Triangulation of parametric surfaces is important for visualization and finite element analysis. The wrong choice of parameterization can spoil a triangulation or even cause the algorithm to fail. We present a new method that uses a local inverse mapping for almost uniformly sampling and triangulating a surface, so that its parameterization becomes irrelevant; differential geometry provides alm...

The two main approaches for the description of sound fields are methods derived from solutions of the wave equation and geometric methods based on analogies to ray optics. Their mathematical representations are reviewed and it is shown that representations by projective geometry and descriptions by Fourier acoustics lead to similar parametric representations of sound fields.

We demonstrate rapid vibrational imaging based on sum frequency generation (SFG) microscopy with a collinear excitation geometry. Using the tunable picosecond pulses from a high-repetition-rate optical parametric oscillator, vibrationally selective imaging of collagen fibers is achieved with submicrometer lateral resolution. We furthermore show simultaneous SFG and second harmonic generation im...

Curvature formulas for implicit curves and surfaces are derived from the classical curvature formulas in Differential Geometry for parametric curves and surfaces. These closed formulas include curvature for implicit planar curves, curvature and torsion for implicit space curves, and mean and Gaussian curvature for implicit surfaces. Some extensions of these curvature formulas to higher dimensio...

This chapter describes several algorithms for factorization and GCD computation of polynomials over algebraic extension fields. These algorithms are common in using the characteristic set method introduced in the previous chapters. Some performance comparisons between these algorithms are reported. Applications include geometry theorem proving, irreducible decomposition of algebraic variaities,...

The study of the Newton polytope of a parametric hypersurface is currently receiving a lot of attention both because of its computational interest and its connections with Tropical Geometry, Singularity Theory, Intersection Theory and Combinatorics. We introduce the problem and survey the recent progress on it, with emphasis in the case of curves.

Developable surfaces are modelled with pieces of right circular cones. These cone spline surfaces are well-suited for applications: They possess degree two parametric and implicit representations. Bending sequences and the development can be explicitly computed and the offsets are of the same type. The algorithms are based on elementary analytic and constructive geometry. There appear interesti...

Creating 2D and 3D models in CAD and following creation of drawings are standard procedures for constructing in the current industrial practice. Most models of industrially produced machine parts can be made by basic processes of modeling using construction or hybrid methods. Listed procedures are based on input of a basic – primitive body and its shaping using predefined design elements, combi...

We discuss the theoretical and experimental investigation of acousto-optic dispersive tunable filters, based on quasi-collinear geometry of light-sound interaction in a tellurium dioxide single crystal. The geometry uses the effect of strong acoustic anisotropy in the paratellurite as well as peculiarities of acoustic wave reflections at the free boundary of the crystal. A mathematical concept ...

Evolution of Lindenmayer Systems (L-Systems) provides a powerful method for creating complex computer graphics and animations. This paper describes an interactive modelling system for computer graphics in which the user is able to “evolve” grammatical rules and surface equations. Starting from any initial L-System grammar the evolution proceeds via repeated random mutation and user selection. S...