Description

Dynamic NURBS (NonUniform Rational B-Spline) or D-NURBS are a useful tool for modeling free-form shapes and curves. My goal for this project is to use D-NURBS to model "moving" curves, or, more specifically, to create an animated model of the behavior of a string. For the purposes of this project, a string is going to be considered to be simply a line in space. The motions of a string can be thought of as just moving curves or transforming curves. A future goal of the project would be to add "thickness" to the string or change the texture and or shape of the string (for example, the string could be modeled as a shoelace, or a thick piece of yarn, etc.).

The use of NURBS in geometric design has many benefits including the ability to represent free-form curves, construct perspective views, and also create shapes such as conics and quadrics. D-NURBS allow a user to create physics-based models that can be changed or sculpted by the modeler by dynamically applying forces instead of relying on adjusting control points and weights. This ability can permit a more intuitive and predictable motion. A string is a very common and important object to model. It's motion or behavior can be used in countless number of places within modeling and animation (for example, modeling hair or a whip). There are many aspects to consider when modeling the motion of a string. The physical and geometrical constraints for the system have to be defined and the hard constraints that cannot be violated must be identified versus the soft constraints that have to be approximately fulfilled. The purpose of the project is to model the motion of a string as realistically as possible. Mimicking its behavior in the real world as closely as possible includes adding the force of gravity to the system and other real world possibilities.

One of the main goals of the project will be to model a wave traveling or propagating across the string. This will first be done with a string that has two fixed endpoints. The next step will be to model this behavior on a string that has one fixed end and one free end. The challenges for these processes will be highly mathematical and physics-based. The wave travelling on the string will basically be a "moving curve." Depending on the setup of the system (for example, one or two fixed endpoints), this will have to be dealt with in different ways. I will be researching and developing ways to do this over the rest of the semester.

Goals

Preliminary goals to be completed by each date:
• October 15: Reading and research on D-NURBS, primarily in math and physics backgrounds; learn and familiarize myself with the relevant equations; determine the physical and geometrical constraints for the system; etc. Begin working on and complete an implementation of a simple NURBS curve, with no motion.

• November 10: Begin implementation of a moving curve across a string. Start with a string that has two fixed endpoints. Learn the physics and math necessary to model a wave propagating across a string.

• November 30: Implementation of a string with one fixed endpoint and one free end. Obey physical and geometric constraints of a string, including gravity, to model as realistically as possible a string's behavior in the real world.

Future Goals, Ideas, and Uses: By realistically modeling the behavior of a string, future possibilities exist which will probably have to be implemented after this semester.

• Add texture and thickness to the string - string can be modeled as a shoelace or a piece of yarn, etc.
• User interaction with string - This would allow the user to specify the motion of the string using either the mouse or keyboard. This simulation could imitate a person dragging a string across a table where the person is holding one end of the string and the rest of the string follows and moves and curves as the person specifies.
• Attach the string to a rigid body - As the rigid body moves, the string must move along with it. This behavior would be similar to hair; As a person moves his/her head, the hair must follow, but it still has it's own independent behavior.
• Whip - One area that I would really like to explore would be to model the behavior of a whip. This would include the whip's or string's interaction with another object and show how the string either collides or wraps itself around the second object.

Novelty

The research I have done so far has shown using D-NURBS to model mostly abstract and conceptual problems instead of real world simulations, such as the motion of a string. But, I still have a lot more research to do to find out the extent to which D-NURBS have been used within modeling. Particularly I am interested to see how or if D-NURBS have been used to model specific objects such as a whip or cloth objects like a flag blowing in the wind.

I have been reading the following:
• ``Dynamic NURBS with Geometric Constraints for Interactive Sculpting,'' D. Terzopoulos and H. Qin, ACM Transactions on Graphics, 13(2): 103--136, 1994 (Special Issue on Interactive Sculpting).

• ``Dynamic Manipulation of Triangular B-Splines,'' H. Qin and D. Terzopoulos, Proceedings of Third Symposium on Solid Modeling and Applications (Solid Modeling'95), Salt Lake City, Utah, May, 1995, ACM Press, 351--360.
  

Some further reading ideas which I still have to find (and I'm sure there will be more to come):
• "Dynamic free-form deformations for animation synthesis," P. Faloutsos, M. van de Panne, D. Terzopoulos, IEEE Transactions on Visualization and Computer Graphics, 3(3), September, 1997, 201-214.

• "Triangular NURBS and their dynamic generalizations," H. Qin, D. Terzopoulos, Computer-Aided Geometric Design, 14, 1996, 325-347.

• "D-NURBS: A physics-based framework for geometric design," H. Qin, D. Terzopoulos, IEEE Transactions on Visualization and Computer Graphics, 2(1), March, 1996, 85-96.