A friend uses a cooking splatter screen (metal screen on a round frame with a handle) on top of a shallow bowl. I started with a shallow baking sheet (bachelors can find one in the drawer under the oven) with a towel lining the bottom. A cookie sheet works too.
I think it is not that important to pour water over the teapot when brewing (except you want to make it absolutely perfect). So you can just pour the water used for warming up the cups and the tea from the first infusion into another bowl standing next to you. I've have seen that method in a teashop in Beijing.
It is hygiene, for it enforces cleanliness; it is economics, for it shows comfort in simplicity rather than in the complex and costly; it is moral geometry, inasmuch as it defines our sense of proportion to the universe.