A. Pasko and V. Savchenko, The University of Aizu, Japan
Our concept of functionally based geometric modeling is an attempt to step further to a more general modeling scheme using real functions. The modeling concept includes sets of objects, operations and relations. Any operation has to be closed on the representation, i.e., to generate a continuous real function as a result. Transformations of a defining function are described for the basic operations: set-theoretic operations, blending, offsetting, bijective mapping, projection, Cartesian product and metamorphosis, inclusion, point membership and intersection relations were introduced. We presented also the following advanced operations: sweeping by a moving solid, deformation with algebraic sums, and three-dimensional texture modeling.
Designing an actual modeling system, some finite set of primitive objects can be defined. However, the modeling concepts do not require it and allow us to have an ``empty" set of objects assuming that the user will define them either in a symbolic manner by formulas or by evaluation procedures. This approach allows us to unify ``under one roof" very different kinds of solid models provided. We manage to find functional representations for them or to convert the existing ones to the desired form. Thus, for example, we can use together in one model CSG primitives, free-form implicits, swept solids and volumetric objects.
The following list of examples mainly presents the state of our project.
Offsetting along the normal is illustrated by Fig.1 where displacements along the normal vector are controlled by depth data.
Sweeping by a moving solid is one of the long standing problems in solid modeling. We have reduced the problem of a swept solid description to a one-dimensional global extremum search by a parameter of movement. It allows the user to apply arbitrary variable-shape and CSG solids as generators, arbitrary parametrized movement and self-intersections (see Fig. 2).
We have developed a method to define the deformation by arbitrary control points linked to the features of the object. An application of this technique to facial expressions simulation is shown in Fig.3 where five control were moved and eight points were fixed to localize the deformation.
Deformations by positions of arbitrary points was applied for the reconstruction of solids from scattered surface points (Fig.4).
Although parametrized basic and advanced operations provide effective shape control, they seem to be indirect for aesthetic design. The points of the stroke can be used to evaluate parameters of operations by solving a system of non-linear equations (see Fig.5 with an example for blending).
Three-dimensional texture modeling is based on the extensive use of the solid noise primitive defined by well-known solid noise functions. Different hairstyles have been modeled by procedurally defined real functions with the use of solid noise, sweep-like technique, offsetting and set-theoretic operations, and non-linear space mappings (Fig. 6).
References:
Figure 1. Offsetting along the normal with the distance controlled by
depth data of a head.
Figure 2. Sweeping by a moving and rotating solid (union of a ball and an
ellipsoid).
Figure 3. (a), (b). An initial volumetric head (a) and a facial expression
modeled with feature-based space mapping (b).
Figure 4. Reconstruction of Phobos (Mars satellite) from scattered surface
points using the deformation defined by an algebraic sum with a volume spline.
Figure 5. (a) The body and the bottom of a wine glass to be connected with an
aesthetic blend defined by the hand-drawn stroke; (b) The result of blending
with the estimated parameters.
Figure 6. Hair growing and styling with non-linear space mappings and
set-theoretic operations.