Resolutions of the Jerk and Snap Vectors for a Quasi Curve in Euclidean 3-Space

This work aims at studying resolutions of the jerk and snap vectors of a point particle moving along a quasi curve in Euclidean 3-space E3. In particular, we obtain the resolution of the jerk and snap vectors along the quasi vectors and offer an alternative resolution of the jerk and snap vectors along the tangential direction and two special radial directions that lie in the osculating and rectifying planes. This alternative resolution for a quasi plane curve in Euclidean 3-space E3 is given as corollary. Moreover, our results are illustrated via some examples.

Higher rank Clifford indices of curves on a K3 surface

Let (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

Detection of Pipeline Deformation Induced by Frost Heave Using OFDR Technology

Oil and gas pipelines are critical structures. For pipelines in the seasonal frozen soil area, frost heave of the ground will result in deformation of the pipeline. If the deformation continually increases, it will seriously threaten the pipeline safety. Therefore, it is important to monitor the deformation of the pipeline in the frozen soil area. Since optic frequency–domain reflectometer (OFDR) technology has many advantages in distributed strain measurement, this paper utilized the OFDR technology to measure the distributed strain and use the plane curve reconstruction algorithm to calculate the deformed pipeline shape. To verify the feasibility of this approach, a test was conducted to simulate the pipeline deformation induced by frost heave. Test results showed that the pipeline shape can be reconstructed well via the combination of the OFDR and curve reconstruction algorithm, providing a valuable approach for pipeline deformation monitoring.

Plane Curves Which are Quantum Homogeneous Spaces

Let $\mathcal {C}$ C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, $\mathcal {C}$ C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring $\mathcal {O}(\mathcal {C})$ O ( C ) of $\mathcal {C}$ C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.

Convexity limit angles for isoptics

Given an oval C in the plane, the $$\alpha $$ α -isoptic $$C_\alpha $$ C α of C is the plane curve composed of the points from which C can be seen under the angle $$\pi -\alpha $$ π - α . We consider isoptics of ovals parametrized with the support function $$p(t)=a+\cos n t$$ p ( t ) = a + cos n t , $$n\in \mathbb {N}$$ n ∈ N , and present an example of an oval such that when $$\alpha $$ α increases, the $$\alpha $$ α -isoptics begin to be convex, then lose their convexity and finally are convex again along a curve intersecting the isoptics orthogonally. Next we give an example of a curve from the same family, for which the curvature of the isoptics changes its sign three times. These changes occur on the symmetry axes of the oval C and coincide with the orthogonal trajectories which start at the points with extremal curvature. Finally, we formulate the hypothesis concerning the general case where we expect $$n-1$$ n - 1 convexity limit angles for the isoptics of an oval parametrized by $$p(t)=a+\cos n t$$ p ( t ) = a + cos n t .

Sweeping surfaces according to type-2 Bishop frame in Euclidean 3-space

This work is concerned with the study of the kinematic-geometry of a special kind of tube surfaces, so-called sweeping surface in Euclidean 3-space [Formula: see text]. It is generated by a plane curve moving through space such that the movement of any point on the surface is always orthogonal to the plane. In particular, the type-2 Bishop frame is considered and some important theorems are obtained. Also, the problem of singularity and convexity of such sweeping surface is discussed. Finally, an application is presented and plotted using computer aided geometric design.

A Convex Cover for Closed Unit Curves has Area at Least 0.0975

We combine geometric methods with a numerical box search algorithm to show that the minimal area of a convex set in the plane which can cover every closed plane curve of unit length is at least [Formula: see text]. This improves the best previous lower bound of [Formula: see text]. In fact, we show that the minimal area of the convex hull of circle, equilateral triangle, and rectangle of perimeter [Formula: see text] is between [Formula: see text] and [Formula: see text].

Hirzebruch-Type Inequalities Viewed as Tools in Combinatorics

The main purpose of this survey is to provide an introduction, algebro-topological in nature, to Hirzebuch-type inequalities for plane curve arrangements in the complex projective plane. These inequalities gain more and more interest due to their utility in many combinatorial problems related to point or line arrangements in the plane. We would like to present a summary of the technicalities and also some recent applications, for instance in the context of the Weak Dirac Conjecture. We also advertise some open problems and questions.