Blending techniques in Curve and Surface constructions

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B\'ezier curves that are close to elastica

We study the problem of identifying those cubic B\'ezier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are difficult to work with in a digital environment. We seek a sub-class of special B\'ezier curves as a proxy. We identify an easily computable quantity, which we call the lambda-residual, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a B\'ezier curve has lambda-residual below 0.4, which effectively implies that the curve is within 1 percent of its arc-length to an elastic curve in the L2 norm. Finally we give two projection algorithms that take an input B\'ezier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure

Statistical Estimation and Inference Improvements for Exoplanet Discovery

The radial velocity method has been widely used by astronomers since the 1990\u27s for discovering extra-solar planets, often referred to as simply exoplanets . This method involves estimating the radial velocity of a distant star over time using the stellar light, followed by modeling such radial velocity estimates as a function of time using Keplerian-orbital equations with parameters that describe the exoplanet. While a number of approaches exist for estimating the radial velocity from the stellar light, we introduce a new approach for this that uses Hermite-Gaussian functions to reduce the estimation to linear least-squares regression. Furthermore, we demonstrate that this new approach, compared to the commonly used cross-correlation approach, provides an approximate 21% reduction of statistical risk in simulation studies as well as in applications to recently collected data. We then extend this linear model to include additional terms that represent the effect of stellar activity on the observed light, an effect known to both hide and imitate the signal of exoplanets. The F-statistic for the fitted coefficients of these additional terms is found to have higher statistical power than many traditional stellar activity indicators at detecting the presence of stellar activity. Finally, we also use the linear model in a Bayesian framework to merge both traditional steps of the radial velocity method into one that estimates the exoplanet\u27s orbital parameters directly from the time series of observed stellar light

Applying inversion to construct rational spiral curves

A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of boundary conditions, namely, tangents and curvatures at the endpoints, including those, assuming inflection.Comment: 18 pages, 11 figure