Abstract Bezier is one of the influent polynomial and important tool for interpolation because it is easy to compute and is also very stable. In this paper, we develop very simpler constraints for Quadratic and Cubic Bezier curve which they ensure to constrained by a line. The end control points of the Quadratic and Cubic Bezier curve will be left on user desire and only middle control points o...

A run is an inclusion maximal occurrence in a string (as a subinterval) of a factor in which the period repeats at least twice. The maximal number of runs in a string of length n has been thoroughly studied, and is known to be between 0.944n and 1.029n. The proofs are very technical. In this paper we investigate cubic runs, in which the period repeats at least three times. We show the upper bou...

Since the mode of Pythagorean-Hodograph curve’s derivate is a polynomial, its involute should be a rational polynomial. We study the expression form of PH cubic’s involute, discovering that its control points and weights are all constructed by corresponding PH cubic’s geometry property. Moreover, we prove that cubic PH curves have neither cusp nor inflection; there are two points coincidence in...

This paper proposes an algorithm for generating irreducible cubic trinomials in the form x + ax + b, b ∈ Fp, where a is a certain fixed non-zero element in the prime field Fp. The proposed algorithm needs a certain irreducible cubic trinomial over Fp to be previously given as a generator; however, the proposed algorithm can generate irreducible cubic polynomials one after another by changing a ...

A method for constructing the isosurface from a data set on a face-centered-cubic lattice has been developed. Generally, a three-dimensional scalar field is sampled on an orthogonal cubic lattice. And an algorithm called Marching Cubes is widely used for constructing the isosurface on this lattice. It is known, however, that data sampling of a three-dimensional scalar field on a face-centered-c...

It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan–Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan–Raspaud colorings for some selected classes of cubic graphs. The p...

In early 70s Berge conjectured that any bridgeless cubic graph contains five perfect matchings such that each edge belongs to at least one of them. In 1972 Fulkerson conjectured that, in fact, we can find six perfect matchings containing each edge exactly twice. By introducing the concept of an r-graph (a remarkable generalization of one of bridgeless cubic graph) Seymour in 1979 conjectured th...

A Fano colouring is a colouring of the edges of a cubic graph by points of the Fano plane such that the colours of any three mutually adjacent edges form a line of the Fano plane. It has recently been shown by Holroyd and Škoviera (J. Combin. Theory Ser. B, to appear) that a cubic graph has a Fano colouring if and only if it is bridgeless. In this paper we prove that six, and conjecture that fo...

An explicit projective embedding of the moduli space of marked cubic surfaces is given. This embedding is equivariant under the Weyl group of type E6. The image is defined by a system of linear and cubic equations. To express the embedding in a most symmetric way, the target would be 79-dimensional, however the image lies in a 9-dimensional linear subspace. 1 The moduli space of cubic surfaces ...

Black phosphorus was compressed at room temperature across the A17, A7 and simple-cubic phases up to 30 GPa, using a diamond anvil cell and He as pressure transmitting medium. Synchrotron X-ray diffraction showed the persistence of two previously unreported peaks related to the A7 structure in the pressure range of the simple-cubic phase. The Rietveld refinement of the data demonstrates the occ...