In this paper, we introduce the adapted frame of framed curves and give relations between Frenet type curve in four-dimensional Euclidean space. Moreover, define normal We obtain some characterizations terms their curvature functions. Furthermore, necessary sufficient condition for a to be curve.

The aim of this project is to implement the operations of Elliptic Curve Cryptography built over GF(2) represented with optimal normal basis in VIVA System. These operations include multiplication, inverse, point addition, point doubling and scalar multiplication. The target device is Xilinx 4062 embedded in the FAI board of HAL-15 hypercomputer. The synthesis results using VIVA and VHDL are co...

Isogenies of elliptic curves over finite fields have been well-studied, in part because there are several cryptographic applications. Using Vélu’s formula, isogenies can be constructed explicitly given their kernel. Vélu’s formula applies to elliptic curves given by a Weierstrass equation. In this paper we show how to similarly construct isogenies on Edwards curves and Huff curves. Edwards and ...

In order to achieve a smooth surface while rendering a triangle-based mesh, we need to eliminate the mismatch between the smoothness of the shading and the non-smoothness of the geometry that is particularly visible at silhouettes. To eliminate these artifacts, we substitute the geometry of a cubic triangular Bézier curved patch for a triangular flat geometry. Meanwhile a subdivision algorithm ...

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In recent years, the problem of symbolic summation has received much attention due to the exciting applications of Zeilberger’s method for definite hypergeometric summation. This lead to renewed interest in the central part of the algorithmic machinery, Gosper’s “classical” method for indefinite hypergeometric summation. We review some of the recent (partly unpublished as yet) work done in this...

A new approach to estimate the surface curvatures from 3D triangular mesh surfaces with Gaussian curvature’s geometry interpretation is proposed in this work. Unlike previous work, the proposed method does not use local surface fitting, partial derivative computation, or oriented normal vector recovery. Instead, the Gaussian curvature is estimated at a vertex as the area of its small neighborho...

We report the observation and measurement of second-harmonic generation in self-assembled ultra thin film nonlinear optical materials using a femtosecond high repetition rate laser system. Second-harmonic intensity, as a function of the incident angle in these films, has been measured using incident p-polarized and s-polarized optical beam components. The second-order nonlinear optical suscepti...

The increased vulnerability of network attached computers has elevated the need for strong security methods. At the same time, advancements in the the area of cryptography have risen to meet this challenge. This report concerns one particular form of cryptography which uses the mathematics of elliptic curves. In particular, the suitability of enhancing current methods with specialized hardware ...

We present a system for Computer Aided Diagnosis in Virtual Colonography based on simple geometric model fitting. Our approach extends surface normal analysis and sphere fitting methods. We label locations in the volume data, which have a high probability of being colonic polyps, and present them in a user-friendly way. The method was tested on a study group of 30 patients. Using normal colonos...