Reconstruction approximating method by biquadratic splines of offset surfaces holes

Standard Offset surfaces are defined as locus of the points which are at constant distance along the unit normal direction from the generator surfaces. Offset are widely used in various practical applications, such as tolerance analysis, geometric optics and robot path-planning. In some of the engineering applications, we need to extend the concept of standard offset to the generalized offset where distance offset is not necessarily constant and offset direction are not necessarily along the normal direction. Normally, a generalized offset is functionally more complex than its progenitor because of the square root appears in the expression of the unit normal vector. For this, an approximation method of its construction is necessary. In many situation it is necessary to fill or reconstruct certain function defined in a domain in which there is a lack of information inside one or several sub-domains (holes). In some practical cases, we may have some specific geometrical constrains, of industrial or design type, for example, the case of a specified volume inside each one of these holes. The problem of filling holes or completing a 3D surface arises in all sorts of computational graphics areas, like CAGD, CAD-CAM, Earth Sciences, computer vision in robotics, image reconstruction from satellite and radar information, etc. In this work we present an approximation method of filling holes of the generalized offset of a surface when there is a lack information in a sub-domain of the function that define it. We prove the existence and uniqueness of solution of this problem, we show how to compute it and we establish a convergence result of this approximation method. Finally, we give some graphical and numerical examples.

A reciprocal relation for Hermite polynomials

For $x\in \mathbb{R}$, the ordinary Hermite polynomial $H_k(x)$ can be written\begin{eqnarray*}\displaystyleH_k(x)= \mathbb{E} \left[ (x + {\rm i} N)^k \right] =\sum_{j=0}^k {k\choose j} x^{k-j} {\rm i}^j \mathbb{E} \left[ N^j \right],\end{eqnarray*}where ${\rm i} = \sqrt{-1}$ and $N$ is a unit normal random variable. We prove the reciprocal relation\begin{eqnarray*}\displaystylex^k=\sum_{j=0}^k {k\choose j} H_{k-j}(x)\ \mathbb{E} \left[ N^j \right].\end{eqnarray*}A similar result is given for the multivariate Hermite polynomial.

Direction Curves Associated with Darboux Vectors Fields and Their Characterizations

In this paper, we consider the Darboux frame of a curve α lying on an arbitrary regular surface and we use its unit osculator Darboux vector D ¯ o , unit rectifying Darboux vector D ¯ r , and unit normal Darboux vector D ¯ n to define some direction curves such as D ¯ o -direction curve, D ¯ r -direction curve, and D ¯ n -direction curve, respectively. We prove some relationships between α and these associated curves. Especially, the necessary and sufficient conditions for each direction curve to be a general helix, a spherical curve, and a curve with constant torsion are found. In addition to this, we have seen the cases where the Darboux invariants δ o , δ r , and δ n are, respectively, zero. Finally, we enrich our study by giving some examples.

Harmonic Evolute Surface of Tubular Surfaces via B -Darboux Frame in Euclidean 3-Space

In this article, we look at a surface associated with real-valued functions. The surface is known as a harmonic surface, and its unit normal vector and mean curvature have been used to characterize it. We use the Bishop-Darboux frame ( B -Darboux frame) in Euclidean 3-space E 3 to study and explain the geometric characteristics of the harmonic evolute surfaces of tubular surfaces. The characterizations of the harmonic evolute surface’s ϱ and ς parameter curves are evaluated, and then, they are compared. Finally, an example of a tubular surface’s harmonic evolute surface is presented, along with visuals of these surfaces.

Design and Analysis of Posterior Semicircular Canal BPPV Diagnostic Test Based on Physical Simulation

To discusses and analyzes how to realize the design of posterior semicircular canal BPPV diagnostic maneuver. First, measure the spatial attitude of the human semicircular canal, establish a BPPV virtual simulation platform, then analyze the key positions of the maneuver, and finally design a new diagnostic maneuver according to the demand, and perform physical simulation verification. The average value of the unit normal vector of the right posterior semicircular plane is [ 0.660, 0.702, 0.266], after rotate -46.8 ° around Z axis and 15.4 ° around Y axis, it parallel to the X axis. After that, when the tilt back angle reaches 70 °, the free otoconia in the left utricle will fall into the common crus; when bend forward 53.3°, the unit normal vector of the crista ampullaris plane of the posterior semicircular canal to the XY plane; when bend forward angle reaches 30°, the otoconia slides to the opening of the ampulla; when bend forward angle reaches 70°, the otoconia slides to the bottom of the crista ampullaris. The shallow pitching Yang maneuver is designed as turn head 45° to the one side, bend forward 45°, tilt back 90°, and bend forward 90°. The deep pitching Yang maneuver is designed as bend forward 90°, turn head 45° to one side, tilt back 135°, and bend forward 90°. A new posterior semicircular BPPV diagnostic test is designed to make the induced nystagmus have the characteristics of long latency, reversal, and repeatability, will not cause the inhibitory stimulation of the contralateral superior semicircular canal, and has good operation fault tolerance, which is of great value for clinical and scientific research.

Revisit the Wishart Distributionm

If S_pxp can be written as S=X' X , where X_nxp is a data matrix from N_p(0,V) , then S is said to have a Wishart distribution with scale matrix V of degree of freedom parameter n. We write S~W_p(V,n). When V=I,  the distribution is said to be in standard form. When p=1, the W_1(σ^2, n)  distribution is found to be Σ^n_i=1(x^2_i) , where the elements of x_i  are identically independently distributed unit normal variables; being the σ^2(x_n)^2 distribution. Although Anderson (1984, p248~249) has presented two theorems for the Wishart distribution. In the following we give an alternative proof.

Constructing a New Frame of Spacelike Curves on Timelike Surfaces in Minkowski 3-Space

In this paper, we define a relativeMinkowski normal plane and relative tangent vector TM:We construct a new relative S-frame (Shnoda-Saad frame) of regular spacelike curves on timelike surfaces. It depends only on the curve lies on the surface, Euclidean and Minkowski unit normal vectors. Also, we define S-curve according to this frame with some related theorems.