Implicit linear interval estimations
SCCG '02 Proceedings of the 18th spring conference on Computer graphics
Comparison of interval methods for plotting algebraic curves
Computer Aided Geometric Design
Near-optimal parameterization of the intersection of quadrics
Proceedings of the nineteenth annual symposium on Computational geometry
Topology and arrangement computation of semi-algebraic planar curves
Computer Aided Geometric Design
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Interval computations mostly fall into two categories: those using small intervals to track the accuracy of a computation (typically one that uses floating-point arithmetic), and those using large intervals to approximate the value of some function over a box-shaped region of a space. This paper concentrates on the second category of interval computations and its applications in location, simplification, and root finding for multivariate implicit functions that are used as shape primitives in a set-theoretic (that is, a CSG) geometric modeler. Three problems are discussed, and solutions to them presented: 1. The location and simplification of the surfaces of semi-algebraic sets (surfaces involving some transcendental functions will be dealt with as well);2. The generalization of Newton-Raphson using intervals; and3. Interval ray tracing.Examples are presented for both conventional three-dimensional geometric models and for CSG models in higher dimensions representing configuration-space maps for moving and colliding three-dimensional objects.