GENERALIZED CONIC CURVES AND THEIR APPLICATIONS IN CURVE APPROXIMAION

The appriximation properties of generalized conic curves are studied in this paper. A generalized conic curve is defined as one of the following curves or their affine and translation e-quivalent curves:(i) conic curves i including parabolas, hyperbolas and ellipses;(ii) generalized monomial curves, including curves of the form x=yr,.r R.r=0,1, in the x-y Cartesian coordinate system;(iii) exponential spiral curves of the form p=Apolar coordinate system.This type of curves has many important properties such as convexity , approximation property, effective numerical computation property and the subdivision property etc. Applications of these curves in both interpolation and approximations using piecewise generalized conic segment are also developed. It is shown that these generalized conic splines are very similar to the cubic polynomial splines and the best error of approximation is or at least in general provided appropriate procedures are used. Finally some numerical examples of interpolation and approximations with generalized conic splines are given.

A New Method for Generating Conics

In this paper,based on the mean value theorem of differential,a new method of generating conics such as circles and parabolas is given,and the related algorithm for generating conics is designed.

The improved QV signature scheme based on conic curves over Z_n

The classical RSA is vulnerable to low private exponent attacks （LPEA） and has homomorphism. KMOV based on elliptic curve En（a,b） over Zn can resist LPEA but still has homomorphism. QV over En（a,b） not only can resist LPEA but also has no homomorphism. However, QV over En（a,b） requires the existence of points whose order is Mn= 1cm{#Ep（a,b）, #Eq（a,b）}. This requirement is impractical for all general elliptic curves. Besides, the computation over En（a,b） is quite complicated. In this paper, we further study conic curve Cn（a,b） over Zn and its corresponding properties, and advance several key theorems and corollaries for designing digital signature schemes, and point out that Cn（a,b） always has some points whose order is Mn： 1cm{#Ep（a,b）,#Eq（a,b））. Thereby we present an improved QV signature over Cn（a,b）, which inherits the property of non-homomorphism and can resist the Wiener attack. Furthermore, under the same security requirements, the improved QV scheme is easier than that over En（a,b）, with respect plaintext embedding, inverse elements computation, points computation and points＇ order calculation. Especially, it is applicable to general conic curves, which is of great significance to the application of QV schemes.

Finite element simulation and microstructure of two-pass inner spinning process of curved-generatrix cone cylindrical parts with annealing/quenching

A two-pass annealing/quenching internal spinning process with small-end rotations is proposed to form a curved generatrix conical thin-walled shell.That is,annealing at 360°C for 2 h followed by the 1st pass spinning,and finally quenching in ice water after holding for 1 h at 498°C followed by the 2nd pass spinning.ABAQUS finite element software is used to simulate the internal spinning process of the products formed under different forming parameters.The distribution laws of spinning force,the stress and strain under different forming processes were compared and analyzed.The mechanical properties and microstructure of the products are subsequently analyzed.The results show that the strain and the residual stress in the skin area of the formed products under two-pass spinning process more uniform,and the hardness and the mechanical performance are improved.The microstructure of the products formed with the 0.15 mm thickness reduction at the 2nd pass is excellent.And the second phase grain size distributed uniformly in the range of 36μm.Whereas,the second phase particles are broken seriously and the size distribution inhomogeneity is increased when the thickness reduction in the skin area is greater than 0.20 mm at the 2nd pass spinning process.

Four Conic Section Curves

Mathematicians are constantly constructing and exploring the properties of abstract objects only because they find them beautiful and interesting. Later, sometimes centuries later, the objects may turn out to be e-normously useful when they are applied to the physical world. There are no more elegant examples of this than the work done in ancient Greece on the four conic-section curve. If a right circular cone is sliced by a plane parallel to its base, the cross section is a circle. Tip the