It still looks to me like a dog's or fox's urine - the concentrated urine burns the grass but around the edges where the urine isn't so concentrated the nitrogen in the urine acts as a fertiliser making the grass grow darker and taller - your photographs are text book examples.
I think Buddyboy may be thinking of Fusariam patch lawn disease, which is now called Michrodochium nivale. If that's what it is the brown patches will get bigger and bigger and eventually join up. It's most commonly seen in the winter, particularly after snow has been laying on the grass - not something that's happened recently. Also it doesn't have the longer darker grass around the patches which are shown clearly in JanBi's photographs.