Methods for simulating and visually representing objects using particles

1. A method for visually representing an object by simulating the object as a spherical particle with fewer degrees of freedom than those visually represented, the method comprising:

simulating an object as a spherical particle having an orientation represented by only three degrees of freedom based on the linear motion of the spherical particle;

varying a radius of the spherical particle in each simulation time step based on a shape of the object and a position of the object on a contact surface;

deriving, via a specialized controller, an angular component of the spherical particle based on the radius of the spherical particle in each simulation step;

determining three rotational degrees of freedom for the object based on the derived angular component of the spherical particle;

determining a modified orientation for the object based on the three linear degrees of freedom associated with the spherical particle and the three rotational degrees of freedom determined for the object; and

generating a visual display of the object having six degrees of freedom based on the modified orientation for the object.

2. The method of claim 1 , further including deriving the angular component of the spherical particle based on at least one of the linear motion of the spherical particle, a position of the spherical particle, a nature of a contact surface that that the spherical particle may rest on, and a size of the spherical particle.

3. The method of claim 1 , wherein the step of determining a modified motion for the object comprises the step of determining a new orientation of the spherical particle based on a current orientation of the spherical particle and the derived angular component of the spherical particle.

4. A method for visually representing an object in contact with a moving surface by simulating the object as a spherical particle with fewer degrees of freedom than those visually represented, the method comprising:

simulating an object as a spherical particle having an orientation represented by only three degrees of freedom based on the linear motion of the spherical particle;

determining a linear motion of a moving surface in contact with the object;

determining a relative motion between the linear motion of the spherical particle and the linear motion of the moving surface;

varying a radius of the spherical particle in each simulation time step based on a shape of the object and a position of the object on a contact surface to generate a set of radii;

deriving, via a specialized controller, an angular component of the spherical particle based on the radius of the spherical particle in each simulation step;

determining three rotational degrees of freedom for the object based on the derived angular component of the spherical particle;

determining a modified orientation for the object based on the three linear degrees of freedom associated with the spherical particle and the three rotational degrees of freedom determined for the object; and

generating a visual display of the object having six degrees of freedom based on the modified orientation for the object.

5. The method of claim 4 , further including simulating the moving surface as a rigid body.

6. The method of claim 4 , further including deriving the angular component of the spherical particle based on at least one of a size of the spherical particle and a nature of the moving surface.

7. The method of claim 4 , wherein the step of determining a modified orientation of the spherical particle comprises the step of determining a new orientation of the spherical particle based on a current orientation of the spherical particle and the derived angular component of the spherical particle.

8. A method for visually representing a non-spherical object as a spherical particle with fewer degrees of freedom than those visually represented, the method comprising:

simulating an non-spherical object as a spherical particle having an orientation represented by only three degrees of freedom based on a linear motion of the spherical particle;

varying a radius of the spherical particle in each simulation time step based on a shape of the object and a position of the object on a contact surface to generate a set of radii;

deriving, via a specialized controller, an angular component of the spherical particle based on the radius of the spherical particle in each simulation step;

determining an orientation based on the linear motion and the derived angular component;

determining a local up axis of the spherical particle based on the determined orientation;

determining a correction vector based on the determined local up axis and a world up axis;

determining three rotational degrees of freedom for the object based on the derived angular component of the spherical particle;

determining a modified orientation for the object based on the three linear degrees of freedom associated with the spherical particle and the three rotational degrees of freedom determined for the object; and

generating a visual display of the object having six degrees of freedom based on the modified orientation for the object.

9. The method of claim 8 , wherein determining the local up axis includes determining a local up axis from a collection of local up axes such that the determined local up axis is closest to the world up axis.

10. The method of claim 8 , wherein determining a correction vector includes computing a cross product of the determined local up axis and the world up axis.

11. The method of claim 8 , wherein the step of determining the modified orientation includes the step of aligning the determined local up axis with the world up axis based on a length and a direction of the correction vector.

12. The method of claim 11 , wherein the length of the correction vector is an angle by which the determined local up axis is rotated to align the determined local up axis with the world up axis.

13. The method of claim 11 , wherein the direction of the correction vector is a rotation axis along which the determined local up axis is rotated to align the determined local up axis with the world up axis.

14. The method of claim 8 , further comprising the step of clamping a length of the correction vector to form a clamped correction vector, wherein a length of the clamped correction vector is less than a length of the correction vector by a predetermined amount.

15. The method of claim 14 , wherein the modified orientation is also determined based on the clamped correction vector.