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How to get the points from a contour plot object?

I've defined a plot object for the hyperboloid in R^(-1,1).

hyperboloid = region_plot(-x^2+y^2 == 1, (x,-3,3),(y,-3,3))

I want to reflect this hyperboloid across a plane, and to do that I can't think of a better way than to somehow extract the points from the ContourPlot object behind the hyperboloid and then apply a reflection transformation to each point, then plot the points I get back. How do I get the points backing the plot? If that's not possible, or if there is a better way, how can I achieve my desired reflection?