^2-21*k^2*l*z^2-30*l^2*z^2-3*k*l^2*z^2+9*l^3*z^2-16*z^3-24*k*z^3-12*k^2*z^3-2*k^3*z^3+24*l*z^3+24*k*l*z^3+6*k^2*l*z^3-12*l^2*z^3-6*k*l^2*z^3+2*l^3*z^3+Sqrt[(146+60*k-57*k^2-25*k^3-18*l+90*k*l+54*k^2*l-135*l^2-81*k*l^2+54*l^3-84*z-282*k*z-204*k^2*z-42*k^3*z-102*l*z+84*k*l*z+60*k^2*l*z+180*l^2*z+36*k*l^2*z-54*l^3*z+24*z^2+84*k*z^2+66*k^2*z^2+15*k^3*z^2+12*l*z^2-36*k*l*z^2-21*k^2*l*z^2-30*l^2*z^2-3*k*l^2*z^2+9*l^3*z^2-16*z^3-24*k*z^3-12*k^2*z^3-2*k^3*z^3+24*l*z^3+24*k*l*z^3+6*k^2*l*z^3-12*l^2*z^3-6*k*l^2*z^3+2*l^3*z^3)^2+4*(-(5+2*k-3*l-4*z-2*k*z+2*l*z)^2+3*(-2-k+l)*(-8-4*k+6*z+4*k*z-2*l*z-2*z^2-k*z^2+l*z^2))^3])^(1/3);