A Different View on Dynamics of Space Curves Geometry

In this study, we define the X-torque curves, $$X-$$ X - equilibrium curves, X-moment conservative curves, $$X-$$ X - gyroscopic curves as new curves derived from a regular space curve by using the Frenet vectors of a space curve and its position vector, where $$X\in \left\{ T\left( s\right) , N\left( s\right) , B\left( s\right) \right\} $$ X ∈ T s , N s , B s and we examine these curves and we give their properties.

Epihypocurves and epihypocyclic surfaces with arbitrary base curve

If a circle rolls around another motionless circle then a point bind with the rolling circle forms a curve. It is called epicycloid, if a circle is rolling outside the motionless circle; it is called hypocycloid if the circle is rolling inside the motionless circle. The point bind to the rolling circle forms a space curve if the rolling circle has the constant incline to the plane of the motionless circle. The cycloid curve is formed when the circle is rolling along a straight line. The geometry of the curves formed by the point bind to the circle rolling along some base curve is investigated at this study. The geometry of the surfaces formed when the circle there is rolling along some curve and rotates around the tangent to the curve is considered as well. Since when the circle rotates in the normal plane of the base curve, a point rigidly connected to the rotating circle arises the circle, then an epihypocycloidal cyclic surface is formed. The vector equations of the epihypocycloid curve and epihypocycloid cycle surfaces with any base curve are established. The figures of the epihypocycloids with base curves of ellipse and sinus are got on the base of the equations obtained. These figures demonstrate the opportunities of form finding of the surfaces arised by the cycle rolling along different base curves. Unlike epihypocycloidal curves and surfaces with a base circle, the shape of epihypocycloidal curves and surfaces with a base curve other than a circle depends on the initial rolling point of the circle on the base curve.

Some new characterizations of a space curve due to a modified frame N⃗,C⃗,W⃗$\left\{ \overrightarrow {N},\; \overrightarrow {C},\; \overrightarrow {W}\right\}$ in Euclidean 3-space

In our paper, we computed some new characterizations due to an alternative modified frame N ⃗ , C ⃗ , W ⃗ $\left\{ \overrightarrow {N}, \overrightarrow {C}, \overrightarrow {W}\right\}$ in Euclidean 3-space and we get general differential equation characterizations of a space curve due to the vectors N ⃗ , C ⃗ , W ⃗ $ \overrightarrow {N}, \overrightarrow {C}, \overrightarrow {W}$ . Furthermore, we investigated some differential equations characterizations of the harmonic and harmonic 1-type curves.

Slant Helices of (k,m)-Type According to the ED-Frame in Minkowski 4-Space

In differential geometry, relations between curves are a large and important area of study for many researchers. Frame areas are an important tool when studying curves, specially the Frenet–Serret frame along a space curve and the Darboux frame along a surface curve in differential geometry. In this paper, we obtain slant helices of k-type according to the extended Darboux frame (or, for brevity, ED-frame) field by using the ED-frame field of the first kind (or, for brevity, EDFFK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} and the ED-frame field of the second kind (or, for brevity, EDFSK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} in four-dimensional Minkowski space E14. In addition, we present some characterizations of slant helices and determine (k,m)-type slant helices for the EDFFK and EDFSK in Minkowski 4-space.

Bridging the gap between rectifying developables and tangent developables: a family of developable surfaces associated with a space curve

There are two familiar constructions of a developable surface from a space curve. The tangent developable is a ruled surface for which the rulings are tangent to the curve at each point and relative to this surface the absolute value of the geodesic curvature κ g of the curve equals the curvature κ . The alternative construction is the rectifying developable. The geodesic curvature of the curve relative to any such surface vanishes. We show that there is a family of developable surfaces that can be generated from a curve, one surface for each function k that is defined on the curve and satisfies | k | ≤ κ , and that the geodesic curvature of the curve relative to each such constructed surface satisfies κ g = k .

A new quaternion valued frame of curves with an application

The aim of the paper is to obtain a new version of Serret-Frenet formulae for a quaternionic curve in R4 by using the method given by Bharathi and Nagaraj. Then, we define quaternionic helices in H named as quaternionic right and left X-helix with the help of given a unit vector field X. Since the quaternion product is not commutative, the authors ([4], [7]) have used by one-sided multiplication to find a space curve related to a given quaternionic curve in previous studies. Firstly, we obtain new expressions by using the right product and the left product for quaternions. Then, we generalized the construction of Serret-Frenet formulae of quaternionic curves. Finally, as an application, we obtain an example that supports the theory of this paper.

MicroRNA-431 Promotes Progress of Breast Cancer by Aiming Rejection Targeting Molecule A and Activating Epithelial-Mesenchymal Transition

Objective: To conduct mathematical expression of the law of internal fascicular groups of peripheral nerves in spatial extension, so as to reveal the universal law of fascicular groups during the process of spatial extension. Methods: The centroid of each fascicular group shown on each Micro-CT image of the peripheral nerves was extracted, and these centroids were connected to form the centroid space curve of each fascicular group respectively based on preliminary studies. Results: The mean value of relative offset of fascicular groups in space was 3.7 μm while its maximum was: when a distance of 5 μm was extended, the fascicular groups centroid would have an offset of 21.4 μm. The accuracy of fitting of the centroid spatial curve of the fascicular groups using the 4th-order Fourier model could be up to 98%. Each parameter in the model obeyed the t distribution with position/dimension parameters. The dimension of parameters in the 1st-order component was obviously greater than that of the components of the other orders, indicating that the probability density function of harmonic component parameter showed an obvious peak shape. Conclusions: The centroid space curve of the Fascicular groups could express the extension of fascicular groups in space truly and exactly. The extension process of fascicular groups in space could be expressed accurately by the 4th-order Fourier model. The reason for using the 4th-order model was that a better balance could be obtained in model complexity and accuracy of fitting.

The Path Planning of Synchronous Cooperative Motion Control between Robot and Positioner for Complex Space Curve Processing

In this paper, we propose a synchronous cooperative path planning (SCPP) algorithm for the robot and the positioner to process complex space curve workpieces. The specific algorithm design is illustrated by using the intersecting line welding as an example. The robot and positioner are regarded as an 8-degree-of-freedom (DOF) system to plan the whole synchronous cooperative motion path, and the constraint for the Y-axis of the welding torch coordination system is added for solving the intersecting line orientation information. SCPP is used to process the intersecting line. The changes of welding torch orientation and robot joint rotation angles during welding of the intersecting line by using the improved method and compared to the traditional method. The experimental results show that 8-DOF keeps synchronous cooperative in the whole movement. There is no interference happening during the entire cooperative movement and the welding torch’s orientation remains basically unchanged during the welding intersecting line.