The successive simplification of closed, smooth surfaces is the subject of many approaches. Some methods work directly with geometric data; others transform the data into a new representation in which the reduction is executed. A common example is the simplification in wavelet space [53, 209].
The classic method, which works directly with geometric data, is described by Hoppe et al. [93, 94]. An initial surface is successively simplified by removing triangle edges. If an edge has to be deleted, the end points are moved to the middle point of the edge, and all adjacent triangles are modified accordingly. This is called an “edge collapse”. The deletion of the edges is repeated until a sufficiently simple base geometry is yielded. Hereby those triangle edges are coarsened that generate the smallest errors during the removal process.
Edge collapse: deletion of an edge, two vertices are merged into one
For representing the data, the base geometry and the succession of the “edge collapse” steps is stored with the respective data. Interestingly, through efficient storage management, a representation can be found that almost uses not much more memory than the initial geometry itself.
In Fig. 10.1 an “edge collapse” operation is illustrated. The thin lines are the initial geometry, which is transformed into a simplified net through the collapse of an edge (thick lines).
The base geometry can now be used for representing the distant object. If the virtual viewer moves towards the object, the inverse operation to the edge collapse is performed. Thereby a vertex is doubled, the new vertices are moved to the position of the original vertices, and the connection structure is updated
due to the inserted edge. In this way, a more and more refined description is developed until the initial geometry is won back.
A similar simplification method for height fields (see Chap. 7) is discussed in . Here triangles are simplified if the occurring error in the height is below a given threshold that is defined by the distance from the viewer to the actual geometry. The technique allows a fluent display of complex terrains with applications in real-time rendering, such as in flight simulators.