How can I plot the hyperbola represented by the following complex equation:
$$\left|z-a\right|-\left|z-b\right|=2t\tag{1}$$
Should I convert (1) into the following form?
ContourPlot[x^2 - y^2 == 1, {x, -2, 2}, {y, -2, 2}]
Since you're trying to plot an implicit equation, you would normally use ContourPlot
. For example, to get two branches satisfying the equation for $t = 1$, just do this:
With[{a = 2, b = 1},
ContourPlot[
Abs[x + I y - a] - Abs[x + I y - b], {x, -1, 3}, {y, -2, 2},
Contours -> {1/2, -1/2}, ContourShading -> False,
ContourStyle -> ColorData[1][1]
]
]
ComplexExpand[Abs[x + I y - a] - Abs[x + I y - b], TargetFunctions -> {Re, Im}]
, which doe not give an hyperbola though. $\endgroup$ParametricPlot[cc[u, 3/2], {u, 0, 2 Pi}, Axes -> False, PlotStyle -> Thick, ColorFunction -> Function[{x, y, u}, Hue[u/(2 Pi)]], ColorFunctionScaling -> False, Exclusions -> {Cos[u] == 2/3, Cos[u] == -2/3}]
$\endgroup$