is flat ships (Kim 1963)。 The criteria for a flat ship    are:

B B

D\\L;         ffi    :

k L

A flat has a shallow draft and its length and breadth are of the same order。 It is very sensitive to the variation of sea surface。 A raft is a flat ship according to these criteria。 The last type is slender ships (Vossers 1962)。 Their shapes meet the following conditions:

B\\L;     D\\L;     k ffi L:

Slender ships are the most used ships。 They include, for example, war ships and cargo ships。 Slender ships are narrow and their draft are deep。 They are less sensitive to sea waves。 Since wave length is considered in the classification, ship type is not invariant。 When encountering wave of large amplitude and length, a slender ship may become a thin ship。 By using this ship classification, it is easier to assign char- acters to ships。 In our work, a mechanism is developed for specifying ship parameters via a graphical user interface。 Therefore, ships’ behaviors will not go wild while ships can still possess certain maneuvering characters。

2。3Wave model

The wave model used in our system is based on the work of (Fournier and Reeves 1986; Peachy 1986)。 To increase fidelity, we assume the sea wave is composed of    multiple

sinusoidal waves of different frequencies, amplitudes, and directions。 Our wave model is similar to that of (Masten and Watterberg 1987)。 To model the animation of the sea surface, the sea surface is treated as a height field which can be computed by using the following   equation:

n

hðx; z; tÞ¼ X fi sin

i¼1

2p

ððx cos hi  — z sin hiÞ— xitÞ;

i

where fi, ki, and xi are the amplitude, wave length, and speed of the ith wave, respectively, and hi is the angle between X axis of the world coordinate system and the incoming direction of this wave, and t is the time variable。 To render the scene, a regular grid is super-imposed on the sea surface。 The X and Z coordinates of the grid points are substituted into the equation to calculate the Y coordinates。 Then, by using finite difference method, the gradients (normals) of the points can be computed, and the sea sur- face can be shaded。 In previous sub-section, wave length is treated as a factor for ship classification。 Though there may be multiple waves present on the sea surface, only the wave with the largest amplitude is taken into consideration for ship classification。

3Motions induced by sea waves

The motions induced by sea waves include heave, pitch, and roll。 In ship theory literatures, these motions are regarded as oscillations with damping effects。 Each motion can have an exact solution if the frequency of the wave is known and the vibration frequency of the ship in still water is available (Rawson and Tupper 2001)。 However, in our system, waves are comprised with multiple waves  and ships are virtually fabricated, both information is not available。 Therefore we go for other numerical procedures for computing these motions。 We assume that the swellness of water under a ship causes the ship to perform these three movements  based  on  Archimedes’  principle。  In  order to

Fig。 2  The computational grid for ship   motion

Subtracting resistance force from the force of water, the net force is obtained, and accelerations are calculated by applying Newton’s law。 Then velocities are updated and the magnitudes of motions are obtained by multiplying the velocities with the time step  size。

3。1Heave

Assume that the variation of sea surface in vertical direction produces force for heave。 To calculate the vertical variation of sea surface, the average height field, Ha of the grid is com- puted。 If the resulted height field is not zero, the swell of sea surface produces force to raise up (or pull down) the ship。 We assume the force is related to the volume of water swelled under the ship。 The other force for heave is the resistance of water acting on the ship hull。 The resistance is the anti-force for heave。 It is related to the current distance of heave and the displacement force。 Since the heave distance is computed by multiplying heave velocity with time step size and the dis- placement force is related to the ship mass, we assume the resistance force is related to the moment of the ship。 There- fore, the net force for heave is estimated by: