I have two functions f[x,y,z,a,b] and g[x,y,z,a,b], where the variables are restricted to a set of values: x $\in$ [x0, x1], y $\in$ [y0, y1], z $\in$ [z0, z1], a $\in$ [a0, a1], b $\in$ [b0, b1]. I can find the minimum and maximum values within the restrictions using Minimize and Maximize.

These two functions are the X and Y coordinates of a point, I am using them to show some paths using ParametricPlot and to draw graphics using Graphics. The thing is, I need to know the boundaries which define the possible locations of the point.

How could I find the contour of the area for which the (X,Y) plot of these two functions is defined?


I have tried to do it in two different ways.

1. Using ParametricRegion:

R = ParametricRegion[{Xt[X2, Th3, La1, La2, La3, La4, La5] /. {X2 -> X2in, La3 -> La3in} /. {Th3 -> Th3in, La4 -> La4in, La5 -> La5in}, Yt[X2, Th3, La1, La2, La3, La4, La5] /. {X2 -> X2in, La3 -> La3in} /. {Th3 -> Th3in, La4 -> La4in, La5 -> La5in}}, {{La1, dA1min, dA1max}, {La2, dA2min, dA2max}}];

And trying to represent it by:


This is not able to give a result (it is still running after about 10h and 8GB of RAM).

2. Using ParametricPlot

ParametricPlot[{{Xt[X2, Th3, La1, La2, La3, La4, La5], Yt[X2, Th3, La1, La2, La3, La4, La5]}} /. {X2 -> X2min, Th3 -> Th3min, La3 -> dA3min, La4 -> dA4min, La5 -> dA5min}, {La1, dA1min, dA1max}, {La2, dA2min, dA2max}, AspectRatio -> Automatic]

This is able to give a result relatively fast (40s and <2GB of RAM).

Parametric Plot

But unfortunately, ParametricPlot can't plot regions depending on more than 2 variables.

Just to give some insight into the problem, (Xt, Yt) give the position of a point from a body in a machine. I need to obtain the 'range of action' of this point. I need to find the contour of the area (or the area itself) in which the point can be located. The problem is that the point depends on 5 (6) variables.

  • $\begingroup$ I guess ParametricRegion is the way to go but hard to say without example. Maybe it will take more time than to generate point cloud for random point from the domain. $\endgroup$
    – Kuba
    Mar 22, 2018 at 10:49
  • $\begingroup$ I have been trying to use ParametricRegion, but it takes an awful lot of time to give the results with 2/5 variables. Any other suggestions? $\endgroup$ Mar 22, 2018 at 20:18
  • $\begingroup$ It is kind of strange that using ParametricPlot with 2 variables is much faster than ParametricRegion with the same 2 variables. $\endgroup$ Mar 22, 2018 at 23:28

1 Answer 1


The dark blue region is the place where different combinations of parameter values give the same (X,Y) combination. Maybe this superposition/overlap of points is the reason why ParametricRegion is not able to give an output.

I have more or less found the solution to this: plotting variables in pairs with the correct choice of maximum and minimum values and overlapping them.

The output is:

enter image description here

Once checked that this visual representation covers the whole region, it is now possible to identify the curves that define the edges and draw only them.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.