B-splines, Pólya curves, and duality

AbstractLocal duality between B-splines and Pólya curves is examined, mostly from the viewpoint of computer-aided geometric design. Certain known results for the two curve types are shown to be related. A few new results for Pólya curves and a curve scheme related to B-splines also follow from these investigations

Covariance matrices for variance-suppressed simulations

Cosmological $N$-body simulations provide numerical predictions of the structure of the universe against which to compare data from ongoing and future surveys. The growing volume of the surveyed universe, however, requires increasingly large simulations. It was recently proposed to reduce the variance in simulations by adopting fixed-amplitude initial conditions. This method has been demonstrated not to introduce bias in various statistics, including the two-point statistics of galaxy samples typically used for extracting cosmological parameters from galaxy redshift survey data. However, we must revisit current methods for estimating covariance matrices for these simulations to be sure that we can properly use them. In this work, we find that it is not trivial to construct the covariance matrix analytically, but we demonstrate that EZmock, the most efficient method for constructing mock catalogues with accurate two- and three-point statistics, provides reasonable covariance matrix estimates for variance-suppressed simulations. We further investigate the behavior of the variance suppression by varying galaxy bias, three-point statistics, and small-scale clustering.Comment: 9 pages, 7 figure

A Novel Drill Set for the Enhancement and Assessment of Robotic Surgical Performance

Background: There currently exist several training modules to improve performance during video-assisted surgery. The unique characteristics of robotic surgery make these platforms an inadequate environment for the development and assessment of robotic surgical performance. Methods: Expert surgeons (n=4) (greater than 50 clinical robotic procedures and greater than 2 years of clinical robotic experience) were compared to novice surgeons (n=17) (less than 5 clinical cases and limited laboratory experience) using the da Vinci Surgical System. Seven drills were designed to simulate clinical robotic surgical tasks. Performance score was calculated by the equation Time to Completion + (minor error) x 5 + (major error) x 10. The Robotic Learning Curve (RLC) was expressed as a trend line of the performance scores corresponding to each repeated drill. Results: Performance scores for experts were better than novices in all 7 drills (p less than 0.05). The RLC for novices reflected an improvement in scores (p less than 0.05). In contrast, experts demonstrated a flat RLC for 6 drills and an improvement in one drill (p=0.027). Conclusion: This new drill set provides a framework for performance assessment during robotic surgery. The inclusion of particular drills and their role in training robotic surgeons of the future awaits larger validation studies

Math in the Movies

Film making is undergoing a digital revolution brought on by advances in areas such as computer technology, computational physics, geometry, and approximation theory. Using numerous examples drawn from Pixar\u27s feature films, this talk will provide a behind the scenes look at the role that math plays in the revolution. Tony DeRose is currently a Senior Scientist and lead of the Research Group at Pixar Animation Studios. He received a B.S. in Physics from the University of California, Davis, and a Ph.D. in Computer Science from the University of California, Berkeley. From 1986 to 1995 Dr. DeRose was a Professor of Computer Science and Engineering at the University of Washington. In 1998 he was a major contributor to the Oscar-winning short film “Geri\u27s game”, in 1999 he received the ACM SIGGRAPH Computer Graphics Achievement Award, and in 2006 he received an Academy Award for his work on surface representations.https://egrove.olemiss.edu/math_dalrymple/1008/thumbnail.jp

Computing values and derivatives of Bézier and B-spline tensor products

: We give an efficient algorithm for evaluating B'ezier and B-spline tensor products for both positions and normals. The algorithm is an extension of a method for computing the position and tangent to a B'ezier curve, and is asymptotically twice as fast as the standard bilinear algorithm. Keywords: Tensor product surfaces, blossoms, evaluation, rendering Abbreviated title: Tensor Product Evaluation Many applications, such as rendering a surface using Phong shading, require evaluating both the value and derivatives of a surface. Repeated bilinear interpolation can be used to compute values and derivatives of n \Theta n tensor product B'ezier surfaces (see Figure 1a and Farin [Far93]). The final computed point is a point on the surface, and the points in the next to last step can be used to calculate the derivatives of the surface. However, this algorithm cannot be used for an n \Theta m tensor product surface with m 6= n. In this paper we develop an algorithm to handle the general c..