The geometric fish models are constructed manually using the *
Alias* 3D modeler. We model the
shape of any given fish's body in accordance with the shape evident in
digitized color pictures of the animal
(Fig. ). We employ two juxtaposed NURBS
surfaces, one for the left half and the other for the right half of
the fish body. The NURBS surfaces are of order 3 (or of degree
4) along both the and axes ( and represent the two axes of the material coordinates of a
surface). Each NURBS surface has control points
which, when connected, form a surface mesh as shown in
Fig. . We will refer to this mesh model as the *
control-point mesh*. The control points must be updated at each
display time step such that the geometric fish model moves and deforms
in accordance with the underlying physics-based fish model (the coming
Section gives the details).

We choose to use a moderate number of control points (*u*=9; *v*=21)
for all the geometric fish models in order to achieve satisfactory
rendering speed while capturing the characteristic shape of different
species of fishes. The NURBS surface generated from the
control-point mesh of Fig. is shown in
Fig. .

The dorsal and ventral fins are also NURBS surfaces each of which has , control points and is of order 1 along the axis and order 3 along the axis. Note that the lower boundary of the dorsal fin coincides with the upper edge of the fish body and the same relationship holds between the upper boundary of the ventral fin and the lower edge of the fish body. This is achieved by simply making the corresponding control points coincide. The left and right pectoral fins are modeled as polygonal surfaces each with five vertices. Fig. shows the geometric fish model with dorsal and ventral fins (top) and with all fins (bottom). Finally, Fig. shows the complete geometric models of four different kinds of fishes patterned after the four kinds of natural fishes shown in Fig. .

**Figure:** The shaded NURBS surface of the left half of a fish body.

**Figure:** Control-point mesh fish models.

Xiaoyuan Tu | January 1996 |