# Digital Geometry Processing with Topological Guarantees

## Abstract

We describe novel approaches to compute reliable solutions for many non-linear geometric problems that arise in geometric modeling, computer graphics and robotics. Specifically, we focus on problems that can be formulated as *surface extraction problems*. These include Boolean operations and Minkowski sums of polyhedral or higher models as well as reliable polygonization of general implicit surfaces. All these problems reduce to computing a topology preserving isosurface from a volumetric grid, i.e. the zero set of a scalar field. A common way of representing a scalar field is to discretize the continuous scalar field into discrete samples – to compute the value of the scalar field at the vertices of a volumetric grid. We refer to this step as a sampling of the scalar field. The grid is an approximate representation of the scalar field; the accuracy of the approximate representation depends on the rate of sampling – the resolution of the grid. An explicit boundary representation of the implicit surface can be obtained by extracting the zero-level isosurface using Marching Cubes or any of its variants. We refer to these isosurface extraction algorithms collectively as *MC-like algorithms*. The output of an MC-like algorithm is an approximation – usually a polygonal approximation – of the implicit surface. We refer to this step as reconstruction of the implicit surface.

## References

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