Geometric Modeling without Coordinates and Indices

Since Descartes, coordinates have been the standard mathematical construct for describing and manipulating geometric objects. In addition, indices are commonly used to label elements of discrete geometric structures, such as polygon meshes. Since the coordinates and indices are numbers, they can be manipulated algebraically. This sets the framework for the handling of geometric objects in algebraic terms that is the essence of analytic geometry. In some modeling applications, however, the use of coordinates leads to complications that are not inherent in the modeling problems themselves. For example, DeRose pointed to both confusion and errors stemming from different geometric interpretations of the same transformation matrix [3, 4]. We will focus on dynamic geometric objects, which can be seen to develop over time. Examples include fractals, subdivision curves and surfaces, and models of growing biological structures. Coordinates and indices do not identify elements of these objects in a convenient, time-invariant way. For instance, the coordinates of a cell in a growing biological tissue may change over time, even though it materially is the same cell. Furthermore, if the cells are indexed, the indices may have to be updated following cell division in order to preserve consecutive numbering of the adjacent cells. Similarly, in the case of subdivision curves and surfaces, the existing points may change position over time, while the insertion of new points may require all points to be reindexed. We can address these difficulties using formalisms that hide coordinates and indices from the modeler’s perspective. Such formalism are expressed in local terms, which means that: