We present algorithms to compute the topology of 2D and 3D hyperelliptic
curves. The algorithms are based on the fact that 2D and 3D hyperelliptic
curves can be seen as the image of a planar curve (the Weierstrass form of the
curve), whose topology is easy to compute, under a birational mapping of the
plane or the space. We report on a {\tt Maple} implementation of these
algorithms, and present several examples. Complexity and certification issues
are also discussed.Comment: 34 pages, lot of figure