### F-Reps/Implicits

By far the most commonly used representation scheme in conventional solid modeling is Boundary Representation (B-Rep), where an object is defined as the union of surfaces that form the object’s boundary.  F-Reps (function or implicit representation) describe the geometry of a solid with a single implicit function G(x,y,z).  Point membership classification is determined by the sign of the implicit defining function with the convention G < 0 on the interior and G > 0 on the exterior of the object. The surface of the solid lies on the kernel of the implicit function (i.e. the G=0 isosurface).  Many researchers have been and continue to be involved in the development of implicit modeling techniques. To demonstrate F-Rep/ISM, we present two examples – one simple and one more complicated:

Gs(x,y,z) = x^2+y^2+z^2-1

(1)

Ge(x,y,z) = max(x^2+y^2-196,-x^2-y^2+16,-min(max(x^2+y^2-138.0625000,-x^2-y^2+16,1/12*abs(z+20)-1,-z-20,-max(-x^2-y^2+64,x^2+y^2-625/4,1/8*abs(z+20)-1)),max(x^2+y^2-138.0625000,-x^2-y^2+16,1/12*abs(z-20)-1,z-20,-max(-x^2-y^2+64,x^2+y^2-625/4,1/8*abs(z-20)-1))),1/14*abs(z)-1)

(2)

The function Gs in Eq. (1) corresponds to the familiar - implicit description a unit sphere, and Ge in Eq. (2) defines the geometry of the elastomer center of a   W. M.  Berg  "Flexible Coupling". While the implicit defining function may be somewhat more complicated than the description of a parametric patch employed in a B-Rep system, the F-Rep/ISM representation is concise and efficient because it only takes one such equation to describe an entire object.

### 3D Volume Spaces

A new formulation of solid modeling that addresses the issue of including parts whose geometry is determined from volumetric scans (CT, MRI, PET, etc.) along with parts whose geometry is designed by traditional computer-aided design (CAD) operations. Such issues arise frequently in the design of medical devices or prostheses where fit and/or interference between man-made artifacts and existing anatomy are essential considerations, but the modeling formulation presented is not limited to medical applications and can be applied to any parts whose volume can be actually or virtually scanned. Scanner data typically comprises a grid of intensity values and segmentation must be performed to determine the extent of the part. In current practice, the segmented scanner data is run through a polygonizer to obtain an approximate tessellation of the object’s surface (such triangulated models can be problematic due to excessive complexity).

An alternative approach based on recent advances in segmentation with level set and interval methods. The output of the level set computation is a grid of approximate values for the signed distance from each grid point to the nearest point on the surface of the scanned object. We propose interpolating the grid of signed distance values to obtain an implicit or function-based representation (f-rep) for the object, and we introduce appropriate wavelets to effectively perform the interpolation while also providing a number of other useful properties including data compression, inherently multi-scale modeling, and capabilities for skeletal-based modeling operations. We are investigating these new spaces using concepts of interval methods.

Currently, we are joining a team at the VA hospital to explore patient-specific modeling issues.

### 3D Printing/Ceramic Printing

While 3D printing is not a new technology, we have been working on new material/binder systems for use in current 3D printing hardware for the creation of mid-fire to high-fire ceramics.  3D printing is one of a variety of techniques in which objects are produced by printing binder onto a layer of powder (approximately 0.1 mm thick).  Our process involves 3D printing of an object, depowdering, sintering, and finishing.  In some application an infiltration step is performed with colloidal silica after sintering to decrease porosity and increase strength.  Several available dry clay bodies were adapted for use in an existing commercial 3D printer.   We a testing these new materials to provide engineering data (presented as graphs) on sintering temperature verses shrinkage, flexural strength, and porosity for the various clay bodies demonstrated in 3D printing.   Our resulting material is a porous ceramic sponge-like body.