Reflection by a curve

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Consider point $A(x_1, y_1)$ and a line $y = x.$ After reflecting point $A$ over $y = x$ we get new point, $A'(y_1, x_1).$ Is there general approach to this? Which point we get after refleting it by line $y = mx$? What about $y = mx + b$? What about reflecting any set of points by any continuous curve?