Abstract
Geometry of living things is one of the most attractive subjects for researchers in the field of biomechanisms. Recently, not only anatomists, but mathematicians and engineers are tackling the problem of clarifying how living organs geometry is optimized to its functions. Such study was accelerated by the development of shape-measurement equipment, such as the CT scanner, MRI, and 3-D digitizer.
Dental prostheses are among oldest artificial organs, having been used by edentulous patients for hundreds of years. Since the design and manufacturing of dental prostheses require advanced skills, their quality cannot easily be maintained within acceptable limits. Attempts have been made to apply advanced CAD/CAM technology to this production field. Currently, several dental CAD systems are commercially available. Their detailed and optimum design, however, cannot easily be accomplished under present CAD circumstances, because conventional manufacturing techniques matured in dentistry have yet to be fully computerized.
In an attempt to realize a more sophisticated dental CAD, this paper proposes a novel tooth model with both surface and solid descriptions, and an innovative partial metamorphosis operator, to which the Hit-or-Miss transform developed in the field of mathematical morphology was applied.
Our tooth model consists of a solid description of the occlusal surface and a surface description of the entire crown. The use of a solid model enables easy detection of the contact and interference of the opposing teeth. In order to make the two different models agree with each other in corresponding portions, we introduced a mutual transform between them, which is always determined uniquely if one of the models is deformed infinitesimally. Our partial deformation operator, on the other hand, can partially dilate or erode the contour around the deformation center without loss of smoothness, by adjusting only two parameter values representing the amount and extension of the deformation. Large deformation is always accomplished by repeating a minute deformation, thus maintaining the agreement of the two models.
In order to evaluate the effectiveness of this model and deformation operator, we applied them to the positional adjustment of tooth cusps and the one-point contact formation to neighboring teeth in a dental 3-D CAD, Vocs1/B. In experiments of designing a lower first molar, tooth cusps was successfully modified semiautomatically into a required form, without loss of natural tooth contour, and the contact to a neighboring tooth was also created at a point without interference or loss of smoothness. Thus, our fusion model and partial deformation operator were both verified to be effective for detailed optimum design of a dental crown.
