Function Based Terrain

This topic isn''t about how to draw the terrain exactly, but more on how to generate it. Although bitmaps are the most commonly used methods of obtaining the height of a specific vertex, wouldn''t it be easier to do with a function based on the x and y coords. The function would be complicated, but would save on memory for large terrains, where a heightmap isn''t easy to implement. Also using some fractal methods(I have a few ideas, but haven''t worked out how to use them yet) you could put as much detail as you want if necessary. Using a system like ROAM you could get VERY large terrains, with very detailed surfaces. This detail of course couldn''t be defined by the mapper, but then again, what sort of mapper would have time to put that much detail in. Creating a hill would be easy, with a function like f(x,y) = k / (1 + dist^2), where dist is the distance from a defined point. By varying 1(yes I know, but you know what I mean) and k, you could created blunted and sharp shapes. By using a different distance(not linear), you could vary the general shape of the hill. I''ve thought of many other functions for other purposes, such as making a section of ground flat. When wanting a height value, a function would take the x and y coord and stick them in the different functions in a predefined way and order and out would come a value. This wouldn''t be quick for many vertices, but shouldn''t be too bad, if new ones are spread out over time. If this is an old idea then please tell me and point me to some reading on it. I''d like your views and opinions and any potential problem you''ve seen.