# Working Precision warnings for ParametricPlot

I wish to plot a series of parametric curves for the following function called poly that has five parameters (Γ, κ1, κ2, g1, g2). My parameterization is such that x = g1 and y = g2 while the other three parameters eventually taking on constant values. I first have my equation:

27 (256 g1^6 + (4 g2^2 + (Γ - κ1) (Γ - κ2))^2 (4 g2 + κ1 - κ2) (4 g2 - κ1 + κ2) + 16 g1^4 (48 g2^2 - Γ^2 + 10 Γ κ1 - κ1^2 - 8 (Γ + κ1) κ2 + 8 κ2^2) + 8 g1^2 (96 g2^4 + 4 g2^2 (-10 Γ^2 + Γ κ1 - κ1^2 + 19 Γ κ2 + κ1 κ2 - 10 κ2^2) + (Γ - κ2) (-κ1 + κ2) (Γ^2 - 4 Γ κ1 + κ1^2 + 2 (Γ + κ1) κ2 - 2 κ2^2))) = 0


Which I then call all of the left hand side as poly:

poly = 27 (256 g1^6 + (4 g2^2 + (Γ - κ1) (Γ - κ2))^2 (4 g2 + κ1 - κ2) (4 g2 - κ1 + κ2) + 16 g1^4 (48 g2^2 - Γ^2 + 10 Γ κ1 - κ1^2 - 8 (Γ + κ1) κ2 + 8 κ2^2) + 8 g1^2 (96 g2^4 + 4 g2^2 (-10 Γ^2 + Γ κ1 - κ1^2 + 19 Γ κ2 + κ1 κ2 - 10 κ2^2) + (Γ - κ2) (-κ1 + κ2) (Γ^2 - 4 Γ κ1 + κ1^2 + 2 (Γ + κ1) κ2 - 2 κ2^2)))


Since I need my g2 in terms of g1 for my y-axis, I solve for g2 in terms of all the other parameters (note that g2 is a polynomial of order 6 so I should expect 6 solutions)

{g21p, g21n, g22p, g22n, g23p, g23n} = g2 /. (Solve[poly == 0, g2] /. {Γ -> 1/100, κ1 -> 2.0, κ2 ->3.25});


Where I have named each of the 6 solutions as g21p, g21n, g22p, g22n, g23p, g23n respectively and the parameters Γ, κ1, κ2 all take on 0.01, 2, and 3.25 respectively. I now proceed to produce my parametric plot:

ParametricPlot[{{g1, g21p}, {g1, g21n}, {g1, g22p}, {g1, g22n}, {g1, g23p}, {g1, g23n}}, {g1, 0, 10}, PlotRange -> All, PlotLegends -> "Expressions", PlotPoints -> 100, Exclusions -> None]


And I am returned with:

Notice that there's a gap in between the curves. This could be slightly improved by setting WorkingPrecision -> 10 (there is still a tiny gap in between)

However, I am returned with a bunch of warnings along the lines of The precision argument function (one of the six g2 solutions) is less than WorkingPrecision (10.) What is going on here and how can I remedy the warnings while fixing the gap?

• One thing that helps is to use exact parameters in for x1 and x2. (It fixes the problem with WorkingPrecision.) Note that MachinePrecision is considered "less" than any arbitrary precision setting (see for instance this commnet). – Michael E2 Sep 27 '18 at 20:47
• The gaps come from not hitting the exact points where the solutions become complex. – Michael E2 Sep 27 '18 at 20:51

Using exact parameters and the correct endpoint

{g21p, g21n, g22p, g22n, g23p, g23n} =
g2 /. (Solve[poly == 0, g2] /. {Γ -> 1/100, κ1 -> 2, κ2 -> 325/100});

ParametricPlot[
{{g1, g21p}, {g1, g21n}, {g1, g22p}, {g1, g22n}, {g1, g23p}, {g1, g23n}},
{g1, 0, (523 Sqrt[523/3])/10800},
PlotRange -> All, PlotLegends -> "Expressions", PlotPoints -> 100,
Exclusions -> None, WorkingPrecision -> 16]


• Thanks! This is very much what I needed. How did you know that the endpoint is at (523 Sqrt[523/3])/10800? That is an oddly specific number of around 0.6. One would think that increasing the range up to a large amount (say {g1, 0, 10}) would include 0.6`. How did you figure out the endpoint? – kowalski Sep 27 '18 at 21:50
• @kowalski Solved for where the expression under the radical was zero. Would've said something but I got interrupted while trying to figure why @halirutan's buttons got disabled by an upgrade to Safari. – Michael E2 Sep 27 '18 at 23:43