Adding a curved line to a DensityPlot

Question

Suppose I am plotting some two-dimensional function, using DensityPlot, and that I would like to add the outer line of a circle to this figure. I found one way to do it:

pt1 = DensityPlot[x y, {x, -5, 5}, {y, -5, 5}];
pt2 = RegionPlot[x^2 + y^2 < 1, {x, -5, 5}, {y, -5, 5}, 
BoundaryStyle -> Black, PlotStyle -> White];
Show[pt1, pt2]

However, 1) I suspect that this can be done in an easier way; 2) The inner part of the circle is white now, whereas I would like to so the function value of the first figure in the center. Any ideas on how to do this?

EDIT: more generally, I would like to plot a function which can become zero, e.g. sin[x]sin[y] using DensityPlot and then plot a black line connecting all the points where this is zero. Any thoughts on how to do this?

Answer 1

In general for adding a graphics overlay to a plot your best options is Epilog:

DensityPlot[x y, {x, -5, 5}, {y, -5, 5}, Epilog -> Circle[{0, 0}, 1]]

For your more specific case you could plot the zero contour using ContourPlot and pass that to Epilog:

ZeroDensityPlot[f_, args__] := 
 DensityPlot[f, args, 
  Epilog -> First@ContourPlot[f == 0, args, ContourStyle -> Black]
 ]

ZeroDensityPlot[Sin[x] Sin[y], {x, -5, 5}, {y, -5, 5}]