This is more of an extended comment/question. Are you looking for an interactive 3D representation? Below is a very incomplete example:
Manipulate[
aa = Table[-10 + 20 i/100, {i, 0, 100}];
a = (1 + abar - bbar)/2;
b = (1 - abar + bbar)/2;
Show[Plot3D[(aa - abar) (bb - bbar), {aa, -10, 10}, {bb, -10, 10},
BoundaryStyle -> None, MeshStyle -> LightGray,
PlotStyle -> Opacity[0.2], SphericalRegion -> True,
RotationAction -> "Clip", AspectRatio -> Full,
PlotRange -> {{-10, 10}, {-10, 10}, All}],
Graphics3D[{Thickness[0.01],
Line[Table[{aa[[i]],
1 - aa[[i]], (aa[[i]] - abar) (1 - aa[[i]] - bbar)}, {i, 1,
101}]]}],
ListPointPlot3D[{{abar, bbar, -10}}, PlotStyle -> PointSize[0.03]],
ListPointPlot3D[{{a, b, (a - abar) (b - bbar)}},
PlotStyle -> {Red, PointSize[0.03]}]],
{{abar, 0, "\!\(\*OverscriptBox[\(a\), \(_\)]\)"}, -10, 10,
Appearance -> "Labeled"},
{{bbar, 0, "\!\(\*OverscriptBox[\(b\), \(_\)]\)"}, -10, 10,
Appearance -> "Labeled"},
TrackedSymbols :> {abar, bbar}]
