# I need to remove a line from a ParametricPlot

What happens is that when I make this ParametricPlot:

ClearAll [f, g]

f2 [t_]:= t + 1 / t
g2 [t_]:= t - 1 / t
ParametricPlot [{f2 [t], g2 [t]}, {t, 0.00001, 20},
PlotRange -> {{0, 5}, {-4, 1}}, ImageSize -> 200]
ParametricPlot [{f2 [t], g2 [t]}, {t, -80, 80},
PlotRange -> {{-4, 4}, {-2, 4}}, ImageSize -> 200]

The graph of the parametric function appears and additionally appears a straight line through the origin. How I can remove it?

ParametricPlot[{f2[t], g2[t]},
{t, -80, 80},
PlotRange -> {{-4, 4}, {-2, 4}},
Exclusions -> {f2[t] == 0},
ImageSize -> 400]

Because

FunctionDomain[f2[t], t]

Another possibility:

ParametricPlot[{f2[t], g2[t]},
{t, -80, 80},
PlotRange -> {{-4, 4}, {-2, 4}},
Exclusions -> {f2[t] == 0},
ExclusionsStyle -> Directive[Red, Dashed],
ImageSize -> 400]

Update

Based upon Karsten's comment and Michael's review a complete solution set could look like this:

pg = Point[Transpose[{Re@#, Im@#}]] &[t /. Solve[g2[t] == 0, t]]

Point[{{-1, 0}, {1, 0}}]

pf = Point[Transpose[{Re@#, Im@#}]] &[t /. Solve[f2[t] == 0, t]]

Point[{{0, -1}, {0, 1}}]

par =
ParametricPlot[{f2[t], g2[t]},
{t, -80, 80},
PlotRange -> {{-4, 4}, {-2, 4}},
Exclusions -> t == 0,
Epilog -> {PointSize[0.02], {Blue, pf}, {Red, pg}},
PlotStyle -> Green,
ImageSize -> 400];

leg =
SwatchLegend[{Green, Blue, Red}, {"{f2[t], g2[t]} != 0", "f2[t] == 0", "g2[t] == 0"}];

Legended[par, leg]

• Or Exclusions -> t == 0 Aug 30, 2014 at 13:22
• (+1) very nice answer. Aug 30, 2014 at 16:47
• The only solutions to f2[t] == 0 are ± I. Why would that exclude t == 0, which needs to be excluded according to FunctionDomain? Aug 31, 2014 at 13:54
• @MichaelE2 (a) FunctionDomain[f2[t], t, Complexes] gives t != 0. (b) Solve[f2[t] == 0, t] gives, as you wisely remarked, {{t -> -I}, {t -> I}}.
– eldo
Aug 31, 2014 at 14:10
• You seem to agree that the Exclusions condition in your answer does not correspond to the FunctionDomain condition. It seems to me that the Exclusions setting is wrong and the FunctionDomain is correct. Is it an accident that Exclusions -> {f2[t] == 0} works on the ParametricPlot, or is there some explanation that I'm not seeing? Aug 31, 2014 at 14:27

This is not really any advance on eldo's answer. I post it for illustration:

Manipulate[
ParametricPlot[{t + 1/t, t - 1/t}, {t, -10, 10}, Exclusions -> {0},
Epilog -> {Red, PointSize[0.02],
Point[{p + 1/p, p - 1/p}]}], {p, -10, 10}]

• Nice jump ! Amazing how the point accelerates on the long diagonals :)
– eldo
Aug 31, 2014 at 12:25
• @eldo must admit...jump major reason i posted :) Aug 31, 2014 at 12:29
• Exclusions -> {0} seems a quite clear way to address the OP's problem. Aug 31, 2014 at 13:45
r = ParametricRegion[{t + 1/t , t - 1/t }, {{t, -80, 80}}];
RegionPlot[r, PlotRange -> {{-4, 4}, {-2, 4}}, Frame -> False,
Axes -> True]

• As unusual as interesting :)
– eldo
Aug 30, 2014 at 17:48